full_name
stringlengths 3
121
| state
stringlengths 7
9.32k
| tactic
stringlengths 3
5.35k
| target_state
stringlengths 7
19k
| url
stringclasses 1
value | commit
stringclasses 1
value | file_path
stringlengths 21
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tendsto_div_of_monotone_of_exists_subseq_tendsto_div
|
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
∀ (a : ℝ),
1 < a →
∃ c,
(∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d < l
⊢ Tendsto (fun ε => d + ε * (1 + l)) (𝓝 0) (𝓝 (d + 0 * (1 + l)))
|
exact tendsto_const_nhds.add (tendsto_id.mul <a>tendsto_const_nhds</a>)
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/SpecificLimits/FloorPow.lean
|
tendsto_div_of_monotone_of_exists_subseq_tendsto_div
|
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
∀ (a : ℝ),
1 < a →
∃ c,
(∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d < l
ε : ℝ
hε : d + ε * (1 + l) < l
εpos : 0 < ε
n : ℕ
hn : ↑n * l - u n ≤ ε * (1 + l) * ↑n
npos : n ∈ Set.Ioi 0
⊢ d < (↑n)⁻¹ * ↑n * (l - ε * (1 + l))
|
rw [<a>inv_mul_cancel</a>, <a>one_mul</a>]
|
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
∀ (a : ℝ),
1 < a →
∃ c,
(∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d < l
ε : ℝ
hε : d + ε * (1 + l) < l
εpos : 0 < ε
n : ℕ
hn : ↑n * l - u n ≤ ε * (1 + l) * ↑n
npos : n ∈ Set.Ioi 0
⊢ d < l - ε * (1 + l)
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
∀ (a : ℝ),
1 < a →
∃ c,
(∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d < l
ε : ℝ
hε : d + ε * (1 + l) < l
εpos : 0 < ε
n : ℕ
hn : ↑n * l - u n ≤ ε * (1 + l) * ↑n
npos : n ∈ Set.Ioi 0
⊢ ↑n ≠ 0
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/SpecificLimits/FloorPow.lean
|
tendsto_div_of_monotone_of_exists_subseq_tendsto_div
|
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
∀ (a : ℝ),
1 < a →
∃ c,
(∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d < l
ε : ℝ
hε : d + ε * (1 + l) < l
εpos : 0 < ε
n : ℕ
hn : ↑n * l - u n ≤ ε * (1 + l) * ↑n
npos : n ∈ Set.Ioi 0
⊢ d < l - ε * (1 + l)
|
linarith only [hε]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/SpecificLimits/FloorPow.lean
|
tendsto_div_of_monotone_of_exists_subseq_tendsto_div
|
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
∀ (a : ℝ),
1 < a →
∃ c,
(∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d < l
ε : ℝ
hε : d + ε * (1 + l) < l
εpos : 0 < ε
n : ℕ
hn : ↑n * l - u n ≤ ε * (1 + l) * ↑n
npos : n ∈ Set.Ioi 0
⊢ ↑n ≠ 0
|
exact <a>Nat.cast_ne_zero</a>.2 (<a>ne_of_gt</a> npos)
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/SpecificLimits/FloorPow.lean
|
tendsto_div_of_monotone_of_exists_subseq_tendsto_div
|
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
∀ (a : ℝ),
1 < a →
∃ c,
(∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d < l
ε : ℝ
hε : d + ε * (1 + l) < l
εpos : 0 < ε
n : ℕ
hn : ↑n * l - u n ≤ ε * (1 + l) * ↑n
npos : n ∈ Set.Ioi 0
⊢ (↑n)⁻¹ * ↑n * (l - ε * (1 + l)) = (↑n)⁻¹ * (↑n * l - ε * (1 + l) * ↑n)
|
ring
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/SpecificLimits/FloorPow.lean
|
tendsto_div_of_monotone_of_exists_subseq_tendsto_div
|
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
∀ (a : ℝ),
1 < a →
∃ c,
(∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d < l
ε : ℝ
hε : d + ε * (1 + l) < l
εpos : 0 < ε
n : ℕ
hn : ↑n * l - u n ≤ ε * (1 + l) * ↑n
npos : n ∈ Set.Ioi 0
⊢ (↑n)⁻¹ * (↑n * l - ε * (1 + l) * ↑n) ≤ (↑n)⁻¹ * u n
|
gcongr
|
case h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
∀ (a : ℝ),
1 < a →
∃ c,
(∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d < l
ε : ℝ
hε : d + ε * (1 + l) < l
εpos : 0 < ε
n : ℕ
hn : ↑n * l - u n ≤ ε * (1 + l) * ↑n
npos : n ∈ Set.Ioi 0
⊢ ↑n * l - ε * (1 + l) * ↑n ≤ u n
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/SpecificLimits/FloorPow.lean
|
tendsto_div_of_monotone_of_exists_subseq_tendsto_div
|
case h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
∀ (a : ℝ),
1 < a →
∃ c,
(∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d < l
ε : ℝ
hε : d + ε * (1 + l) < l
εpos : 0 < ε
n : ℕ
hn : ↑n * l - u n ≤ ε * (1 + l) * ↑n
npos : n ∈ Set.Ioi 0
⊢ ↑n * l - ε * (1 + l) * ↑n ≤ u n
|
linarith only [hn]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/SpecificLimits/FloorPow.lean
|
tendsto_div_of_monotone_of_exists_subseq_tendsto_div
|
case refine_2
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
∀ (a : ℝ),
1 < a →
∃ c,
(∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
⊢ ∀ᶠ (b : ℕ) in atTop, u b / ↑b < d
|
obtain ⟨ε, hε, εpos⟩ : ∃ ε : ℝ, l + ε * (1 + ε + l) < d ∧ 0 < ε := by have L : <a>Filter.Tendsto</a> (fun ε => l + ε * (1 + ε + l)) (𝓝[>] 0) (𝓝 (l + 0 * (1 + 0 + l))) := by apply <a>Filter.Tendsto.mono_left</a> _ <a>nhdsWithin_le_nhds</a> exact tendsto_const_nhds.add (tendsto_id.mul ((tendsto_const_nhds.add <a>Filter.tendsto_id</a>).<a>Filter.Tendsto.add</a> <a>tendsto_const_nhds</a>)) simp only [<a>MulZeroClass.zero_mul</a>, <a>add_zero</a>] at L exact (((<a>tendsto_order</a>.1 L).2 d hd).<a>Filter.Eventually.and</a> <a>self_mem_nhdsWithin</a>).<a>Filter.Eventually.exists</a>
|
case refine_2.intro.intro
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
∀ (a : ℝ),
1 < a →
∃ c,
(∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
ε : ℝ
hε : l + ε * (1 + ε + l) < d
εpos : 0 < ε
⊢ ∀ᶠ (b : ℕ) in atTop, u b / ↑b < d
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/SpecificLimits/FloorPow.lean
|
tendsto_div_of_monotone_of_exists_subseq_tendsto_div
|
case refine_2.intro.intro
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
∀ (a : ℝ),
1 < a →
∃ c,
(∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
ε : ℝ
hε : l + ε * (1 + ε + l) < d
εpos : 0 < ε
⊢ ∀ᶠ (b : ℕ) in atTop, u b / ↑b < d
|
filter_upwards [A ε εpos, <a>Filter.Ioi_mem_atTop</a> 0] with n hn (npos : 0 < n)
|
case h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
∀ (a : ℝ),
1 < a →
∃ c,
(∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
ε : ℝ
hε : l + ε * (1 + ε + l) < d
εpos : 0 < ε
n : ℕ
hn : u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
npos : 0 < n
⊢ u n / ↑n < d
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/SpecificLimits/FloorPow.lean
|
tendsto_div_of_monotone_of_exists_subseq_tendsto_div
|
case h
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
∀ (a : ℝ),
1 < a →
∃ c,
(∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
ε : ℝ
hε : l + ε * (1 + ε + l) < d
εpos : 0 < ε
n : ℕ
hn : u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
npos : 0 < n
⊢ u n / ↑n < d
|
calc u n / n ≤ (n * l + ε * (1 + ε + l) * n) / n := by gcongr; linarith only [hn] _ = (l + ε * (1 + ε + l)) := by field_simp; ring _ < d := hε
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/SpecificLimits/FloorPow.lean
|
tendsto_div_of_monotone_of_exists_subseq_tendsto_div
|
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
∀ (a : ℝ),
1 < a →
∃ c,
(∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
⊢ ∃ ε, l + ε * (1 + ε + l) < d ∧ 0 < ε
|
have L : <a>Filter.Tendsto</a> (fun ε => l + ε * (1 + ε + l)) (𝓝[>] 0) (𝓝 (l + 0 * (1 + 0 + l))) := by apply <a>Filter.Tendsto.mono_left</a> _ <a>nhdsWithin_le_nhds</a> exact tendsto_const_nhds.add (tendsto_id.mul ((tendsto_const_nhds.add <a>Filter.tendsto_id</a>).<a>Filter.Tendsto.add</a> <a>tendsto_const_nhds</a>))
|
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
∀ (a : ℝ),
1 < a →
∃ c,
(∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
L : Tendsto (fun ε => l + ε * (1 + ε + l)) (𝓝[>] 0) (𝓝 (l + 0 * (1 + 0 + l)))
⊢ ∃ ε, l + ε * (1 + ε + l) < d ∧ 0 < ε
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/SpecificLimits/FloorPow.lean
|
tendsto_div_of_monotone_of_exists_subseq_tendsto_div
|
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
∀ (a : ℝ),
1 < a →
∃ c,
(∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
L : Tendsto (fun ε => l + ε * (1 + ε + l)) (𝓝[>] 0) (𝓝 (l + 0 * (1 + 0 + l)))
⊢ ∃ ε, l + ε * (1 + ε + l) < d ∧ 0 < ε
|
simp only [<a>MulZeroClass.zero_mul</a>, <a>add_zero</a>] at L
|
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
∀ (a : ℝ),
1 < a →
∃ c,
(∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
L : Tendsto (fun ε => l + ε * (1 + ε + l)) (𝓝[>] 0) (𝓝 l)
⊢ ∃ ε, l + ε * (1 + ε + l) < d ∧ 0 < ε
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/SpecificLimits/FloorPow.lean
|
tendsto_div_of_monotone_of_exists_subseq_tendsto_div
|
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
∀ (a : ℝ),
1 < a →
∃ c,
(∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
L : Tendsto (fun ε => l + ε * (1 + ε + l)) (𝓝[>] 0) (𝓝 l)
⊢ ∃ ε, l + ε * (1 + ε + l) < d ∧ 0 < ε
|
exact (((<a>tendsto_order</a>.1 L).2 d hd).<a>Filter.Eventually.and</a> <a>self_mem_nhdsWithin</a>).<a>Filter.Eventually.exists</a>
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/SpecificLimits/FloorPow.lean
|
tendsto_div_of_monotone_of_exists_subseq_tendsto_div
|
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
