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Theorem znval2 20238
 Description: Self-referential expression for the ℤ/nℤ structure. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 13-Jun-2019.)
Hypotheses
Ref Expression
znval2.s 𝑆 = (RSpan‘ℤring)
znval2.u 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁})))
znval2.y 𝑌 = (ℤ/nℤ‘𝑁)
znval2.l = (le‘𝑌)
Assertion
Ref Expression
znval2 (𝑁 ∈ ℕ0𝑌 = (𝑈 sSet ⟨(le‘ndx), ⟩))

Proof of Theorem znval2
StepHypRef Expression
1 znval2.s . . 3 𝑆 = (RSpan‘ℤring)
2 znval2.u . . 3 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁})))
3 znval2.y . . 3 𝑌 = (ℤ/nℤ‘𝑁)
4 eqid 2824 . . 3 ((ℤRHom‘𝑈) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))) = ((ℤRHom‘𝑈) ↾ if(𝑁 = 0, ℤ, (0..^𝑁)))
5 eqid 2824 . . 3 if(𝑁 = 0, ℤ, (0..^𝑁)) = if(𝑁 = 0, ℤ, (0..^𝑁))
6 eqid 2824 . . 3 ((((ℤRHom‘𝑈) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))) ∘ ≤ ) ∘ ((ℤRHom‘𝑈) ↾ if(𝑁 = 0, ℤ, (0..^𝑁)))) = ((((ℤRHom‘𝑈) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))) ∘ ≤ ) ∘ ((ℤRHom‘𝑈) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))))
71, 2, 3, 4, 5, 6znval 20236 . 2 (𝑁 ∈ ℕ0𝑌 = (𝑈 sSet ⟨(le‘ndx), ((((ℤRHom‘𝑈) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))) ∘ ≤ ) ∘ ((ℤRHom‘𝑈) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))))⟩))
8 znval2.l . . . . 5 = (le‘𝑌)
91, 2, 3, 4, 5, 8znle 20237 . . . 4 (𝑁 ∈ ℕ0 = ((((ℤRHom‘𝑈) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))) ∘ ≤ ) ∘ ((ℤRHom‘𝑈) ↾ if(𝑁 = 0, ℤ, (0..^𝑁)))))
109opeq2d 4796 . . 3 (𝑁 ∈ ℕ0 → ⟨(le‘ndx), ⟩ = ⟨(le‘ndx), ((((ℤRHom‘𝑈) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))) ∘ ≤ ) ∘ ((ℤRHom‘𝑈) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))))⟩)
1110oveq2d 7167 . 2 (𝑁 ∈ ℕ0 → (𝑈 sSet ⟨(le‘ndx), ⟩) = (𝑈 sSet ⟨(le‘ndx), ((((ℤRHom‘𝑈) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))) ∘ ≤ ) ∘ ((ℤRHom‘𝑈) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))))⟩))
127, 11eqtr4d 2862 1 (𝑁 ∈ ℕ0𝑌 = (𝑈 sSet ⟨(le‘ndx), ⟩))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2115  ifcif 4450  {csn 4550  ⟨cop 4556  ◡ccnv 5542   ↾ cres 5545   ∘ ccom 5547  ‘cfv 6345  (class class class)co 7151  0cc0 10537   ≤ cle 10676  ℕ0cn0 11896  ℤcz 11980  ..^cfzo 13039  ndxcnx 16482   sSet csts 16483  lecple 16574   /s cqus 16780   ~QG cqg 18277  RSpancrsp 19945  ℤringzring 20172  ℤRHomczrh 20202  ℤ/nℤczn 20205 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614  ax-addf 10616  ax-mulf 10617 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6137  df-ord 6183  df-on 6184  df-lim 6185  df-suc 6186  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7577  df-1st 7686  df-2nd 7687  df-wrecs 7945  df-recs 8006  df-rdg 8044  df-1o 8100  df-oadd 8104  df-er 8287  df-en 8508  df-dom 8509  df-sdom 8510  df-fin 8511  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11637  df-2 11699  df-3 11700  df-4 11701  df-5 11702  df-6 11703  df-7 11704  df-8 11705  df-9 11706  df-n0 11897  df-z 11981  df-dec 12098  df-uz 12243  df-fz 12897  df-struct 16487  df-ndx 16488  df-slot 16489  df-base 16491  df-sets 16492  df-ress 16493  df-plusg 16580  df-mulr 16581  df-starv 16582  df-tset 16586  df-ple 16587  df-ds 16589  df-unif 16590  df-0g 16717  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-grp 18108  df-minusg 18109  df-subg 18278  df-cmn 18910  df-mgp 19242  df-ur 19254  df-ring 19301  df-cring 19302  df-subrg 19535  df-cnfld 20101  df-zring 20173  df-zn 20209 This theorem is referenced by:  znbaslem  20239
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Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  znval2 Structured version   Visualization version   GIF version

