znle - Intuitionistic Logic Explorer

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Theorem znle 14944

Description: The value of the ℤ/nℤ structure. It is defined as the quotient ring ℤ / 𝑛ℤ, with an "artificial" ordering added. (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 AV, 13-Jun-2019.)

**Hypotheses**
Ref	Expression
znval.s	⊢ 𝑆 = (RSpan‘ℤ_ring)
znval.u	⊢ 𝑈 = (ℤ_ring /_s (ℤ_ring ~_QG (𝑆‘{𝑁})))
znval.y	⊢ 𝑌 = (ℤ/nℤ‘𝑁)
znval.f	⊢ 𝐹 = ((ℤRHom‘𝑈) ↾ 𝑊)
znval.w	⊢ 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁))
znle.l	⊢ ≤ = (le‘𝑌)

**Assertion**
Ref	Expression
znle	⊢ (𝑁 ∈ ℕ₀ → ≤ = ((𝐹 ∘ ≤ ) ∘ ^◡𝐹))

**Proof of Theorem znle**
Step	Hyp	Ref	Expression
1		znval.s	. . . 4 ⊢ 𝑆 = (RSpan‘ℤ_ring)
2		znval.u	. . . 4 ⊢ 𝑈 = (ℤ_ring /_s (ℤ_ring ~_QG (𝑆‘{𝑁})))
3		znval.y	. . . 4 ⊢ 𝑌 = (ℤ/nℤ‘𝑁)
4		znval.f	. . . 4 ⊢ 𝐹 = ((ℤRHom‘𝑈) ↾ 𝑊)
5		znval.w	. . . 4 ⊢ 𝑊 = if(𝑁 = 0, ℤ, (0..^𝑁))
6		eqid 2238	. . . 4 ⊢ ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) = ((𝐹 ∘ ≤ ) ∘ ^◡𝐹)
7	1, 2, 3, 4, 5, 6	znval 14943	. . 3 ⊢ (𝑁 ∈ ℕ₀ → 𝑌 = (𝑈 sSet ⟨(le‘ndx), ((𝐹 ∘ ≤ ) ∘ ^◡𝐹)⟩))
8	7	fveq2d 5694	. 2 ⊢ (𝑁 ∈ ℕ₀ → (le‘𝑌) = (le‘(𝑈 sSet ⟨(le‘ndx), ((𝐹 ∘ ≤ ) ∘ ^◡𝐹)⟩)))
9		znle.l	. . 3 ⊢ ≤ = (le‘𝑌)
10	9	a1i 9	. 2 ⊢ (𝑁 ∈ ℕ₀ → ≤ = (le‘𝑌))
11		zringring 14900	. . . . 5 ⊢ ℤ_ring ∈ Ring
12		rspex 14783	. . . . . . . . 9 ⊢ (ℤ_ring ∈ Ring → (RSpan‘ℤ_ring) ∈ V)
13	11, 12	ax-mp 5	. . . . . . . 8 ⊢ (RSpan‘ℤ_ring) ∈ V
14	1, 13	eqeltri 2311	. . . . . . 7 ⊢ 𝑆 ∈ V
15		snexg 4316	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → {𝑁} ∈ V)
16		fvexg 5709	. . . . . . 7 ⊢ ((𝑆 ∈ V ∧ {𝑁} ∈ V) → (𝑆‘{𝑁}) ∈ V)
17	14, 15, 16	sylancr 418	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → (𝑆‘{𝑁}) ∈ V)
18		eqgex 14001	. . . . . 6 ⊢ ((ℤ_ring ∈ Ring ∧ (𝑆‘{𝑁}) ∈ V) → (ℤ_ring ~_QG (𝑆‘{𝑁})) ∈ V)
19	11, 17, 18	sylancr 418	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (ℤ_ring ~_QG (𝑆‘{𝑁})) ∈ V)
20		qusex 13623	. . . . 5 ⊢ ((ℤ_ring ∈ Ring ∧ (ℤ_ring ~_QG (𝑆‘{𝑁})) ∈ V) → (ℤ_ring /_s (ℤ_ring ~_QG (𝑆‘{𝑁}))) ∈ V)
21	11, 19, 20	sylancr 418	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → (ℤ_ring /_s (ℤ_ring ~_QG (𝑆‘{𝑁}))) ∈ V)
22	2, 21	eqeltrid 2325	. . 3 ⊢ (𝑁 ∈ ℕ₀ → 𝑈 ∈ V)
23		eqid 2238	. . . . . . . 8 ⊢ (ℤRHom‘𝑈) = (ℤRHom‘𝑈)
24	23	zrhex 14928	. . . . . . 7 ⊢ (𝑈 ∈ V → (ℤRHom‘𝑈) ∈ V)
25		resexg 5098	. . . . . . 7 ⊢ ((ℤRHom‘𝑈) ∈ V → ((ℤRHom‘𝑈) ↾ 𝑊) ∈ V)
26	22, 24, 25	3syl 17	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → ((ℤRHom‘𝑈) ↾ 𝑊) ∈ V)
27	4, 26	eqeltrid 2325	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → 𝐹 ∈ V)
28		xrex 10237	. . . . . . 7 ⊢ ℝ^* ∈ V
29	28, 28	xpex 4886	. . . . . 6 ⊢ (ℝ^* × ℝ^*) ∈ V
30		lerelxr 8378	. . . . . 6 ⊢ ≤ ⊆ (ℝ^* × ℝ^*)
31	29, 30	ssexi 4266	. . . . 5 ⊢ ≤ ∈ V
32		coexg 5327	. . . . 5 ⊢ ((𝐹 ∈ V ∧ ≤ ∈ V) → (𝐹 ∘ ≤ ) ∈ V)
33	27, 31, 32	sylancl 417	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → (𝐹 ∘ ≤ ) ∈ V)
34		cnvexg 5320	. . . . 5 ⊢ (𝐹 ∈ V → ^◡𝐹 ∈ V)
35	27, 34	syl 14	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → ^◡𝐹 ∈ V)
36		coexg 5327	. . . 4 ⊢ (((𝐹 ∘ ≤ ) ∈ V ∧ ^◡𝐹 ∈ V) → ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) ∈ V)
37	33, 35, 36	syl2anc 415	. . 3 ⊢ (𝑁 ∈ ℕ₀ → ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) ∈ V)
38		pleslid 13533	. . . 4 ⊢ (le = Slot (le‘ndx) ∧ (le‘ndx) ∈ ℕ)
39	38	setsslid 13381	. . 3 ⊢ ((𝑈 ∈ V ∧ ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) ∈ V) → ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) = (le‘(𝑈 sSet ⟨(le‘ndx), ((𝐹 ∘ ≤ ) ∘ ^◡𝐹)⟩)))
40	22, 37, 39	syl2anc 415	. 2 ⊢ (𝑁 ∈ ℕ₀ → ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) = (le‘(𝑈 sSet ⟨(le‘ndx), ((𝐹 ∘ ≤ ) ∘ ^◡𝐹)⟩)))
41	8, 10, 40	3eqtr4d 2281	1 ⊢ (𝑁 ∈ ℕ₀ → ≤ = ((𝐹 ∘ ≤ ) ∘ ^◡𝐹))

