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Theorem znval 19931
 Description: The value of the ℤ/nℤ structure. It is defined as the quotient ring ℤ / 𝑛ℤ, with an "artificial" ordering added to make it a Toset. (In other words, ℤ/nℤ is a ring with an order , but it is not an ordered ring , which as a term implies that the order is compatible with the ring operations in some way.) (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 2-May-2016.) (Revised by AV, 13-Jun-2019.)
Hypotheses
Ref Expression
znval.s 𝑆 = (RSpan‘ℤring)
znval.u 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁})))
znval.y 𝑌 = (ℤ/nℤ‘𝑁)
znval.f 𝐹 = ((ℤRHom‘𝑈) ↾ 𝑊)
znval.w 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁))
znval.l = ((𝐹 ∘ ≤ ) ∘ 𝐹)
Assertion
Ref Expression
znval (𝑁 ∈ ℕ0𝑌 = (𝑈 sSet ⟨(le‘ndx), ⟩))

Proof of Theorem znval
Dummy variables 𝑓 𝑛 𝑠 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 znval.y . 2 𝑌 = (ℤ/nℤ‘𝑁)
2 zringring 19869 . . . . 5 ring ∈ Ring
32a1i 11 . . . 4 (𝑛 = 𝑁 → ℤring ∈ Ring)
4 ovexd 6720 . . . . 5 ((𝑛 = 𝑁𝑧 = ℤring) → (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) ∈ V)
5 id 22 . . . . . . 7 (𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) → 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))))
6 simpr 476 . . . . . . . . 9 ((𝑛 = 𝑁𝑧 = ℤring) → 𝑧 = ℤring)
76fveq2d 6233 . . . . . . . . . . . 12 ((𝑛 = 𝑁𝑧 = ℤring) → (RSpan‘𝑧) = (RSpan‘ℤring))
8 znval.s . . . . . . . . . . . 12 𝑆 = (RSpan‘ℤring)
97, 8syl6eqr 2703 . . . . . . . . . . 11 ((𝑛 = 𝑁𝑧 = ℤring) → (RSpan‘𝑧) = 𝑆)
10 simpl 472 . . . . . . . . . . . 12 ((𝑛 = 𝑁𝑧 = ℤring) → 𝑛 = 𝑁)
1110sneqd 4222 . . . . . . . . . . 11 ((𝑛 = 𝑁𝑧 = ℤring) → {𝑛} = {𝑁})
129, 11fveq12d 6235 . . . . . . . . . 10 ((𝑛 = 𝑁𝑧 = ℤring) → ((RSpan‘𝑧)‘{𝑛}) = (𝑆‘{𝑁}))
136, 12oveq12d 6708 . . . . . . . . 9 ((𝑛 = 𝑁𝑧 = ℤring) → (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})) = (ℤring ~QG (𝑆‘{𝑁})))
146, 13oveq12d 6708 . . . . . . . 8 ((𝑛 = 𝑁𝑧 = ℤring) → (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) = (ℤring /s (ℤring ~QG (𝑆‘{𝑁}))))
15 znval.u . . . . . . . 8 𝑈 = (ℤring /s (ℤring ~QG (𝑆‘{𝑁})))
1614, 15syl6eqr 2703 . . . . . . 7 ((𝑛 = 𝑁𝑧 = ℤring) → (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) = 𝑈)
175, 16sylan9eqr 2707 . . . . . 6 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → 𝑠 = 𝑈)
18 fvex 6239 . . . . . . . . . 10 (ℤRHom‘𝑠) ∈ V
1918resex 5478 . . . . . . . . 9 ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) ∈ V
2019a1i 11 . . . . . . . 8 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) ∈ V)
21 id 22 . . . . . . . . . . . 12 (𝑓 = ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) → 𝑓 = ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))))
2217fveq2d 6233 . . . . . . . . . . . . . 14 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → (ℤRHom‘𝑠) = (ℤRHom‘𝑈))
23 simpll 805 . . . . . . . . . . . . . . . . 17 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → 𝑛 = 𝑁)
2423eqeq1d 2653 . . . . . . . . . . . . . . . 16 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → (𝑛 = 0 ↔ 𝑁 = 0))
2523oveq2d 6706 . . . . . . . . . . . . . . . 16 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → (0..^𝑛) = (0..^𝑁))
2624, 25ifbieq2d 4144 . . . . . . . . . . . . . . 15 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → if(𝑛 = 0, ℤ, (0..^𝑛)) = if(𝑁 = 0, ℤ, (0..^𝑁)))
27 znval.w . . . . . . . . . . . . . . 15 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁))