∀ (a : ℝ),
1 < a →
∃ c,
(∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
⊢ Tendsto (fun ε => l + ε * (1 + ε + l)) (𝓝[>] 0) (𝓝 (l + 0 * (1 + 0 + l)))
|
apply <a>Filter.Tendsto.mono_left</a> _ <a>nhdsWithin_le_nhds</a>
|
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
∀ (a : ℝ),
1 < a →
∃ c,
(∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
⊢ Tendsto (fun ε => l + ε * (1 + ε + l)) (𝓝 0) (𝓝 (l + 0 * (1 + 0 + l)))
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/SpecificLimits/FloorPow.lean
|
tendsto_div_of_monotone_of_exists_subseq_tendsto_div
|
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
∀ (a : ℝ),
1 < a →
∃ c,
(∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
⊢ Tendsto (fun ε => l + ε * (1 + ε + l)) (𝓝 0) (𝓝 (l + 0 * (1 + 0 + l)))
|
exact tendsto_const_nhds.add (tendsto_id.mul ((tendsto_const_nhds.add <a>Filter.tendsto_id</a>).<a>Filter.Tendsto.add</a> <a>tendsto_const_nhds</a>))
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/SpecificLimits/FloorPow.lean
|
tendsto_div_of_monotone_of_exists_subseq_tendsto_div
|
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
∀ (a : ℝ),
1 < a →
∃ c,
(∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
ε : ℝ
hε : l + ε * (1 + ε + l) < d
εpos : 0 < ε
n : ℕ
hn : u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
npos : 0 < n
⊢ u n / ↑n ≤ (↑n * l + ε * (1 + ε + l) * ↑n) / ↑n
|
gcongr
|
case hab
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
∀ (a : ℝ),
1 < a →
∃ c,
(∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
ε : ℝ
hε : l + ε * (1 + ε + l) < d
εpos : 0 < ε
n : ℕ
hn : u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
npos : 0 < n
⊢ u n ≤ ↑n * l + ε * (1 + ε + l) * ↑n
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/SpecificLimits/FloorPow.lean
|
tendsto_div_of_monotone_of_exists_subseq_tendsto_div
|
case hab
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
∀ (a : ℝ),
1 < a →
∃ c,
(∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
ε : ℝ
hε : l + ε * (1 + ε + l) < d
εpos : 0 < ε
n : ℕ
hn : u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
npos : 0 < n
⊢ u n ≤ ↑n * l + ε * (1 + ε + l) * ↑n
|
linarith only [hn]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/SpecificLimits/FloorPow.lean
|
tendsto_div_of_monotone_of_exists_subseq_tendsto_div
|
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
∀ (a : ℝ),
1 < a →
∃ c,
(∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
ε : ℝ
hε : l + ε * (1 + ε + l) < d
εpos : 0 < ε
n : ℕ
hn : u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
npos : 0 < n
⊢ (↑n * l + ε * (1 + ε + l) * ↑n) / ↑n = l + ε * (1 + ε + l)
|
field_simp
|
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
∀ (a : ℝ),
1 < a →
∃ c,
(∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
ε : ℝ
hε : l + ε * (1 + ε + l) < d
εpos : 0 < ε
n : ℕ
hn : u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
npos : 0 < n
⊢ ↑n * l + ε * (1 + ε + l) * ↑n = (l + ε * (1 + ε + l)) * ↑n
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/SpecificLimits/FloorPow.lean
|
tendsto_div_of_monotone_of_exists_subseq_tendsto_div
|
u : ℕ → ℝ
l : ℝ
hmono : Monotone u
hlim :
∀ (a : ℝ),
1 < a →
∃ c,
(∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧
Tendsto c atTop atTop ∧ Tendsto (fun n => u (c n) / ↑(c n)) atTop (𝓝 l)
lnonneg : 0 ≤ l
A : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
B : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, ↑n * l - u n ≤ ε * (1 + l) * ↑n
d : ℝ
hd : d > l
ε : ℝ
hε : l + ε * (1 + ε + l) < d
εpos : 0 < ε
n : ℕ
hn : u n - ↑n * l ≤ ε * (1 + ε + l) * ↑n
npos : 0 < n
⊢ ↑n * l + ε * (1 + ε + l) * ↑n = (l + ε * (1 + ε + l)) * ↑n
|
ring
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/SpecificLimits/FloorPow.lean
|
Monoid.minOrder_eq_top
|
α : Type u_1
inst✝ : Monoid α
a : α
⊢ minOrder α = ⊤ ↔ IsTorsionFree α
|
simp [<a>Monoid.minOrder</a>, <a>Monoid.IsTorsionFree</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/GroupTheory/Order/Min.lean
|
ContDiffBump.zero_of_le_dist
|
E : Type u_1
X : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup X
inst✝¹ : NormedSpace ℝ X
inst✝ : HasContDiffBump E
c : E
f : ContDiffBump c
x : E
n : ℕ∞
hx : f.rOut ≤ dist x c
⊢ ↑f x = 0
|
rwa [← <a>Function.nmem_support</a>, <a>ContDiffBump.support_eq</a>, <a>Metric.mem_ball</a>, <a>not_lt</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/Calculus/BumpFunction/Basic.lean
|
TypeVec.dropFun_of_subtype
|
n : ℕ
α : TypeVec.{u_1} (n + 1)
p : α ⟹ repeat (n + 1) Prop
⊢ dropFun (ofSubtype p) = ofSubtype (dropFun p)
|
ext i : 2
|
case a.h
n : ℕ
α : TypeVec.{u_1} (n + 1)
p : α ⟹ repeat (n + 1) Prop
i : Fin2 n
x✝ : (Subtype_ p).drop i
⊢ dropFun (ofSubtype p) i x✝ = ofSubtype (dropFun p) i x✝
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/TypeVec.lean
|
TypeVec.dropFun_of_subtype
|
case a.h
n : ℕ
α : TypeVec.{u_1} (n + 1)
p : α ⟹ repeat (n + 1) Prop
i : Fin2 n
x✝ : (Subtype_ p).drop i
⊢ dropFun (ofSubtype p) i x✝ = ofSubtype (dropFun p) i x✝
|
induction i <;> simp [<a>TypeVec.dropFun</a>, *] <;> rfl
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/TypeVec.lean
|
measurableSet_region_between_co
|
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f g : α → ℝ
s : Set α
hf : Measurable f
hg : Measurable g
hs : MeasurableSet s
⊢ MeasurableSet {p | p.1 ∈ s ∧ p.2 ∈ Ico (f p.1) (g p.1)}
|
dsimp only [<a>regionBetween</a>, <a>Set.Ico</a>, <a>Set.mem_setOf_eq</a>, <a>Set.setOf_and</a>]
|
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f g : α → ℝ
s : Set α
hf : Measurable f
hg : Measurable g
hs : MeasurableSet s
⊢ MeasurableSet ({a | a.1 ∈ s} ∩ {a | a.2 ∈ {a_1 | f a.1 ≤ a_1} ∩ {a_1 | a_1 < g a.1}})
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean
|
measurableSet_region_between_co
|
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f g : α → ℝ
s : Set α
hf : Measurable f
hg : Measurable g
hs : MeasurableSet s
⊢ MeasurableSet ({a | a.1 ∈ s} ∩ {a | a.2 ∈ {a_1 | f a.1 ≤ a_1} ∩ {a_1 | a_1 < g a.1}})
|
refine <a>MeasurableSet.inter</a> ?_ ((<a>measurableSet_le</a> (hf.comp <a>measurable_fst</a>) <a>measurable_snd</a>).<a>MeasurableSet.inter</a> (<a>measurableSet_lt</a> <a>measurable_snd</a> (hg.comp <a>measurable_fst</a>)))
|
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f g : α → ℝ
s : Set α
hf : Measurable f
hg : Measurable g
hs : MeasurableSet s
⊢ MeasurableSet {a | a.1 ∈ s}
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean
|
measurableSet_region_between_co
|
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f g : α → ℝ
s : Set α
hf : Measurable f
hg : Measurable g
hs : MeasurableSet s
⊢ MeasurableSet {a | a.1 ∈ s}
|
exact <a>measurable_fst</a> hs
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean
|
LieModule.isNilpotent_toEnd_of_isNilpotent₂
|
R : Type u
L : Type v
M : Type w
inst✝¹¹ : CommRing R
inst✝¹⁰ : LieRing L
inst✝⁹ : LieAlgebra R L
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : LieRingModule L M
inst✝⁵ : LieModule R L M
k : ℕ
N : LieSubmodule R L M
M₂ : Type w₁
inst✝⁴ : AddCommGroup M₂
inst✝³ : Module R M₂
inst✝² : LieRingModule L M₂
inst✝¹ : LieModule R L M₂
inst✝ : IsNilpotent R L M
x y : L
⊢ _root_.IsNilpotent ((toEnd R L M) x ∘ₗ (toEnd R L M) y)
|
obtain ⟨k, hM⟩ := <a>LieModule.exists_lowerCentralSeries_eq_bot_of_isNilpotent</a> R L M
|
case intro
R : Type u
L : Type v
M : Type w
inst✝¹¹ : CommRing R
inst✝¹⁰ : LieRing L
inst✝⁹ : LieAlgebra R L
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : LieRingModule L M
inst✝⁵ : LieModule R L M
k✝ : ℕ
N : LieSubmodule R L M
M₂ : Type w₁
inst✝⁴ : AddCommGroup M₂
inst✝³ : Module R M₂
inst✝² : LieRingModule L M₂
inst✝¹ : LieModule R L M₂
inst✝ : IsNilpotent R L M
x y : L
k : ℕ
hM : lowerCentralSeries R L M k = ⊥
⊢ _root_.IsNilpotent ((toEnd R L M) x ∘ₗ (toEnd R L M) y)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Lie/Nilpotent.lean
|
LieModule.isNilpotent_toEnd_of_isNilpotent₂
|
case intro
R : Type u
L : Type v
M : Type w
inst✝¹¹ : CommRing R
inst✝¹⁰ : LieRing L
inst✝⁹ : LieAlgebra R L
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : LieRingModule L M
inst✝⁵ : LieModule R L M
k✝ : ℕ
N : LieSubmodule R L M
M₂ : Type w₁
inst✝⁴ : AddCommGroup M₂
inst✝³ : Module R M₂
inst✝² : LieRingModule L M₂
inst✝¹ : LieModule R L M₂
inst✝ : IsNilpotent R L M
x y : L
k : ℕ
hM : lowerCentralSeries R L M k = ⊥
⊢ _root_.IsNilpotent ((toEnd R L M) x ∘ₗ (toEnd R L M) y)
|
replace hM : <a>LieModule.lowerCentralSeries</a> R L M (2 * k) = ⊥ := by rw [<a>eq_bot_iff</a>, ← hM]; exact <a>LieModule.antitone_lowerCentralSeries</a> R L M (by omega)
|
case intro
R : Type u
L : Type v
M : Type w
inst✝¹¹ : CommRing R
inst✝¹⁰ : LieRing L
inst✝⁹ : LieAlgebra R L
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : LieRingModule L M
inst✝⁵ : LieModule R L M
k✝ : ℕ
N : LieSubmodule R L M
M₂ : Type w₁
inst✝⁴ : AddCommGroup M₂
inst✝³ : Module R M₂
inst✝² : LieRingModule L M₂