Colors of variables: wff set class

Syntax hints: → wi 4 = wceq 1402 ∈ wcel 2209 Vcvv 2821 ifcif 3635 {csn 3705 ⟨cop 3708 × cxp 4767 ^◡ccnv 4768 ↾ cres 4771 ∘ ccom 4773 ‘cfv 5372 (class class class)co 6075 0cc0 8169 ℝ^*cxr 8349 ≤ cle 8351 ℕ₀cn0 9542 ℤcz 9623 ..^cfzo 10527 ndxcnx 13327 sSet csts 13328 lecple 13415 /_s cqus 13600 ~_QG cqg 13949 Ringcrg 14274 RSpancrsp 14777 ℤ_ringczring 14897 ℤRHomczrh 14918 ℤ/nℤczn 14920

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4241 ax-sep 4244 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286 ax-addf 8291 ax-mulf 8292

This theorem depends on definitions: df-bi 117 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3636 df-pw 3687 df-sn 3711 df-pr 3712 df-tp 3713 df-op 3714 df-uni 3931 df-int 3966 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-id 4433 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-1st 6364 df-2nd 6365 df-ec 6799 df-map 6914 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-sub 8489 df-neg 8490 df-reap 8893 df-inn 9284 df-2 9342 df-3 9343 df-4 9344 df-5 9345 df-6 9346 df-7 9347 df-8 9348 df-9 9349 df-n0 9543 df-z 9624 df-dec 9757 df-uz 9901 df-rp 10034 df-fz 10391 df-cj 11585 df-abs 11743 df-struct 13332 df-ndx 13333 df-slot 13334 df-base 13336 df-sets 13337 df-iress 13338 df-plusg 13421 df-mulr 13422 df-starv 13423 df-sca 13424 df-vsca 13425 df-ip 13426 df-tset 13427 df-ple 13428 df-ds 13430 df-unif 13431 df-0g 13589 df-topgen 13591 df-iimas 13601 df-qus 13602 df-mgm 13653 df-sgrp 13694 df-mnd 13707 df-grp 13785 df-minusg 13786 df-subg 13950 df-eqg 13952 df-cmn 14066 df-mgp 14195 df-ur 14238 df-ring 14276 df-cring 14277 df-rhm 14432 df-subrg 14500 df-lsp 14696 df-sra 14744 df-rgmod 14745 df-rsp 14779 df-bl 14855 df-mopn 14856 df-fg 14858 df-metu 14859 df-cnfld 14866 df-zring 14898 df-zrh 14921 df-zn 14923

This theorem is referenced by: znval2 14945 znle2 14959

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