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Theorem znval 21482
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 21392 . . . . 5 ring ∈ Ring
32a1i 11 . . . 4 (𝑛 = 𝑁 → ℤring ∈ Ring)
4 ovexd 7454 . . . . 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 483 . . . . . . . . 9 ((𝑛 = 𝑁𝑧 = ℤring) → 𝑧 = ℤring)
76fveq2d 6900 . . . . . . . . . . . 12 ((𝑛 = 𝑁𝑧 = ℤring) → (RSpan‘𝑧) = (RSpan‘ℤring))
8 znval.s . . . . . . . . . . . 12 𝑆 = (RSpan‘ℤring)
97, 8eqtr4di 2783 . . . . . . . . . . 11 ((𝑛 = 𝑁𝑧 = ℤring) → (RSpan‘𝑧) = 𝑆)
10 simpl 481 . . . . . . . . . . . 12 ((𝑛 = 𝑁𝑧 = ℤring) → 𝑛 = 𝑁)
1110sneqd 4642 . . . . . . . . . . 11 ((𝑛 = 𝑁𝑧 = ℤring) → {𝑛} = {𝑁})
129, 11fveq12d 6903 . . . . . . . . . 10 ((𝑛 = 𝑁𝑧 = ℤring) → ((RSpan‘𝑧)‘{𝑛}) = (𝑆‘{𝑁}))
136, 12oveq12d 7437 . . . . . . . . 9 ((𝑛 = 𝑁𝑧 = ℤring) → (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})) = (ℤring ~QG (𝑆‘{𝑁})))
146, 13oveq12d 7437 . . . . . . . 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 2783 . . . . . . 7 ((𝑛 = 𝑁𝑧 = ℤring) → (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) = 𝑈)
175, 16sylan9eqr 2787 . . . . . 6 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → 𝑠 = 𝑈)
18 fvex 6909 . . . . . . . . . 10 (ℤRHom‘𝑠) ∈ V
1918resex 6034 . . . . . . . . 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 6900 . . . . . . . . . . . . . 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 2727 . . . . . . . . . . . . . . . 16 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → (𝑛 = 0 ↔ 𝑁 = 0))
2523oveq2d 7435 . . . . . . . . . . . . . . . 16 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → (0..^𝑛) = (0..^𝑁))
2624, 25ifbieq2d 4556 . . . . . . . . . . . . . . 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 2783 . . . . . . . . . . . . . 14 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → if(𝑛 = 0, ℤ, (0..^𝑛)) = 𝑊)
2922, 28reseq12d 5986 . . . . . . . . . . . . 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 2783 . . . . . . . . . . . 12 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) = 𝐹)
3221, 31sylan9eqr 2787 . . . . . . . . . . 11 ((((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) ∧ 𝑓 = ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛)))) → 𝑓 = 𝐹)
3332coeq1d 5864 . . . . . . . . . 10 ((((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) ∧ 𝑓 = ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛)))) → (𝑓 ∘ ≤ ) = (𝐹 ∘ ≤ ))
3432cnveqd 5878 . . . . . . . . . 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 5867 . . . . . . . . 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 2783 . . . . . . . 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 3927 . . . . . . 7 (((𝑛 = 𝑁𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) / 𝑓((𝑓 ∘ ≤ ) ∘ 𝑓) = )
3938opeq2d 4882 . . . . . 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 7437 . . . . 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 3927 . . . 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 3927 . . 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 21449 . . 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 7452 . . 3 (𝑈 sSet ⟨(le‘ndx), ⟩) ∈ V
4542, 43, 44fvmpt 7004 . 2 (𝑁 ∈ ℕ0 → (ℤ/nℤ‘𝑁) = (𝑈 sSet ⟨(le‘ndx), ⟩))
461, 45eqtrid 2777 1 (𝑁 ∈ ℕ0𝑌 = (𝑈 sSet ⟨(le‘ndx), ⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  Vcvv 3461  csb 3889  ifcif 4530  {csn 4630  cop 4636  ccnv 5677  cres 5680  ccom 5682  cfv 6549  (class class class)co 7419  0cc0 11140  cle 11281  0cn0 12505  cz 12591  ..^cfzo 13662   sSet csts 17135  ndxcnx 17165  lecple 17243   /s cqus 17490   ~QG cqg 19085  Ringcrg 20185  RSpancrsp 21115  ringczring 21389  ℤRHomczrh 21442  ℤ/nczn 21445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217  ax-addf 11219
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12506  df-z 12592  df-dec 12711  df-uz 12856  df-fz 13520  df-struct 17119  df-sets 17136  df-slot 17154  df-ndx 17166  df-base 17184  df-ress 17213  df-plusg 17249  df-mulr 17250  df-starv 17251  df-tset 17255  df-ple 17256  df-ds 17258  df-unif 17259  df-0g 17426  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-grp 18901  df-minusg 18902  df-subg 19086  df-cmn 19749  df-abl 19750  df-mgp 20087  df-rng 20105  df-ur 20134  df-ring 20187  df-cring 20188  df-subrng 20495  df-subrg 20520  df-cnfld 21297  df-zring 21390  df-zn 21449
This theorem is referenced by:  znle  21483  znval2  21484
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