inst✝¹ : LieModule R L M₂
inst✝ : IsNilpotent R L M
x y : L
k : ℕ
hM : lowerCentralSeries R L M (2 * k) = ⊥
⊢ _root_.IsNilpotent ((toEnd R L M) x ∘ₗ (toEnd R L M) y)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Lie/Nilpotent.lean
|
LieModule.isNilpotent_toEnd_of_isNilpotent₂
|
case intro
R : Type u
L : Type v
M : Type w
inst✝¹¹ : CommRing R
inst✝¹⁰ : LieRing L
inst✝⁹ : LieAlgebra R L
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : LieRingModule L M
inst✝⁵ : LieModule R L M
k✝ : ℕ
N : LieSubmodule R L M
M₂ : Type w₁
inst✝⁴ : AddCommGroup M₂
inst✝³ : Module R M₂
inst✝² : LieRingModule L M₂
inst✝¹ : LieModule R L M₂
inst✝ : IsNilpotent R L M
x y : L
k : ℕ
hM : lowerCentralSeries R L M (2 * k) = ⊥
⊢ _root_.IsNilpotent ((toEnd R L M) x ∘ₗ (toEnd R L M) y)
|
use k
|
case h
R : Type u
L : Type v
M : Type w
inst✝¹¹ : CommRing R
inst✝¹⁰ : LieRing L
inst✝⁹ : LieAlgebra R L
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : LieRingModule L M
inst✝⁵ : LieModule R L M
k✝ : ℕ
N : LieSubmodule R L M
M₂ : Type w₁
inst✝⁴ : AddCommGroup M₂
inst✝³ : Module R M₂
inst✝² : LieRingModule L M₂
inst✝¹ : LieModule R L M₂
inst✝ : IsNilpotent R L M
x y : L
k : ℕ
hM : lowerCentralSeries R L M (2 * k) = ⊥
⊢ ((toEnd R L M) x ∘ₗ (toEnd R L M) y) ^ k = 0
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Lie/Nilpotent.lean
|
LieModule.isNilpotent_toEnd_of_isNilpotent₂
|
case h
R : Type u
L : Type v
M : Type w
inst✝¹¹ : CommRing R
inst✝¹⁰ : LieRing L
inst✝⁹ : LieAlgebra R L
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : LieRingModule L M
inst✝⁵ : LieModule R L M
k✝ : ℕ
N : LieSubmodule R L M
M₂ : Type w₁
inst✝⁴ : AddCommGroup M₂
inst✝³ : Module R M₂
inst✝² : LieRingModule L M₂
inst✝¹ : LieModule R L M₂
inst✝ : IsNilpotent R L M
x y : L
k : ℕ
hM : lowerCentralSeries R L M (2 * k) = ⊥
⊢ ((toEnd R L M) x ∘ₗ (toEnd R L M) y) ^ k = 0
|
ext m
|
case h.h
R : Type u
L : Type v
M : Type w
inst✝¹¹ : CommRing R
inst✝¹⁰ : LieRing L
inst✝⁹ : LieAlgebra R L
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : LieRingModule L M
inst✝⁵ : LieModule R L M
k✝ : ℕ
N : LieSubmodule R L M
M₂ : Type w₁
inst✝⁴ : AddCommGroup M₂
inst✝³ : Module R M₂
inst✝² : LieRingModule L M₂
inst✝¹ : LieModule R L M₂
inst✝ : IsNilpotent R L M
x y : L
k : ℕ
hM : lowerCentralSeries R L M (2 * k) = ⊥
m : M
⊢ (((toEnd R L M) x ∘ₗ (toEnd R L M) y) ^ k) m = 0 m
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Lie/Nilpotent.lean
|
LieModule.isNilpotent_toEnd_of_isNilpotent₂
|
case h.h
R : Type u
L : Type v
M : Type w
inst✝¹¹ : CommRing R
inst✝¹⁰ : LieRing L
inst✝⁹ : LieAlgebra R L
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : LieRingModule L M
inst✝⁵ : LieModule R L M
k✝ : ℕ
N : LieSubmodule R L M
M₂ : Type w₁
inst✝⁴ : AddCommGroup M₂
inst✝³ : Module R M₂
inst✝² : LieRingModule L M₂
inst✝¹ : LieModule R L M₂
inst✝ : IsNilpotent R L M
x y : L
k : ℕ
hM : lowerCentralSeries R L M (2 * k) = ⊥
m : M
⊢ (((toEnd R L M) x ∘ₗ (toEnd R L M) y) ^ k) m = 0 m
|
rw [<a>LinearMap.pow_apply</a>, <a>LinearMap.zero_apply</a>, ← <a>LieSubmodule.mem_bot</a> (R := R) (L := L), ← hM]
|
case h.h
R : Type u
L : Type v
M : Type w
inst✝¹¹ : CommRing R
inst✝¹⁰ : LieRing L
inst✝⁹ : LieAlgebra R L
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : LieRingModule L M
inst✝⁵ : LieModule R L M
k✝ : ℕ
N : LieSubmodule R L M
M₂ : Type w₁
inst✝⁴ : AddCommGroup M₂
inst✝³ : Module R M₂
inst✝² : LieRingModule L M₂
inst✝¹ : LieModule R L M₂
inst✝ : IsNilpotent R L M
x y : L
k : ℕ
hM : lowerCentralSeries R L M (2 * k) = ⊥
m : M
⊢ (⇑((toEnd R L M) x ∘ₗ (toEnd R L M) y))^[k] m ∈ lowerCentralSeries R L M (2 * k)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Lie/Nilpotent.lean
|
LieModule.isNilpotent_toEnd_of_isNilpotent₂
|
case h.h
R : Type u
L : Type v
M : Type w
inst✝¹¹ : CommRing R
inst✝¹⁰ : LieRing L
inst✝⁹ : LieAlgebra R L
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : LieRingModule L M
inst✝⁵ : LieModule R L M
k✝ : ℕ
N : LieSubmodule R L M
M₂ : Type w₁
inst✝⁴ : AddCommGroup M₂
inst✝³ : Module R M₂
inst✝² : LieRingModule L M₂
inst✝¹ : LieModule R L M₂
inst✝ : IsNilpotent R L M
x y : L
k : ℕ
hM : lowerCentralSeries R L M (2 * k) = ⊥
m : M
⊢ (⇑((toEnd R L M) x ∘ₗ (toEnd R L M) y))^[k] m ∈ lowerCentralSeries R L M (2 * k)
|
exact <a>LieModule.iterate_toEnd_mem_lowerCentralSeries₂</a> R L M x y m k
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Lie/Nilpotent.lean
|
LieModule.isNilpotent_toEnd_of_isNilpotent₂
|
R : Type u
L : Type v
M : Type w
inst✝¹¹ : CommRing R
inst✝¹⁰ : LieRing L
inst✝⁹ : LieAlgebra R L
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : LieRingModule L M
inst✝⁵ : LieModule R L M
k✝ : ℕ
N : LieSubmodule R L M
M₂ : Type w₁
inst✝⁴ : AddCommGroup M₂
inst✝³ : Module R M₂
inst✝² : LieRingModule L M₂
inst✝¹ : LieModule R L M₂
inst✝ : IsNilpotent R L M
x y : L
k : ℕ
hM : lowerCentralSeries R L M k = ⊥
⊢ lowerCentralSeries R L M (2 * k) = ⊥
|
rw [<a>eq_bot_iff</a>, ← hM]
|
R : Type u
L : Type v
M : Type w
inst✝¹¹ : CommRing R
inst✝¹⁰ : LieRing L
inst✝⁹ : LieAlgebra R L
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : LieRingModule L M
inst✝⁵ : LieModule R L M
k✝ : ℕ
N : LieSubmodule R L M
M₂ : Type w₁
inst✝⁴ : AddCommGroup M₂
inst✝³ : Module R M₂
inst✝² : LieRingModule L M₂
inst✝¹ : LieModule R L M₂
inst✝ : IsNilpotent R L M
x y : L
k : ℕ
hM : lowerCentralSeries R L M k = ⊥
⊢ lowerCentralSeries R L M (2 * k) ≤ lowerCentralSeries R L M k
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Lie/Nilpotent.lean
|
LieModule.isNilpotent_toEnd_of_isNilpotent₂
|
R : Type u
L : Type v
M : Type w
inst✝¹¹ : CommRing R
inst✝¹⁰ : LieRing L
inst✝⁹ : LieAlgebra R L
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : LieRingModule L M
inst✝⁵ : LieModule R L M
k✝ : ℕ
N : LieSubmodule R L M
M₂ : Type w₁
inst✝⁴ : AddCommGroup M₂
inst✝³ : Module R M₂
inst✝² : LieRingModule L M₂
inst✝¹ : LieModule R L M₂
inst✝ : IsNilpotent R L M
x y : L
k : ℕ
hM : lowerCentralSeries R L M k = ⊥
⊢ lowerCentralSeries R L M (2 * k) ≤ lowerCentralSeries R L M k
|
exact <a>LieModule.antitone_lowerCentralSeries</a> R L M (by omega)
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Lie/Nilpotent.lean
|
LieModule.isNilpotent_toEnd_of_isNilpotent₂
|
R : Type u
L : Type v
M : Type w
inst✝¹¹ : CommRing R
inst✝¹⁰ : LieRing L
inst✝⁹ : LieAlgebra R L
inst✝⁸ : AddCommGroup M
inst✝⁷ : Module R M
inst✝⁶ : LieRingModule L M
inst✝⁵ : LieModule R L M
k✝ : ℕ
N : LieSubmodule R L M
M₂ : Type w₁
inst✝⁴ : AddCommGroup M₂
inst✝³ : Module R M₂
inst✝² : LieRingModule L M₂
inst✝¹ : LieModule R L M₂
inst✝ : IsNilpotent R L M
x y : L
k : ℕ
hM : lowerCentralSeries R L M k = ⊥
⊢ k ≤ 2 * k
|
omega
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Lie/Nilpotent.lean
|
CategoryTheory.isPullback_of_cofan_isVanKampen
|
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.324390, u_1} K
D : Type u_2
inst✝² : Category.{?u.324397, u_2} D
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
i j : ι
inst✝ : DecidableEq ι
⊢ IsPullback (if h : j = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (X i))
(if h : j = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (X j)) (c.inj i) (c.inj j)
|
refine (hc (<a>CategoryTheory.Limits.Cofan.mk</a> (X i) (f := fun k ↦ if k = i then X i else ⊥_ C) (fun k ↦ if h : k = i then (<a>CategoryTheory.eqToHom</a> <| <a>if_pos</a> h) else (<a>CategoryTheory.eqToHom</a> <| <a>if_neg</a> h) ≫ <a>CategoryTheory.Limits.initial.to</a> _)) (<a>CategoryTheory.Discrete.natTrans</a> (fun k ↦ if h : k.1 = i then (<a>CategoryTheory.eqToHom</a> <| (<a>if_pos</a> h).<a>Eq.trans</a> (<a>congr_arg</a> X h.symm)) else (<a>CategoryTheory.eqToHom</a> <| <a>if_neg</a> h) ≫ <a>CategoryTheory.Limits.initial.to</a> _)) (c.inj i) ?_ (<a>CategoryTheory.NatTrans.equifibered_of_discrete</a> _)).<a>Iff.mp</a> ⟨?_⟩ ⟨j⟩
|
case refine_1
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.324390, u_1} K
D : Type u_2
inst✝² : Category.{?u.324397, u_2} D
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
i j : ι
inst✝ : DecidableEq ι
⊢ (Discrete.natTrans fun k => if h : k.as = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to ((Discrete.functor X).obj k)) ≫
c.ι =
(Cofan.mk (X i) fun k => if h : k = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (X i)).ι ≫
(Functor.const (Discrete ι)).map (c.inj i)
case refine_2
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.324390, u_1} K
D : Type u_2
inst✝² : Category.{?u.324397, u_2} D
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
i j : ι
inst✝ : DecidableEq ι
⊢ IsColimit (Cofan.mk (X i) fun k => if h : k = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (X i))
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Limits/VanKampen.lean
|
CategoryTheory.isPullback_of_cofan_isVanKampen
|
case refine_1
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.324390, u_1} K
D : Type u_2
inst✝² : Category.{?u.324397, u_2} D
