Theorem znval 21421

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 ∈ ℕ0 → 𝑌 = (𝑈 sSet ⟨(le‘ndx), ≤ ⟩))

Proof of Theorem znval

Dummy variables 𝑓 𝑛 𝑠 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		znval.y	. 2 ⊢ 𝑌 = (ℤ/nℤ‘𝑁)
2		zringring 21335	. . . . 5 ⊢ ℤring ∈ Ring
3	2	a1i 11	. . . 4 ⊢ (𝑛 = 𝑁 → ℤring ∈ Ring)
4		ovexd 7404	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = 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 484	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → 𝑧 = ℤring)
7	6	fveq2d 6844	. . . . . . . . . . . 12 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → (RSpan‘𝑧) = (RSpan‘ℤring))
8		znval.s	. . . . . . . . . . . 12 ⊢ 𝑆 = (RSpan‘ℤring)
9	7, 8	eqtr4di 2782	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → (RSpan‘𝑧) = 𝑆)
10		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → 𝑛 = 𝑁)
11	10	sneqd 4597	. . . . . . . . . . 11 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → {𝑛} = {𝑁})
12	9, 11	fveq12d 6847	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → ((RSpan‘𝑧)‘{𝑛}) = (𝑆‘{𝑁}))
13	6, 12	oveq12d 7387	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})) = (ℤring ~QG (𝑆‘{𝑁})))
14	6, 13	oveq12d 7387	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = 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})))
16	14, 15	eqtr4di 2782	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = ℤring) → (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛}))) = 𝑈)
17	5, 16	sylan9eqr 2786	. . . . . 6 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → 𝑠 = 𝑈)
18		fvex 6853	. . . . . . . . . 10 ⊢ (ℤRHom‘𝑠) ∈ V
19	18	resex 5989	. . . . . . . . 9 ⊢ ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) ∈ V
20	19	a1i 11	. . . . . . . 8 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = 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))))
22	17	fveq2d 6844	. . . . . . . . . . . . . 14 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → (ℤRHom‘𝑠) = (ℤRHom‘𝑈))
23		simpll 766	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → 𝑛 = 𝑁)
24	23	eqeq1d 2731	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → (𝑛 = 0 ↔ 𝑁 = 0))
25	23	oveq2d 7385	. . . . . . . . . . . . . . . 16 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → (0..^𝑛) = (0..^𝑁))
26	24, 25	ifbieq2d 4511	. . . . . . . . . . . . . . 15 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = 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))
28	26, 27	eqtr4di 2782	. . . . . . . . . . . . . 14 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → if(𝑛 = 0, ℤ, (0..^𝑛)) = 𝑊)
29	22, 28	reseq12d 5940	. . . . . . . . . . . . 13 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = 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)
31	29, 30	eqtr4di 2782	. . . . . . . . . . . 12 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) = 𝐹)
32	21, 31	sylan9eqr 2786	. . . . . . . . . . 11 ⊢ ((((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) ∧ 𝑓 = ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛)))) → 𝑓 = 𝐹)
33	32	coeq1d 5815	. . . . . . . . . 10 ⊢ ((((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) ∧ 𝑓 = ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛)))) → (𝑓 ∘ ≤ ) = (𝐹 ∘ ≤ ))
34	32	cnveqd 5829	. . . . . . . . . 10 ⊢ ((((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) ∧ 𝑓 = ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛)))) → ◡𝑓 = ◡𝐹)
35	33, 34	coeq12d 5818	. . . . . . . . 9 ⊢ ((((𝑛 = 𝑁 ∧ 𝑧 = 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)
37	35, 36	eqtr4di 2782	. . . . . . . 8 ⊢ ((((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) ∧ 𝑓 = ((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛)))) → ((𝑓 ∘ ≤ ) ∘ ◡𝑓) = ≤ )
38	20, 37	csbied 3895	. . . . . . 7 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → ⦋((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) / 𝑓⦌((𝑓 ∘ ≤ ) ∘ ◡𝑓) = ≤ )
39	38	opeq2d 4840	. . . . . 6 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = ℤring) ∧ 𝑠 = (𝑧 /s (𝑧 ~QG ((RSpan‘𝑧)‘{𝑛})))) → ⟨(le‘ndx), ⦋((ℤRHom‘𝑠) ↾ if(𝑛 = 0, ℤ, (0..^𝑛))) / 𝑓⦌((𝑓 ∘ ≤ ) ∘ ◡𝑓)⟩ = ⟨(le‘ndx), ≤ ⟩)
40	17, 39	oveq12d 7387	. . . . 5 ⊢ (((𝑛 = 𝑁 ∧ 𝑧 = 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), ≤ ⟩))
41	4, 40	csbied 3895	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑧 = 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), ≤ ⟩))
42	3, 41	csbied 3895	. . 3 ⊢ (𝑛 = 𝑁 → ⦋ℤring / 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 21392	. . 3 ⊢ ℤ/nℤ = (𝑛 ∈ ℕ0 ↦ ⦋ℤring / 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 7402	. . 3 ⊢ (𝑈 sSet ⟨(le‘ndx), ≤ ⟩) ∈ V
45	42, 43, 44	fvmpt 6950	. 2 ⊢ (𝑁 ∈ ℕ0 → (ℤ/nℤ‘𝑁) = (𝑈 sSet ⟨(le‘ndx), ≤ ⟩))
46	1, 45	eqtrid 2776	1 ⊢ (𝑁 ∈ ℕ0 → 𝑌 = (𝑈 sSet ⟨(le‘ndx), ≤ ⟩))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3444  ⦋csb 3859  ifcif 4484  {csn 4585  ⟨cop 4591  ^◡ccnv 5630   ↾ cres 5633   ∘ ccom 5635  ‘cfv 6499  (class class class)co 7369  0cc0 11044   ≤ cle 11185  ℕ₀cn0 12418  ℤcz 12505  ..^cfzo 13591   sSet csts 17109  ndxcnx 17139  lecple 17203   /_s cqus 17444   ~_QG cqg 19030  Ringcrg 20118  RSpancrsp 21093  ℤ_ringczring 21332  ℤRHomczrh 21385  ℤ/nℤczn 21388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121  ax-addf 11123

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-z 12506  df-dec 12626  df-uz 12770  df-fz 13445  df-struct 17093  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-ress 17177  df-plusg 17209  df-mulr 17210  df-starv 17211  df-tset 17215  df-ple 17216  df-ds 17218  df-unif 17219  df-0g 17380  df-mgm 18543  df-sgrp 18622  df-mnd 18638  df-grp 18844  df-minusg 18845  df-subg 19031  df-cmn 19688  df-abl 19689  df-mgp 20026  df-rng 20038  df-ur 20067  df-ring 20120  df-cring 20121  df-subrng 20431  df-subrg 20455  df-cnfld 21241  df-zring 21333  df-zn 21392

This theorem is referenced by:  znle  21422  znval2  21423