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Theorem znval 20938
Description: The value of the ℤ/n structure. It is defined as the quotient ring ℤ / 𝑛, with an "artificial" ordering added to make it a Toset. (In other words, ℤ/n 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 20872 . . . . 5 ring ∈ Ring
32a1i 11 . . . 4 (𝑛 = 𝑁 → ℤring ∈ Ring)
4 ovexd 7392 . . . . 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 485 . . . . . . . . 9 ((𝑛 = 𝑁𝑧 = ℤring) → 𝑧 = ℤring)
76fveq2d 6846 . . . . . . . . . . . 12 ((𝑛 = 𝑁𝑧 = ℤring) → (RSpan‘𝑧) = (RSpan‘ℤring))
8 znval.s . . . . . . . . . . . 12 𝑆 = (RSpan‘ℤring)
97, 8eqtr4di 2794 . . . . . . . . . . 11 ((𝑛 = 𝑁𝑧 = ℤring) → (RSpan‘𝑧) = 𝑆)
10 simpl 483 . . . . . . . . . . . 12 ((𝑛 = 𝑁𝑧 = ℤring) → 𝑛 = 𝑁)
1110sneqd 4598 . . . . . . . . . . 11 ((𝑛 = 𝑁𝑧 = ℤring) → {𝑛} = {𝑁})
129, 11fveq12d 6849 . . . . . . . . . 10 ((𝑛 = 𝑁𝑧 = ℤring) → ((RSpan‘𝑧)‘{𝑛}) = (𝑆‘{𝑁}))
136, 12oveq12d 7375 . . . . . . . . 9 ((𝑛 = 𝑁𝑧 = ℤring) → (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})) = (ℤring ~QG (𝑆‘{𝑁})))
146, 13oveq12d 7375 . . . . . . . 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, 15eqtr4di 2794 . . . . . . 7 ((𝑛 = 𝑁𝑧 = ℤring) → (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) = 𝑈)
175, 16sylan9eqr 2798 . . . . . 6 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → 𝑠 = 𝑈)
18 fvex 6855 . . . . . . . . . 10 (ℤRHom‘𝑠) ∈ V
1918resex 5985 . . . . . . . . 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 6846 . . . . . . . . . . . . . 14 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → (ℤRHom‘𝑠) = (ℤRHom‘𝑈))
23 simpll 765 . . . . . . . . . . . . . . . . 17 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → 𝑛 = 𝑁)
2423eqeq1d 2738 . . . . . . . . . . . . . . . 16 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → (𝑛 = 0 ↔ 𝑁 = 0))
2523oveq2d 7373 . . . . . . . . . . . . . . . 16 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → (0..^𝑛) = (0..^𝑁))
2624, 25ifbieq2d 4512 . . . . . . . . . . . . . . 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, 27eqtr4di 2794 . . . . . . . . . . . . . 14 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → if(𝑛 = 0, ℤ, (0..^𝑛)) = 𝑊)
2922, 28reseq12d 5938 . . . . . . . . . . . . 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, 30eqtr4di 2794 . . . . . . . . . . . 12 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) = 𝐹)
3221, 31sylan9eqr 2798 . . . . . . . . . . 11 ((((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) ∧ 𝑓 = ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛)))) → 𝑓 = 𝐹)
3332coeq1d 5817 . . . . . . . . . 10 ((((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) ∧ 𝑓 = ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛)))) → (𝑓 ∘ ≤ ) = (𝐹 ∘ ≤ ))
3432cnveqd 5831 . . . . . . . . . 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 5820 . . . . . . . . 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, 36eqtr4di 2794 . . . . . . . 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 3893 . . . . . . 7 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) / 𝑓((𝑓 ∘ ≤ ) ∘ 𝑓) = )
3938opeq2d 4837 . . . . . 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 7375 . . . . 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 3893 . . . 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 3893 . . 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 20907 . . 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 7390 . . 3 (𝑈 sSet ⟨(le‘ndx), ⟩) ∈ V
4542, 43, 44fvmpt 6948 . 2 (𝑁 ∈ ℕ0 → (ℤ/nℤ‘𝑁) = (𝑈 sSet ⟨(le‘ndx), ⟩))
461, 45eqtrid 2788 1 (𝑁 ∈ ℕ0𝑌 = (𝑈 sSet ⟨(le‘ndx), ⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  Vcvv 3445  csb 3855  ifcif 4486  {csn 4586  cop 4592  ccnv 5632  cres 5635  ccom 5637  cfv 6496  (class class class)co 7357  0cc0 11051  cle 11190  0cn0 12413  cz 12499  ..^cfzo 13567   sSet csts 17035  ndxcnx 17065  lecple 17140   /s cqus 17387   ~QG cqg 18924  Ringcrg 19964  RSpancrsp 20632  ringczring 20869  ℤRHomczrh 20900  ℤ/nczn 20903
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-addf 11130  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-subg 18925  df-cmn 19564  df-mgp 19897  df-ur 19914  df-ring 19966  df-cring 19967  df-subrg 20220  df-cnfld 20797  df-zring 20870  df-zn 20907
This theorem is referenced by:  znle  20939  znval2  20940
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