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
i j : ι
inst✝ : DecidableEq ι
⊢ (Discrete.natTrans fun k => if h : k.as = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to ((Discrete.functor X).obj k)) ≫
c.ι =
(Cofan.mk (X i) fun k => if h : k = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (X i)).ι ≫
(Functor.const (Discrete ι)).map (c.inj i)
|
ext ⟨k⟩
|
case refine_1.w.h.mk
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.324390, u_1} K
D : Type u_2
inst✝² : Category.{?u.324397, u_2} D
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
i j : ι
inst✝ : DecidableEq ι
k : ι
⊢ ((Discrete.natTrans fun k =>
if h : k.as = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to ((Discrete.functor X).obj k)) ≫
c.ι).app
{ as := k } =
((Cofan.mk (X i) fun k => if h : k = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (X i)).ι ≫
(Functor.const (Discrete ι)).map (c.inj i)).app
{ as := k }
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Limits/VanKampen.lean
|
CategoryTheory.isPullback_of_cofan_isVanKampen
|
case refine_1.w.h.mk
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.324390, u_1} K
D : Type u_2
inst✝² : Category.{?u.324397, u_2} D
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
i j : ι
inst✝ : DecidableEq ι
k : ι
⊢ ((Discrete.natTrans fun k =>
if h : k.as = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to ((Discrete.functor X).obj k)) ≫
c.ι).app
{ as := k } =
((Cofan.mk (X i) fun k => if h : k = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (X i)).ι ≫
(Functor.const (Discrete ι)).map (c.inj i)).app
{ as := k }
|
simp only [<a>CategoryTheory.Discrete.functor_obj</a>, <a>CategoryTheory.Functor.const_obj_obj</a>, <a>CategoryTheory.NatTrans.comp_app</a>, <a>CategoryTheory.Discrete.natTrans_app</a>, <a>CategoryTheory.Limits.Cofan.mk_pt</a>, <a>CategoryTheory.Limits.Cofan.mk_ι_app</a>, <a>CategoryTheory.Functor.const_map_app</a>]
|
case refine_1.w.h.mk
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.324390, u_1} K
D : Type u_2
inst✝² : Category.{?u.324397, u_2} D
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
i j : ι
inst✝ : DecidableEq ι
k : ι
⊢ (if h : k = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (X k)) ≫ c.ι.app { as := k } =
(if h : k = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (X i)) ≫ c.inj i
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Limits/VanKampen.lean
|
CategoryTheory.isPullback_of_cofan_isVanKampen
|
case refine_1.w.h.mk
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.324390, u_1} K
D : Type u_2
inst✝² : Category.{?u.324397, u_2} D
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
i j : ι
inst✝ : DecidableEq ι
k : ι
⊢ (if h : k = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (X k)) ≫ c.ι.app { as := k } =
(if h : k = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (X i)) ≫ c.inj i
|
split
|
case refine_1.w.h.mk.isTrue
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.324390, u_1} K
D : Type u_2
inst✝² : Category.{?u.324397, u_2} D
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
i j : ι
inst✝ : DecidableEq ι
k : ι
h✝ : k = i
⊢ eqToHom ⋯ ≫ c.ι.app { as := k } = eqToHom ⋯ ≫ c.inj i
case refine_1.w.h.mk.isFalse
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.324390, u_1} K
D : Type u_2
inst✝² : Category.{?u.324397, u_2} D
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
i j : ι
inst✝ : DecidableEq ι
k : ι
h✝ : ¬k = i
⊢ (eqToHom ⋯ ≫ initial.to (X k)) ≫ c.ι.app { as := k } = (eqToHom ⋯ ≫ initial.to (X i)) ≫ c.inj i
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Limits/VanKampen.lean
|
CategoryTheory.isPullback_of_cofan_isVanKampen
|
case refine_1.w.h.mk.isTrue
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.324390, u_1} K
D : Type u_2
inst✝² : Category.{?u.324397, u_2} D
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
i j : ι
inst✝ : DecidableEq ι
k : ι
h✝ : k = i
⊢ eqToHom ⋯ ≫ c.ι.app { as := k } = eqToHom ⋯ ≫ c.inj i
|
subst ‹k = i›
|
case refine_1.w.h.mk.isTrue
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.324390, u_1} K
D : Type u_2
inst✝² : Category.{?u.324397, u_2} D
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
j : ι
inst✝ : DecidableEq ι
k : ι
⊢ eqToHom ⋯ ≫ c.ι.app { as := k } = eqToHom ⋯ ≫ c.inj k
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Limits/VanKampen.lean
|
CategoryTheory.isPullback_of_cofan_isVanKampen
|
case refine_1.w.h.mk.isTrue
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.324390, u_1} K
D : Type u_2
inst✝² : Category.{?u.324397, u_2} D
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
j : ι
inst✝ : DecidableEq ι
k : ι
⊢ eqToHom ⋯ ≫ c.ι.app { as := k } = eqToHom ⋯ ≫ c.inj k
|
rfl
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Limits/VanKampen.lean
|
CategoryTheory.isPullback_of_cofan_isVanKampen
|
case refine_1.w.h.mk.isFalse
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.324390, u_1} K
D : Type u_2
inst✝² : Category.{?u.324397, u_2} D
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
i j : ι
inst✝ : DecidableEq ι
k : ι
h✝ : ¬k = i
⊢ (eqToHom ⋯ ≫ initial.to (X k)) ≫ c.ι.app { as := k } = (eqToHom ⋯ ≫ initial.to (X i)) ≫ c.inj i
|
simp
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Limits/VanKampen.lean
|
CategoryTheory.isPullback_of_cofan_isVanKampen
|
case refine_2
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.324390, u_1} K
D : Type u_2
inst✝² : Category.{?u.324397, u_2} D
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
i j : ι
inst✝ : DecidableEq ι
⊢ IsColimit (Cofan.mk (X i) fun k => if h : k = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (X i))
|
refine <a>CategoryTheory.Limits.mkCofanColimit</a> _ (fun t ↦ (<a>CategoryTheory.eqToHom</a> (<a>if_pos</a> <a>rfl</a>).<a>Eq.symm</a>) ≫ t.inj i) ?_ ?_
|
case refine_2.refine_1
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.324390, u_1} K
D : Type u_2
inst✝² : Category.{?u.324397, u_2} D
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
i j : ι
inst✝ : DecidableEq ι
⊢ ∀ (t : Cofan fun k => if k = i then X i else ⊥_ C) (j : ι),
(Cofan.mk (X i) fun k => if h : k = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (X i)).inj j ≫
(fun t => eqToHom ⋯ ≫ t.inj i) t =
t.inj j
case refine_2.refine_2
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.324390, u_1} K
D : Type u_2
inst✝² : Category.{?u.324397, u_2} D
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
i j : ι
inst✝ : DecidableEq ι
⊢ ∀ (t : Cofan fun k => if k = i then X i else ⊥_ C)
(m : (Cofan.mk (X i) fun k => if h : k = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (X i)).pt ⟶ t.pt),
(∀ (j : ι),
(Cofan.mk (X i) fun k => if h : k = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (X i)).inj j ≫ m = t.inj j) →
m = (fun t => eqToHom ⋯ ≫ t.inj i) t
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Limits/VanKampen.lean
|
CategoryTheory.isPullback_of_cofan_isVanKampen
|
case refine_2.refine_1
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.324390, u_1} K
D : Type u_2
inst✝² : Category.{?u.324397, u_2} D
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
i j : ι
inst✝ : DecidableEq ι
⊢ ∀ (t : Cofan fun k => if k = i then X i else ⊥_ C) (j : ι),
(Cofan.mk (X i) fun k => if h : k = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (X i)).inj j ≫
(fun t => eqToHom ⋯ ≫ t.inj i) t =
t.inj j
|
intro t j
|
case refine_2.refine_1
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.324390, u_1} K
D : Type u_2
inst✝² : Category.{?u.324397, u_2} D
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
i j✝ : ι
inst✝ : DecidableEq ι
t : Cofan fun k => if k = i then X i else ⊥_ C
j : ι
⊢ (Cofan.mk (X i) fun k => if h : k = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (X i)).inj j ≫
(fun t => eqToHom ⋯ ≫ t.inj i) t =
t.inj j
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Limits/VanKampen.lean
|
CategoryTheory.isPullback_of_cofan_isVanKampen
|
case refine_2.refine_1
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.324390, u_1} K
D : Type u_2
inst✝² : Category.{?u.324397, u_2} D
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
i j✝ : ι
inst✝ : DecidableEq ι
t : Cofan fun k => if k = i then X i else ⊥_ C
j : ι
⊢ (Cofan.mk (X i) fun k => if h : k = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (X i)).inj j ≫
(fun t => eqToHom ⋯ ≫ t.inj i) t =
t.inj j
|
simp only [<a>CategoryTheory.Limits.Cofan.mk_pt</a>, <a>CategoryTheory.Limits.cofan_mk_inj</a>]
|
case refine_2.refine_1
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.324390, u_1} K
D : Type u_2
inst✝² : Category.{?u.324397, u_2} D
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
i j✝ : ι
inst✝ : DecidableEq ι
t : Cofan fun k => if k = i then X i else ⊥_ C
j : ι
⊢ (if h : j = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (X i)) ≫ eqToHom ⋯ ≫ t.inj i = t.inj j
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Limits/VanKampen.lean
|