2826, 27syl6eqr 2703 . . . . . . . . . . . . . 14 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → if(𝑛 = 0, ℤ, (0..^𝑛)) = 𝑊)
2922, 28reseq12d 5429 . . . . . . . . . . . . 13 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) = ((ℤRHom‘𝑈) ↾ 𝑊))
30 znval.f . . . . . . . . . . . . 13 𝐹 = ((ℤRHom‘𝑈) ↾ 𝑊)
3129, 30syl6eqr 2703 . . . . . . . . . . . 12 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) = 𝐹)
3221, 31sylan9eqr 2707 . . . . . . . . . . 11 ((((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) ∧ 𝑓 = ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛)))) → 𝑓 = 𝐹)
3332coeq1d 5316 . . . . . . . . . 10 ((((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) ∧ 𝑓 = ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛)))) → (𝑓 ∘ ≤ ) = (𝐹 ∘ ≤ ))
3432cnveqd 5330 . . . . . . . . . 10 ((((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) ∧ 𝑓 = ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛)))) → 𝑓 = 𝐹)
3533, 34coeq12d 5319 . . . . . . . . 9 ((((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) ∧ 𝑓 = ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛)))) → ((𝑓 ∘ ≤ ) ∘ 𝑓) = ((𝐹 ∘ ≤ ) ∘ 𝐹))
36 znval.l . . . . . . . . 9 = ((𝐹 ∘ ≤ ) ∘ 𝐹)
3735, 36syl6eqr 2703 . . . . . . . 8 ((((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) ∧ 𝑓 = ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛)))) → ((𝑓 ∘ ≤ ) ∘ 𝑓) = )
3820, 37csbied 3593 . . . . . . 7 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) / 𝑓((𝑓 ∘ ≤ ) ∘ 𝑓) = )
3938opeq2d 4440 . . . . . 6 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → ⟨(le‘ndx), ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) / 𝑓((𝑓 ∘ ≤ ) ∘ 𝑓)⟩ = ⟨(le‘ndx), ⟩)
4017, 39oveq12d 6708 . . . . 5 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → (𝑠 sSet ⟨(le‘ndx), ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) / 𝑓((𝑓 ∘ ≤ ) ∘ 𝑓)⟩) = (𝑈 sSet ⟨(le‘ndx), ⟩))
414, 40csbied 3593 . . . 4 ((𝑛 = 𝑁𝑧 = ℤring) → (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) / 𝑠(𝑠 sSet ⟨(le‘ndx), ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) / 𝑓((𝑓 ∘ ≤ ) ∘ 𝑓)⟩) = (𝑈 sSet ⟨(le‘ndx), ⟩))
423, 41csbied 3593 . . 3 (𝑛 = 𝑁ring / 𝑧(𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) / 𝑠(𝑠 sSet ⟨(le‘ndx), ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) / 𝑓((𝑓 ∘ ≤ ) ∘ 𝑓)⟩) = (𝑈 sSet ⟨(le‘ndx), ⟩))
43 df-zn 19903 . . 3 ℤ/nℤ = (𝑛 ∈ ℕ0ring / 𝑧(𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) / 𝑠(𝑠 sSet ⟨(le‘ndx), ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) / 𝑓((𝑓 ∘ ≤ ) ∘ 𝑓)⟩))
44 ovex 6718 . . 3 (𝑈 sSet ⟨(le‘ndx), ⟩) ∈ V
4542, 43, 44fvmpt 6321 . 2 (𝑁 ∈ ℕ0 → (ℤ/nℤ‘𝑁) = (𝑈 sSet ⟨(le‘ndx), ⟩))
461, 45syl5eq 2697 1 (𝑁 ∈ ℕ0𝑌 = (𝑈 sSet ⟨(le‘ndx), ⟩))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  Vcvv 3231  ⦋csb 3566  ifcif 4119  {csn 4210  ⟨cop 4216  ◡ccnv 5142   ↾ cres 5145   ∘ ccom 5147  ‘cfv 5926  (class class class)co 6690  0cc0 9974   ≤ cle 10113  ℕ0cn0 11330  ℤcz 11415  ..^cfzo 12504  ndxcnx 15901   sSet csts 15902  lecple 15995   /s cqus 16212   ~QG cqg 17637  Ringcrg 18593  RSpancrsp 19219  ℤringzring 19866  ℤRHomczrh 19896  ℤ/nℤczn 19899 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-addf 10053  ax-mulf 10054 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473  df-subg 17638  df-cmn 18241  df-mgp 18536  df-ur 18548  df-ring 18595  df-cring 18596  df-subrg 18826  df-cnfld 19795  df-zring 19867  df-zn 19903 This theorem is referenced by:  znle  19932  znval2  19933
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