Theorem znval2 20238
 Description: Self-referential expression for the ℤ/nℤ structure. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 13-Jun-2019.)
Hypotheses
Ref Expression
znval2.s 𝑆 = (RSpan‘ℤring)
znval2.u 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁})))
znval2.y 𝑌 = (ℤ/nℤ‘𝑁)
znval2.l = (le‘𝑌)
Assertion
Ref Expression
znval2 (𝑁 ∈ ℕ0𝑌 = (𝑈 sSet ⟨(le‘ndx), ⟩))

Proof of Theorem znval2
StepHypRef Expression
1 znval2.s . . 3 𝑆 = (RSpan‘ℤring)
2 znval2.u . . 3 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁})))
3 znval2.y . . 3 𝑌 = (ℤ/nℤ‘𝑁)
4 eqid 2824 . . 3 ((ℤRHom‘𝑈) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))) = ((ℤRHom‘𝑈) ↾ if(𝑁 = 0, ℤ, (0..^𝑁)))
5 eqid 2824 . . 3 if(𝑁 = 0, ℤ, (0..^𝑁)) = if(𝑁 = 0, ℤ, (0..^𝑁))
6 eqid 2824 . . 3 ((((ℤRHom‘𝑈) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))) ∘ ≤ ) ∘ ((ℤRHom‘𝑈) ↾ if(𝑁 = 0, ℤ, (0..^𝑁)))) = ((((ℤRHom‘𝑈) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))) ∘ ≤ ) ∘ ((ℤRHom‘𝑈) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))))
71, 2, 3, 4, 5, 6znval 20236 . 2 (𝑁 ∈ ℕ0𝑌 = (𝑈 sSet ⟨(le‘ndx), ((((ℤRHom‘𝑈) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))) ∘ ≤ ) ∘ ((ℤRHom‘𝑈) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))))⟩))
8 znval2.l . . . . 5 = (le‘𝑌)
91, 2, 3, 4, 5, 8znle 20237 . . . 4 (𝑁 ∈ ℕ0 = ((((ℤRHom‘𝑈) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))) ∘ ≤ ) ∘ ((ℤRHom‘𝑈) ↾ if(𝑁 = 0, ℤ, (0..^𝑁)))))
109opeq2d 4796 . . 3 (𝑁 ∈ ℕ0 → ⟨(le‘ndx), ⟩ = ⟨(le‘ndx), ((((ℤRHom‘𝑈) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))) ∘ ≤ ) ∘ ((ℤRHom‘𝑈) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))))⟩)
1110oveq2d 7167 . 2 (𝑁 ∈ ℕ0 → (𝑈 sSet ⟨(le‘ndx), ⟩) = (𝑈 sSet ⟨(le‘ndx), ((((ℤRHom‘𝑈) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))) ∘ ≤ ) ∘ ((ℤRHom‘𝑈) ↾ if(𝑁 = 0, ℤ, (0..^𝑁))))⟩))
127, 11eqtr4d 2862 1 (𝑁 ∈ ℕ0𝑌 = (𝑈 sSet ⟨(le‘ndx), ⟩))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2115  ifcif 4450  {csn 4550  ⟨cop 4556  ◡ccnv 5542   ↾ cres 5545   ∘ ccom 5547  ‘cfv 6345  (class class class)co 7151  0cc0 10537   ≤ cle 10676  ℕ0cn0 11896  ℤcz 11980  ..^cfzo 13039  ndxcnx 16482   sSet csts 16483  lecple 16574   /s cqus 16780   ~QG cqg 18277  RSpancrsp 19945  ℤringzring 20172  ℤRHomczrh 20202  ℤ/nℤczn 20205 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614  ax-addf 10616  ax-mulf 10617 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6137  df-ord 6183  df-on 6184  df-lim 6185  df-suc 6186  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7577  df-1st 7686  df-2nd 7687  df-wrecs 7945  df-recs 8006  df-rdg 8044  df-1o 8100  df-oadd 8104  df-er 8287  df-en 8508  df-dom 8509  df-sdom 8510  df-fin 8511  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11637  df-2 11699  df-3 11700  df-4 11701  df-5 11702  df-6 11703  df-7 11704  df-8 11705  df-9 11706  df-n0 11897  df-z 11981  df-dec 12098  df-uz 12243  df-fz 12897  df-struct 16487  df-ndx 16488  df-slot 16489  df-base 16491  df-sets 16492  df-ress 16493  df-plusg 16580  df-mulr 16581  df-starv 16582  df-tset 16586  df-ple 16587  df-ds 16589  df-unif 16590  df-0g 16717  df-mgm 17854  df-sgrp 17903  df-mnd 17914  df-grp 18108  df-minusg 18109  df-subg 18278  df-cmn 18910  df-mgp 19242  df-ur 19254  df-ring 19301  df-cring 19302  df-subrg 19535  df-cnfld 20101  df-zring 20173  df-zn 20209 This theorem is referenced by:  znbaslem  20239
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The provided text is a formal proof in a mathematical logic system, specifically the Metamath Proof Explorer. It defines and proves a theorem about the structure of the ring of integers modulo n, denoted as ℤ/nℤ.

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The text does not contain any mathematical question or word problem. It is a formal proof of a theorem, which is a statement of mathematical fact, not a question. No answer to a question is provided because no question is asked.