CategoryTheory.isPullback_of_cofan_isVanKampen
|
case refine_2.refine_1
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.324390, u_1} K
D : Type u_2
inst✝² : Category.{?u.324397, u_2} D
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
i j✝ : ι
inst✝ : DecidableEq ι
t : Cofan fun k => if k = i then X i else ⊥_ C
j : ι
⊢ (if h : j = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (X i)) ≫ eqToHom ⋯ ≫ t.inj i = t.inj j
|
split
|
case refine_2.refine_1.isTrue
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.324390, u_1} K
D : Type u_2
inst✝² : Category.{?u.324397, u_2} D
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
i j✝ : ι
inst✝ : DecidableEq ι
t : Cofan fun k => if k = i then X i else ⊥_ C
j : ι
h✝ : j = i
⊢ eqToHom ⋯ ≫ eqToHom ⋯ ≫ t.inj i = t.inj j
case refine_2.refine_1.isFalse
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.324390, u_1} K
D : Type u_2
inst✝² : Category.{?u.324397, u_2} D
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
i j✝ : ι
inst✝ : DecidableEq ι
t : Cofan fun k => if k = i then X i else ⊥_ C
j : ι
h✝ : ¬j = i
⊢ (eqToHom ⋯ ≫ initial.to (X i)) ≫ eqToHom ⋯ ≫ t.inj i = t.inj j
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Limits/VanKampen.lean
|
CategoryTheory.isPullback_of_cofan_isVanKampen
|
case refine_2.refine_1.isTrue
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.324390, u_1} K
D : Type u_2
inst✝² : Category.{?u.324397, u_2} D
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
i j✝ : ι
inst✝ : DecidableEq ι
t : Cofan fun k => if k = i then X i else ⊥_ C
j : ι
h✝ : j = i
⊢ eqToHom ⋯ ≫ eqToHom ⋯ ≫ t.inj i = t.inj j
|
subst ‹j = i›
|
case refine_2.refine_1.isTrue
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.324390, u_1} K
D : Type u_2
inst✝² : Category.{?u.324397, u_2} D
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
j✝ : ι
inst✝ : DecidableEq ι
j : ι
t : Cofan fun k => if k = j then X j else ⊥_ C
⊢ eqToHom ⋯ ≫ eqToHom ⋯ ≫ t.inj j = t.inj j
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Limits/VanKampen.lean
|
CategoryTheory.isPullback_of_cofan_isVanKampen
|
case refine_2.refine_1.isTrue
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.324390, u_1} K
D : Type u_2
inst✝² : Category.{?u.324397, u_2} D
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
j✝ : ι
inst✝ : DecidableEq ι
j : ι
t : Cofan fun k => if k = j then X j else ⊥_ C
⊢ eqToHom ⋯ ≫ eqToHom ⋯ ≫ t.inj j = t.inj j
|
simp
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Limits/VanKampen.lean
|
CategoryTheory.isPullback_of_cofan_isVanKampen
|
case refine_2.refine_1.isFalse
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.324390, u_1} K
D : Type u_2
inst✝² : Category.{?u.324397, u_2} D
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
i j✝ : ι
inst✝ : DecidableEq ι
t : Cofan fun k => if k = i then X i else ⊥_ C
j : ι
h✝ : ¬j = i
⊢ (eqToHom ⋯ ≫ initial.to (X i)) ≫ eqToHom ⋯ ≫ t.inj i = t.inj j
|
rw [<a>CategoryTheory.Category.assoc</a>, ← <a>CategoryTheory.IsIso.eq_inv_comp</a>]
|
case refine_2.refine_1.isFalse
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.324390, u_1} K
D : Type u_2
inst✝² : Category.{?u.324397, u_2} D
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
i j✝ : ι
inst✝ : DecidableEq ι
t : Cofan fun k => if k = i then X i else ⊥_ C
j : ι
h✝ : ¬j = i
⊢ initial.to (X i) ≫ eqToHom ⋯ ≫ t.inj i = inv (eqToHom ⋯) ≫ t.inj j
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Limits/VanKampen.lean
|
CategoryTheory.isPullback_of_cofan_isVanKampen
|
case refine_2.refine_1.isFalse
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.324390, u_1} K
D : Type u_2
inst✝² : Category.{?u.324397, u_2} D
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
i j✝ : ι
inst✝ : DecidableEq ι
t : Cofan fun k => if k = i then X i else ⊥_ C
j : ι
h✝ : ¬j = i
⊢ initial.to (X i) ≫ eqToHom ⋯ ≫ t.inj i = inv (eqToHom ⋯) ≫ t.inj j
|
exact initialIsInitial.hom_ext _ _
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Limits/VanKampen.lean
|
CategoryTheory.isPullback_of_cofan_isVanKampen
|
case refine_2.refine_2
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.324390, u_1} K
D : Type u_2
inst✝² : Category.{?u.324397, u_2} D
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
i j : ι
inst✝ : DecidableEq ι
⊢ ∀ (t : Cofan fun k => if k = i then X i else ⊥_ C)
(m : (Cofan.mk (X i) fun k => if h : k = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (X i)).pt ⟶ t.pt),
(∀ (j : ι),
(Cofan.mk (X i) fun k => if h : k = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (X i)).inj j ≫ m = t.inj j) →
m = (fun t => eqToHom ⋯ ≫ t.inj i) t
|
intro t m hm
|
case refine_2.refine_2
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.324390, u_1} K
D : Type u_2
inst✝² : Category.{?u.324397, u_2} D
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
i j : ι
inst✝ : DecidableEq ι
t : Cofan fun k => if k = i then X i else ⊥_ C
m : (Cofan.mk (X i) fun k => if h : k = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (X i)).pt ⟶ t.pt
hm :
∀ (j : ι), (Cofan.mk (X i) fun k => if h : k = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (X i)).inj j ≫ m = t.inj j
⊢ m = (fun t => eqToHom ⋯ ≫ t.inj i) t
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Limits/VanKampen.lean
|
CategoryTheory.isPullback_of_cofan_isVanKampen
|
case refine_2.refine_2
J : Type v'
inst✝⁵ : Category.{u', v'} J
C : Type u
inst✝⁴ : Category.{v, u} C
K : Type u_1
inst✝³ : Category.{?u.324390, u_1} K
D : Type u_2
inst✝² : Category.{?u.324397, u_2} D
inst✝¹ : HasInitial C
ι : Type u_3
X : ι → C
c : Cofan X
hc : IsVanKampenColimit c
i j : ι
inst✝ : DecidableEq ι
t : Cofan fun k => if k = i then X i else ⊥_ C
m : (Cofan.mk (X i) fun k => if h : k = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (X i)).pt ⟶ t.pt
hm :
∀ (j : ι), (Cofan.mk (X i) fun k => if h : k = i then eqToHom ⋯ else eqToHom ⋯ ≫ initial.to (X i)).inj j ≫ m = t.inj j
⊢ m = (fun t => eqToHom ⋯ ≫ t.inj i) t
|
simp [← hm i]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Limits/VanKampen.lean
|
AddLECancellable.tsub_add_eq_add_tsub
|
α : Type u_1
inst✝⁵ : AddCommSemigroup α
inst✝⁴ : PartialOrder α
inst✝³ : ExistsAddOfLE α
inst✝² : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1
inst✝¹ : Sub α
inst✝ : OrderedSub α
a b c d : α
hb : AddLECancellable b
h : b ≤ a
⊢ a - b + c = a + c - b
|
rw [<a>add_comm</a> a, hb.add_tsub_assoc_of_le h, <a>add_comm</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Order/Sub/Canonical.lean
|
Filter.eq_top_of_neBot
|
α : Type u
β : Type v
γ : Type w
δ : Type u_1
ι : Sort x
f g : Filter α
s t : Set α
inst✝¹ : Subsingleton α
l : Filter α
inst✝ : l.NeBot
⊢ l = ⊤
|
refine <a>top_unique</a> fun s hs => ?_
|
α : Type u
β : Type v
γ : Type w
δ : Type u_1
ι : Sort x
f g : Filter α
s✝ t : Set α
inst✝¹ : Subsingleton α
l : Filter α
inst✝ : l.NeBot
s : Set α
hs : s ∈ l
⊢ s ∈ ⊤
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Order/Filter/Basic.lean
|
Filter.eq_top_of_neBot
|
α : Type u
β : Type v
γ : Type w
δ : Type u_1
ι : Sort x
f g : Filter α
s✝ t : Set α
inst✝¹ : Subsingleton α
l : Filter α
inst✝ : l.NeBot
s : Set α
hs : s ∈ l
⊢ s ∈ ⊤
|
obtain rfl : s = <a>Set.univ</a> := <a>Subsingleton.eq_univ_of_nonempty</a> (<a>Filter.nonempty_of_mem</a> hs)
|
α : Type u
β : Type v
γ : Type w
δ : Type u_1
ι : Sort x
f g : Filter α
s t : Set α
inst✝¹ : Subsingleton α
l : Filter α
inst✝ : l.NeBot
hs : univ ∈ l
⊢ univ ∈ ⊤
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Order/Filter/Basic.lean
|
Filter.eq_top_of_neBot
|
α : Type u
β : Type v
γ : Type w
δ : Type u_1
ι : Sort x
f g : Filter α
s t : Set α
inst✝¹ : Subsingleton α
l : Filter α
inst✝ : l.NeBot
hs : univ ∈ l
⊢ univ ∈ ⊤
|
exact <a>Filter.univ_mem</a>
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Order/Filter/Basic.lean
|
Pell.y_mul_dvd
|
a : ℕ
a1 : 1 < a
n k : ℕ
⊢ yn a1 n ∣ yn a1 (n * (k + 1))
|
rw [<a>Nat.mul_succ</a>, <a>Pell.yn_add</a>]
|
a : ℕ
a1 : 1 < a
n k : ℕ
⊢ yn a1 n ∣ xn a1 (n * k) * yn a1 n + yn a1 (n * k) * xn a1 n
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/NumberTheory/PellMatiyasevic.lean
|
Pell.y_mul_dvd
|
a : ℕ
a1 : 1 < a
n k : ℕ
⊢ yn a1 n ∣ xn a1 (n * k) * yn a1 n + yn a1 (n * k) * xn a1 n
|
exact <a>dvd_add</a> (<a>dvd_mul_left</a> _ _) ((y_mul_dvd _ k).<a>Dvd.dvd.mul_right</a> _)
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/NumberTheory/PellMatiyasevic.lean
|
RatFunc.mk_zero
|
K : Type u
inst✝¹ : CommRing K
inst✝ : IsDomain K
p : K[X]
⊢ RatFunc.mk p 0 = { toFractionRing := 0 }
|
rw [<a>RatFunc.mk_eq_div'</a>, <a>RingHom.map_zero</a>, <a>div_zero</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/FieldTheory/RatFunc/Defs.lean
|
LinearMap.iterate_surjective
|
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
S : Type u_4
M : Type u_5
M₁ : Type u_6
M₂ : Type u_7
M₃ : Type u_8
N : Type u_9
N₁ : Type u_10
N₂ : Type u_11
inst✝⁷ : Semiring R
inst✝⁶ : AddCommMonoid M
inst✝⁵ : AddCommGroup N₁
inst✝⁴ : Module R M
inst✝³ : Module R N₁
inst✝² : Monoid S
inst✝¹ : DistribMulAction S M
inst✝ : SMulCommClass R S M
f' : M →ₗ[R] M
h : Surjective ⇑f'
n : ℕ
⊢ Surjective ⇑(f' ^ (n + 1))
|
rw [<a>LinearMap.iterate_succ</a>]
|
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
S : Type u_4
M : Type u_5
M₁ : Type u_6
M₂ : Type u_7
M₃ : Type u_8
N : Type u_9
N₁ : Type u_10
N₂ : Type u_11
inst✝⁷ : Semiring R
inst✝⁶ : AddCommMonoid M
inst✝⁵ : AddCommGroup N₁
inst✝⁴ : Module R M
inst✝³ : Module R N₁
inst✝² : Monoid S
inst✝¹ : DistribMulAction S M
inst✝ : SMulCommClass R S M
f' : M →ₗ[R] M
h : Surjective ⇑f'
n : ℕ
⊢ Surjective ⇑((f' ^ n) ∘ₗ f')
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Module/LinearMap/End.lean
|
LinearMap.iterate_surjective
|
R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
S : Type u_4
M : Type u_5
M₁ : Type u_6
M₂ : Type u_7
M₃ : Type u_8
N : Type u_9
N₁ : Type u_10
N₂ : Type u_11
inst✝⁷ : Semiring R
inst✝⁶ : AddCommMonoid M
inst✝⁵ : AddCommGroup N₁
inst✝⁴ : Module R M
inst✝³ : Module R N₁
inst✝² : Monoid S
inst✝¹ : DistribMulAction S M
inst✝ : SMulCommClass R S M
f' : M →ₗ[R] M
h : Surjective ⇑f'
n : ℕ
⊢ Surjective ⇑((f' ^ n) ∘ₗ f')
|
exact (iterate_surjective h n).<a>Function.Surjective.comp</a> h
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Module/LinearMap/End.lean
|
GaloisConnection.upperBounds_l_image
|
α : Type u
β : Type v
γ : Type w
ι : Sort x
κ : ι → Sort u_1
a a₁ a₂ : α
b✝ b₁ b₂ : β
inst✝¹ : Preorder α
inst✝ : Preorder β
l : α → β
u : β → α
gc : GaloisConnection l u
s : Set α
b : β
⊢ b ∈ upperBounds (l '' s) ↔ b ∈ u ⁻¹' upperBounds s
|
simp [<a>upperBounds</a>, gc _ _]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Order/GaloisConnection.lean
|
MeasureTheory.hasFiniteIntegral_iff_ofNNReal
|
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
m : MeasurableSpace α
μ ν : Measure α
inst✝² : MeasurableSpace δ
inst✝¹ : NormedAddCommGroup β
inst✝ : NormedAddCommGroup γ
f : α → ℝ≥0
⊢ HasFiniteIntegral (fun x => ↑(f x)) μ ↔ ∫⁻ (a : α), ↑(f a) ∂μ < ⊤
|
simp [<a>MeasureTheory.hasFiniteIntegral_iff_norm</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/MeasureTheory/Function/L1Space.lean
|
Complex.I_mul
|
z : ℂ
⊢ (I * z).re = { re := -z.im, im := z.re }.re ∧ (I * z).im = { re := -z.im, im := z.re }.im
|
simp
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Complex/Basic.lean
|
MvPolynomial.weightedHomogeneousComponent_mem
|
R : Type u_1
M : Type u_2
inst✝¹ : CommSemiring R
σ : Type u_3
inst✝ : AddCommMonoid M
w✝ : σ → M
n : M
φ✝ ψ : MvPolynomial σ R
w : σ → M
φ : MvPolynomial σ R
m : M
⊢ (weightedHomogeneousComponent w m) φ ∈ weightedHomogeneousSubmodule R w m
|
rw [<a>MvPolynomial.mem_weightedHomogeneousSubmodule</a>]
|
R : Type u_1
M : Type u_2
inst✝¹ : CommSemiring R
σ : Type u_3
inst✝ : AddCommMonoid M
w✝ : σ → M
n : M
φ✝ ψ : MvPolynomial σ R
w : σ → M
φ : MvPolynomial σ R
m : M
⊢ IsWeightedHomogeneous w ((weightedHomogeneousComponent w m) φ) m
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean
|
MvPolynomial.weightedHomogeneousComponent_mem
|
R : Type u_1
M : Type u_2
inst✝¹ : CommSemiring R
σ : Type u_3
inst✝ : AddCommMonoid M
w✝ : σ → M
n : M
φ✝ ψ : MvPolynomial σ R
w : σ → M
φ : MvPolynomial σ R
m : M
⊢ IsWeightedHomogeneous w ((weightedHomogeneousComponent w m) φ) m
|
exact <a>MvPolynomial.weightedHomogeneousComponent_isWeightedHomogeneous</a> m φ
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean
|
List.fst_lt_add_of_mem_enumFrom
|
α : Type u_1
β : Type u_2
x : ℕ × α
n : ℕ
l : List α
h : x ∈ enumFrom n l
⊢ x.1 < n + l.length
|
rcases <a>List.mem_iff_get</a>.1 h with ⟨i, rfl⟩
|
case intro
α : Type u_1
β : Type u_2
n : ℕ
l : List α
i : Fin (enumFrom n l).length
h : (enumFrom n l).get i ∈ enumFrom n l
⊢ ((enumFrom n l).get i).1 < n + l.length
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/List/Enum.lean
|
List.fst_lt_add_of_mem_enumFrom
|
case intro
α : Type u_1
β : Type u_2
n : ℕ
l : List α
i : Fin (enumFrom n l).length
h : (enumFrom n l).get i ∈ enumFrom n l
⊢ ((enumFrom n l).get i).1 < n + l.length
|
simpa using i.is_lt
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/List/Enum.lean
|
CauSeq.Completion.ofRat_div
|
α : Type u_1
inst✝² : LinearOrderedField α
β : Type u_2
inst✝¹ : DivisionRing β
abv : β → α
inst✝ : IsAbsoluteValue abv
x y : β
⊢ ofRat (x / y) = ofRat x / ofRat y
|
simp only [<a>div_eq_mul_inv</a>, <a>CauSeq.Completion.ofRat_inv</a>, <a>CauSeq.Completion.ofRat_mul</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Order/CauSeq/Completion.lean
|
Filter.tendsto_nhds
|
ι : Sort u_1
α : Type u_2
β : Type u_3
X : Type u_4
Y : Type u_5
la : Filter α
lb : Filter β
f : α → Filter β
⊢ Tendsto f la (𝓝 lb) ↔ ∀ s ∈ lb, ∀ᶠ (a : α) in la, s ∈ f a
|
simp only [<a>Filter.nhds_eq'</a>, <a>Filter.tendsto_lift'</a>, <a>Set.mem_setOf_eq</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Topology/Filter.lean
|
Finpartition.card_parts_equitabilise
|
α : Type u_1
inst✝ : DecidableEq α
s t : Finset α
m n a b : ℕ
P : Finpartition s
h : a * m + b * (m + 1) = s.card
hm : m ≠ 0
⊢ (equitabilise h).parts.card = a + b
|
rw [← <a>Finset.filter_true_of_mem</a> fun x => <a>Finpartition.card_eq_of_mem_parts_equitabilise</a>, <a>Finset.filter_or</a>, <a>Finset.card_union_of_disjoint</a>, P.card_filter_equitabilise_small _ hm, P.card_filter_equitabilise_big]
|
α : Type u_1
inst✝ : DecidableEq α
s t : Finset α
m n a b : ℕ
P : Finpartition s
h : a * m + b * (m + 1) = s.card
hm : m ≠ 0
⊢ Disjoint (filter (fun x => x.card = m) (equitabilise h).parts)
(filter (fun x => x.card = m + 1) (equitabilise h).parts)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Combinatorics/SimpleGraph/Regularity/Equitabilise.lean
|
Finpartition.card_parts_equitabilise
|
α : Type u_1
inst✝ : DecidableEq α
s t : Finset α
m n a b : ℕ
P : Finpartition s
h : a * m + b * (m + 1) = s.card
hm : m ≠ 0
⊢ Disjoint (filter (fun x => x.card = m) (equitabilise h).parts)
(filter (fun x => x.card = m + 1) (equitabilise h).parts)
|
exact <a>Finset.disjoint_filter</a>.2 fun x _ h₀ h₁ => <a>Nat.succ_ne_self</a> m <| h₁.symm.trans h₀
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Combinatorics/SimpleGraph/Regularity/Equitabilise.lean
|
Nat.set_eq_univ
|
a b m n k : ℕ
S : Set ℕ
⊢ S = Set.univ → 0 ∈ S ∧ ∀ k ∈ S, k + 1 ∈ S
|
rintro rfl
|
a b m n k : ℕ
⊢ 0 ∈ Set.univ ∧ ∀ k ∈ Set.univ, k + 1 ∈ Set.univ
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Nat/Order/Lemmas.lean
|
Nat.set_eq_univ
|
a b m n k : ℕ
⊢ 0 ∈ Set.univ ∧ ∀ k ∈ Set.univ, k + 1 ∈ Set.univ
|
simp
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Data/Nat/Order/Lemmas.lean
|
ProbabilityTheory.measurable_condexpKernel
|
Ω : Type u_1
F : Type u_2
m mΩ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
μ : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : NormedAddCommGroup F
f : Ω → F
s : Set Ω
hs : MeasurableSet s
⊢ Measurable fun ω => ((condexpKernel μ m) ω) s
|
simp_rw [<a>ProbabilityTheory.condexpKernel_apply_eq_condDistrib</a>]
|
Ω : Type u_1
F : Type u_2
m mΩ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
μ : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : NormedAddCommGroup F
f : Ω → F
s : Set Ω
hs : MeasurableSet s
⊢ Measurable fun ω => ((condDistrib id id μ) (id ω)) s
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Condexp.lean
|
ProbabilityTheory.measurable_condexpKernel
|
Ω : Type u_1
F : Type u_2
m mΩ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
μ : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : NormedAddCommGroup F
f : Ω → F
s : Set Ω
hs : MeasurableSet s
⊢ Measurable fun ω => ((condDistrib id id μ) (id ω)) s
|
convert <a>ProbabilityTheory.measurable_condDistrib</a> (μ := μ) hs
|
case h.e'_3
Ω : Type u_1
F : Type u_2
m mΩ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
μ : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : NormedAddCommGroup F
f : Ω → F
s : Set Ω
hs : MeasurableSet s
⊢ m ⊓ mΩ = MeasurableSpace.comap id (m ⊓ mΩ)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Condexp.lean
|
ProbabilityTheory.measurable_condexpKernel
|
case h.e'_3
Ω : Type u_1
F : Type u_2
m mΩ : MeasurableSpace Ω
inst✝³ : StandardBorelSpace Ω
inst✝² : Nonempty Ω
μ : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : NormedAddCommGroup F
f : Ω → F
s : Set Ω
hs : MeasurableSet s
⊢ m ⊓ mΩ = MeasurableSpace.comap id (m ⊓ mΩ)
|
rw [<a>MeasurableSpace.comap_id</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Probability/Kernel/Condexp.lean
|
CategoryTheory.SimplicialObject.δ_comp_δ_self
|
C : Type u
inst✝ : Category.{v, u} C
X : SimplicialObject C
n : ℕ
i : Fin (n + 2)
⊢ X.δ i.castSucc ≫ X.δ i = X.δ i.succ ≫ X.δ i
|
dsimp [<a>CategoryTheory.SimplicialObject.δ</a>]
|
C : Type u
inst✝ : Category.{v, u} C
X : SimplicialObject C
n : ℕ
i : Fin (n + 2)
⊢ X.map (SimplexCategory.δ i.castSucc).op ≫ X.map (SimplexCategory.δ i).op =
X.map (SimplexCategory.δ i.succ).op ≫ X.map (SimplexCategory.δ i).op
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/AlgebraicTopology/SimplicialObject.lean
|
CategoryTheory.SimplicialObject.δ_comp_δ_self
|
C : Type u
inst✝ : Category.{v, u} C
X : SimplicialObject C
n : ℕ
i : Fin (n + 2)
⊢ X.map (SimplexCategory.δ i.castSucc).op ≫ X.map (SimplexCategory.δ i).op =
X.map (SimplexCategory.δ i.succ).op ≫ X.map (SimplexCategory.δ i).op
|
simp only [← X.map_comp, ← <a>CategoryTheory.op_comp</a>, <a>SimplexCategory.δ_comp_δ_self</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/AlgebraicTopology/SimplicialObject.lean
|
Ordinal.Cardinal.mk_iUnion_Ordinal_le_of_le
|
β : Type u_1
o : Ordinal.{u_1}
c : Cardinal.{u_1}
ho : o.card ≤ c
hc : ℵ₀ ≤ c
A : Ordinal.{u_1} → Set β
hA : ∀ j < o, #↑(A j) ≤ c
⊢ #↑(⋃ j, ⋃ (_ : j < o), A j) ≤ c
|
simp_rw [← <a>Set.mem_Iio</a>, <a>Set.biUnion_eq_iUnion</a>, <a>Set.iUnion</a>, <a>iSup</a>, ← o.enumIsoOut.symm.surjective.range_comp]
|
β : Type u_1
o : Ordinal.{u_1}
c : Cardinal.{u_1}
ho : o.card ≤ c
hc : ℵ₀ ≤ c
A : Ordinal.{u_1} → Set β
hA : ∀ j < o, #↑(A j) ≤ c
⊢ #↑(sSup (range ((fun x => A ↑x) ∘ ⇑o.enumIsoOut.symm))) ≤ c
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/SetTheory/Cardinal/Ordinal.lean
|
Ordinal.Cardinal.mk_iUnion_Ordinal_le_of_le
|
β : Type u_1
o : Ordinal.{u_1}
c : Cardinal.{u_1}
ho : o.card ≤ c
hc : ℵ₀ ≤ c
A : Ordinal.{u_1} → Set β
hA : ∀ j < o, #↑(A j) ≤ c
⊢ #↑(sSup (range ((fun x => A ↑x) ∘ ⇑o.enumIsoOut.symm))) ≤ c
|
apply ((<a>Cardinal.mk_iUnion_le</a> _).<a>LE.le.trans</a> _).<a>LE.le.trans_eq</a> (<a>Cardinal.mul_eq_self</a> hc)
|
β : Type u_1
o : Ordinal.{u_1}
c : Cardinal.{u_1}
ho : o.card ≤ c
hc : ℵ₀ ≤ c
A : Ordinal.{u_1} → Set β
hA : ∀ j < o, #↑(A j) ≤ c
⊢ #(Quotient.out o).α * ⨆ i, #↑(((fun x => A ↑x) ∘ ⇑o.enumIsoOut.symm) i) ≤ c * c
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/SetTheory/Cardinal/Ordinal.lean
|
Ordinal.Cardinal.mk_iUnion_Ordinal_le_of_le
|
β : Type u_1
o : Ordinal.{u_1}
c : Cardinal.{u_1}
ho : o.card ≤ c
hc : ℵ₀ ≤ c
A : Ordinal.{u_1} → Set β
hA : ∀ j < o, #↑(A j) ≤ c
⊢ #(Quotient.out o).α * ⨆ i, #↑(((fun x => A ↑x) ∘ ⇑o.enumIsoOut.symm) i) ≤ c * c
|
rw [<a>Cardinal.mk_ordinal_out</a>]
|
β : Type u_1
o : Ordinal.{u_1}
c : Cardinal.{u_1}
ho : o.card ≤ c
hc : ℵ₀ ≤ c
A : Ordinal.{u_1} → Set β
hA : ∀ j < o, #↑(A j) ≤ c
⊢ o.card * ⨆ i, #↑(((fun x => A ↑x) ∘ ⇑o.enumIsoOut.symm) i) ≤ c * c
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/SetTheory/Cardinal/Ordinal.lean
|
Function.mulSupport_disjoint_iff
|
α : Type u_1
β : Type u_2
A : Type u_3
B : Type u_4
M : Type u_5
N : Type u_6
P : Type u_7
G : Type u_8
inst✝² : One M
inst✝¹ : One N
inst✝ : One P
f : α → M
s : Set α
⊢ Disjoint (mulSupport f) s ↔ EqOn f 1 s
|
simp_rw [← <a>Set.subset_compl_iff_disjoint_right</a>, <a>Function.mulSupport_subset_iff'</a>, <a>Set.not_mem_compl_iff</a>, <a>Set.EqOn</a>, <a>Pi.one_apply</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Group/Support.lean
|
BooleanRing.sup_assoc
|
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : BooleanRing α
inst✝¹ : BooleanRing β
inst✝ : BooleanRing γ
a b c : α
⊢ a + b + a * b + c + (a + b + a * b) * c = a + (b + c + b * c) + a * (b + c + b * c)
|
ring
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Algebra/Ring/BooleanRing.lean
|
affinity_unitClosedBall
|
𝕜 : Type u_1
E : Type u_2
inst✝³ : NormedField 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace ℝ E
r : ℝ
hr : 0 ≤ r
x : E
⊢ x +ᵥ r • closedBall 0 1 = closedBall x r
|
rw [<a>smul_closedUnitBall</a>, <a>Real.norm_of_nonneg</a> hr, <a>vadd_closedBall_zero</a>]
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/Analysis/NormedSpace/Pointwise.lean
|
CategoryTheory.GrothendieckTopology.over_map_compatiblePreserving
|
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (J.over Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : x.Compatible
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
⊢ F.val.map f₁.op (x g₁ hg₁) = F.val.map f₂.op (x g₂ hg₂)
|
let W' : <a>CategoryTheory.Over</a> X := <a>CategoryTheory.Over.mk</a> (f₁.left ≫ Y₁.hom)
|
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (J.over Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : x.Compatible
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
⊢ F.val.map f₁.op (x g₁ hg₁) = F.val.map f₂.op (x g₂ hg₂)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Sites/Over.lean
|
CategoryTheory.GrothendieckTopology.over_map_compatiblePreserving
|
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (J.over Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : x.Compatible
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
⊢ F.val.map f₁.op (x g₁ hg₁) = F.val.map f₂.op (x g₂ hg₂)
|
let g₁' : W' ⟶ Y₁ := <a>CategoryTheory.Over.homMk</a> f₁.left
|
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (J.over Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : x.Compatible
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left ⋯
⊢ F.val.map f₁.op (x g₁ hg₁) = F.val.map f₂.op (x g₂ hg₂)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Sites/Over.lean
|
CategoryTheory.GrothendieckTopology.over_map_compatiblePreserving
|
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (J.over Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : x.Compatible
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left ⋯
⊢ F.val.map f₁.op (x g₁ hg₁) = F.val.map f₂.op (x g₂ hg₂)
|
let g₂' : W' ⟶ Y₂ := <a>CategoryTheory.Over.homMk</a> f₂.left (by simpa using (<a>CategoryTheory.Over.forget</a> _).<a>Prefunctor.congr_map</a> h.symm =≫ Z.hom)
|
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (J.over Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : x.Compatible
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left ⋯
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left ⋯
⊢ F.val.map f₁.op (x g₁ hg₁) = F.val.map f₂.op (x g₂ hg₂)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Sites/Over.lean
|
CategoryTheory.GrothendieckTopology.over_map_compatiblePreserving
|
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (J.over Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : x.Compatible
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left ⋯
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left ⋯
⊢ F.val.map f₁.op (x g₁ hg₁) = F.val.map f₂.op (x g₂ hg₂)
|
let e : (<a>CategoryTheory.Over.map</a> f).<a>Prefunctor.obj</a> W' ≅ W := <a>CategoryTheory.Over.isoMk</a> (<a>CategoryTheory.Iso.refl</a> _) (by simpa [W'] using (<a>CategoryTheory.Over.w</a> f₁).<a>Eq.symm</a>)
|
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (J.over Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : x.Compatible
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left ⋯
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left ⋯
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left) ⋯
⊢ F.val.map f₁.op (x g₁ hg₁) = F.val.map f₂.op (x g₂ hg₂)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Sites/Over.lean
|
CategoryTheory.GrothendieckTopology.over_map_compatiblePreserving
|
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (J.over Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : x.Compatible
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left ⋯
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left ⋯
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left) ⋯
⊢ F.val.map f₁.op (x g₁ hg₁) = F.val.map f₂.op (x g₂ hg₂)
|
convert <a>congr_arg</a> (F.val.map e.inv.op) (hx g₁' g₂' hg₁ hg₂ (by ext; exact (<a>CategoryTheory.Over.forget</a> _).<a>Prefunctor.congr_map</a> h)) using 1
|
case h.e'_2
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (J.over Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : x.Compatible
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left ⋯
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left ⋯
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left) ⋯
⊢ F.val.map f₁.op (x g₁ hg₁) = F.val.map e.inv.op (((Over.map f).op ⋙ F.val).map g₁'.op (x g₁ hg₁))
case h.e'_3
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (J.over Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : x.Compatible
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left ⋯
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left ⋯
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left) ⋯
⊢ F.val.map f₂.op (x g₂ hg₂) = F.val.map e.inv.op (((Over.map f).op ⋙ F.val).map g₂'.op (x g₂ hg₂))
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Sites/Over.lean
|
CategoryTheory.GrothendieckTopology.over_map_compatiblePreserving
|
case h.e'_2
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (J.over Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : x.Compatible
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left ⋯
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left ⋯
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left) ⋯
⊢ F.val.map f₁.op (x g₁ hg₁) = F.val.map e.inv.op (((Over.map f).op ⋙ F.val).map g₁'.op (x g₁ hg₁))
case h.e'_3
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (J.over Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : x.Compatible
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left ⋯
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left ⋯
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left) ⋯
⊢ F.val.map f₂.op (x g₂ hg₂) = F.val.map e.inv.op (((Over.map f).op ⋙ F.val).map g₂'.op (x g₂ hg₂))
|
all_goals dsimp [e, W', g₁', g₂'] rw [← <a>CategoryTheory.FunctorToTypes.map_comp_apply</a>] apply <a>congr_fun</a> congr 1 rw [← <a>CategoryTheory.op_comp</a>] congr 1 ext simp
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Sites/Over.lean
|
CategoryTheory.GrothendieckTopology.over_map_compatiblePreserving
|
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (J.over Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : x.Compatible
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left ⋯
⊢ f₂.left ≫ Y₂.hom = W'.hom
|
simpa using (<a>CategoryTheory.Over.forget</a> _).<a>Prefunctor.congr_map</a> h.symm =≫ Z.hom
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Sites/Over.lean
|
CategoryTheory.GrothendieckTopology.over_map_compatiblePreserving
|
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (J.over Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : x.Compatible
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left ⋯
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left ⋯
⊢ (Iso.refl ((Over.map f).obj W').left).hom ≫ W.hom = ((Over.map f).obj W').hom
|
simpa [W'] using (<a>CategoryTheory.Over.w</a> f₁).<a>Eq.symm</a>
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Sites/Over.lean
|
CategoryTheory.GrothendieckTopology.over_map_compatiblePreserving
|
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (J.over Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : x.Compatible
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left ⋯
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left ⋯
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left) ⋯
⊢ g₁' ≫ g₁ = g₂' ≫ g₂
|
ext
|
case h
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (J.over Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : x.Compatible
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left ⋯
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left ⋯
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left) ⋯
⊢ (g₁' ≫ g₁).left = (g₂' ≫ g₂).left
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Sites/Over.lean
|
CategoryTheory.GrothendieckTopology.over_map_compatiblePreserving
|
case h
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (J.over Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : x.Compatible
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left ⋯
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left ⋯
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left) ⋯
⊢ (g₁' ≫ g₁).left = (g₂' ≫ g₂).left
|
exact (<a>CategoryTheory.Over.forget</a> _).<a>Prefunctor.congr_map</a> h
|
no goals
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Sites/Over.lean
|
CategoryTheory.GrothendieckTopology.over_map_compatiblePreserving
|
case h.e'_3
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (J.over Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : x.Compatible
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left ⋯
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left ⋯
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left) ⋯
⊢ F.val.map f₂.op (x g₂ hg₂) = F.val.map e.inv.op (((Over.map f).op ⋙ F.val).map g₂'.op (x g₂ hg₂))
|
dsimp [e, W', g₁', g₂']
|
case h.e'_3
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (J.over Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : x.Compatible
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left ⋯
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left ⋯
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left) ⋯
⊢ F.val.map f₂.op (x g₂ hg₂) =
F.val.map (Over.isoMk (Iso.refl W.left) ⋯).inv.op
(F.val.map ((Over.map f).map (Over.homMk f₂.left ⋯)).op (x g₂ hg₂))
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Sites/Over.lean
|
CategoryTheory.GrothendieckTopology.over_map_compatiblePreserving
|
case h.e'_3
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (J.over Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : x.Compatible
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left ⋯
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left ⋯
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left) ⋯
⊢ F.val.map f₂.op (x g₂ hg₂) =
F.val.map (Over.isoMk (Iso.refl W.left) ⋯).inv.op
(F.val.map ((Over.map f).map (Over.homMk f₂.left ⋯)).op (x g₂ hg₂))
|
rw [← <a>CategoryTheory.FunctorToTypes.map_comp_apply</a>]
|
case h.e'_3
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (J.over Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : x.Compatible
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left ⋯
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left ⋯
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left) ⋯
⊢ F.val.map f₂.op (x g₂ hg₂) =
F.val.map (((Over.map f).map (Over.homMk f₂.left ⋯)).op ≫ (Over.isoMk (Iso.refl W.left) ⋯).inv.op) (x g₂ hg₂)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Sites/Over.lean
|
CategoryTheory.GrothendieckTopology.over_map_compatiblePreserving
|
case h.e'_3
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (J.over Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : x.Compatible
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left ⋯
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left ⋯
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left) ⋯
⊢ F.val.map f₂.op (x g₂ hg₂) =
F.val.map (((Over.map f).map (Over.homMk f₂.left ⋯)).op ≫ (Over.isoMk (Iso.refl W.left) ⋯).inv.op) (x g₂ hg₂)
|
apply <a>congr_fun</a>
|
case h.e'_3.h
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (J.over Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : x.Compatible
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left ⋯
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left ⋯
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left) ⋯
⊢ F.val.map f₂.op = F.val.map (((Over.map f).map (Over.homMk f₂.left ⋯)).op ≫ (Over.isoMk (Iso.refl W.left) ⋯).inv.op)
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Sites/Over.lean
|
CategoryTheory.GrothendieckTopology.over_map_compatiblePreserving
|
case h.e'_3.h
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (J.over Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : x.Compatible
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left ⋯
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left ⋯
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left) ⋯
⊢ F.val.map f₂.op = F.val.map (((Over.map f).map (Over.homMk f₂.left ⋯)).op ≫ (Over.isoMk (Iso.refl W.left) ⋯).inv.op)
|
congr 1
|
case h.e'_3.h.e_a
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (J.over Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : x.Compatible
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left ⋯
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left ⋯
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left) ⋯
⊢ f₂.op = ((Over.map f).map (Over.homMk f₂.left ⋯)).op ≫ (Over.isoMk (Iso.refl W.left) ⋯).inv.op
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Sites/Over.lean
|
CategoryTheory.GrothendieckTopology.over_map_compatiblePreserving
|
case h.e'_3.h.e_a
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (J.over Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : x.Compatible
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left ⋯
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left ⋯
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left) ⋯
⊢ f₂.op = ((Over.map f).map (Over.homMk f₂.left ⋯)).op ≫ (Over.isoMk (Iso.refl W.left) ⋯).inv.op
|
rw [← <a>CategoryTheory.op_comp</a>]
|
case h.e'_3.h.e_a
C : Type u
inst✝ : Category.{v, u} C
J : GrothendieckTopology C
X Y : C
f : X ⟶ Y
F : SheafOfTypes (J.over Y)
Z : Over X
T : Presieve Z
x : Presieve.FamilyOfElements ((Over.map f).op ⋙ F.val) T
hx : x.Compatible
Y₁ Y₂ : Over X
W : Over Y
f₁ : W ⟶ (Over.map f).obj Y₁
f₂ : W ⟶ (Over.map f).obj Y₂
g₁ : Y₁ ⟶ Z
g₂ : Y₂ ⟶ Z
hg₁ : T g₁
hg₂ : T g₂
h : f₁ ≫ (Over.map f).map g₁ = f₂ ≫ (Over.map f).map g₂
W' : Over X := Over.mk (f₁.left ≫ Y₁.hom)
g₁' : W' ⟶ Y₁ := Over.homMk f₁.left ⋯
g₂' : W' ⟶ Y₂ := Over.homMk f₂.left ⋯
e : (Over.map f).obj W' ≅ W := Over.isoMk (Iso.refl ((Over.map f).obj W').left) ⋯
⊢ f₂.op = ((Over.isoMk (Iso.refl W.left) ⋯).inv ≫ (Over.map f).map (Over.homMk f₂.left ⋯)).op
|
https://github.com/leanprover-community/mathlib4
|
29dcec074de168ac2bf835a77ef68bbe069194c5
|
Mathlib/CategoryTheory/Sites/Over.lean
|
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