znle - Intuitionistic Logic Explorer

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Theorem znle 14650

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 2231	. . . 4 ⊢ ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) = ((𝐹 ∘ ≤ ) ∘ ^◡𝐹)
7	1, 2, 3, 4, 5, 6	znval 14649	. . 3 ⊢ (𝑁 ∈ ℕ₀ → 𝑌 = (𝑈 sSet ⟨(le‘ndx), ((𝐹 ∘ ≤ ) ∘ ^◡𝐹)⟩))
8	7	fveq2d 5643	. 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 14606	. . . . 5 ⊢ ℤ_ring ∈ Ring
12		rspex 14487	. . . . . . . . 9 ⊢ (ℤ_ring ∈ Ring → (RSpan‘ℤ_ring) ∈ V)
13	11, 12	ax-mp 5	. . . . . . . 8 ⊢ (RSpan‘ℤ_ring) ∈ V
14	1, 13	eqeltri 2304	. . . . . . 7 ⊢ 𝑆 ∈ V
15		snexg 4274	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → {𝑁} ∈ V)
16		fvexg 5658	. . . . . . 7 ⊢ ((𝑆 ∈ V ∧ {𝑁} ∈ V) → (𝑆‘{𝑁}) ∈ V)
17	14, 15, 16	sylancr 414	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → (𝑆‘{𝑁}) ∈ V)
18		eqgex 13807	. . . . . 6 ⊢ ((ℤ_ring ∈ Ring ∧ (𝑆‘{𝑁}) ∈ V) → (ℤ_ring ~_QG (𝑆‘{𝑁})) ∈ V)
19	11, 17, 18	sylancr 414	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → (ℤ_ring ~_QG (𝑆‘{𝑁})) ∈ V)
20		qusex 13407	. . . . 5 ⊢ ((ℤ_ring ∈ Ring ∧ (ℤ_ring ~_QG (𝑆‘{𝑁})) ∈ V) → (ℤ_ring /_s (ℤ_ring ~_QG (𝑆‘{𝑁}))) ∈ V)
21	11, 19, 20	sylancr 414	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → (ℤ_ring /_s (ℤ_ring ~_QG (𝑆‘{𝑁}))) ∈ V)
22	2, 21	eqeltrid 2318	. . 3 ⊢ (𝑁 ∈ ℕ₀ → 𝑈 ∈ V)
23		eqid 2231	. . . . . . . 8 ⊢ (ℤRHom‘𝑈) = (ℤRHom‘𝑈)
24	23	zrhex 14634	. . . . . . 7 ⊢ (𝑈 ∈ V → (ℤRHom‘𝑈) ∈ V)
25		resexg 5053	. . . . . . 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 2318	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → 𝐹 ∈ V)
28		xrex 10090	. . . . . . 7 ⊢ ℝ^* ∈ V
29	28, 28	xpex 4842	. . . . . 6 ⊢ (ℝ^* × ℝ^*) ∈ V
30		lerelxr 8241	. . . . . 6 ⊢ ≤ ⊆ (ℝ^* × ℝ^*)
31	29, 30	ssexi 4227	. . . . 5 ⊢ ≤ ∈ V
32		coexg 5281	. . . . 5 ⊢ ((𝐹 ∈ V ∧ ≤ ∈ V) → (𝐹 ∘ ≤ ) ∈ V)
33	27, 31, 32	sylancl 413	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → (𝐹 ∘ ≤ ) ∈ V)
34		cnvexg 5274	. . . . 5 ⊢ (𝐹 ∈ V → ^◡𝐹 ∈ V)
35	27, 34	syl 14	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → ^◡𝐹 ∈ V)
36		coexg 5281	. . . 4 ⊢ (((𝐹 ∘ ≤ ) ∈ V ∧ ^◡𝐹 ∈ V) → ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) ∈ V)
37	33, 35, 36	syl2anc 411	. . 3 ⊢ (𝑁 ∈ ℕ₀ → ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) ∈ V)
38		pleslid 13284	. . . 4 ⊢ (le = Slot (le‘ndx) ∧ (le‘ndx) ∈ ℕ)
39	38	setsslid 13132	. . 3 ⊢ ((𝑈 ∈ V ∧ ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) ∈ V) → ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) = (le‘(𝑈 sSet ⟨(le‘ndx), ((𝐹 ∘ ≤ ) ∘ ^◡𝐹)⟩)))
40	22, 37, 39	syl2anc 411	. 2 ⊢ (𝑁 ∈ ℕ₀ → ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) = (le‘(𝑈 sSet ⟨(le‘ndx), ((𝐹 ∘ ≤ ) ∘ ^◡𝐹)⟩)))
41	8, 10, 40	3eqtr4d 2274	1 ⊢ (𝑁 ∈ ℕ₀ → ≤ = ((𝐹 ∘ ≤ ) ∘ ^◡𝐹))

Colors of variables: wff set class

Syntax hints: → wi 4 = wceq 1397 ∈ wcel 2202 Vcvv 2802 ifcif 3605 {csn 3669 ⟨cop 3672 × cxp 4723 ^◡ccnv 4724 ↾ cres 4727 ∘ ccom 4729 ‘cfv 5326 (class class class)co 6017 0cc0 8031 ℝ^*cxr 8212 ≤ cle 8214 ℕ₀cn0 9401 ℤcz 9478 ..^cfzo 10376 ndxcnx 13078 sSet csts 13079 lecple 13166 /_s cqus 13382 ~_QG cqg 13755 Ringcrg 14008 RSpancrsp 14481 ℤ_ringczring 14603 ℤRHomczrh 14624 ℤ/nℤczn 14626

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-cnex 8122 ax-resscn 8123 ax-1cn 8124 ax-1re 8125 ax-icn 8126 ax-addcl 8127 ax-addrcl 8128 ax-mulcl 8129 ax-mulrcl 8130 ax-addcom 8131 ax-mulcom 8132 ax-addass 8133 ax-mulass 8134 ax-distr 8135 ax-i2m1 8136 ax-0lt1 8137 ax-1rid 8138 ax-0id 8139 ax-rnegex 8140 ax-precex 8141 ax-cnre 8142 ax-pre-ltirr 8143 ax-pre-ltwlin 8144 ax-pre-lttrn 8145 ax-pre-apti 8146 ax-pre-ltadd 8147 ax-pre-mulgt0 8148 ax-addf 8153 ax-mulf 8154

This theorem depends on definitions: df-bi 117 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-tp 3677 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-id 4390 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-riota 5970 df-ov 6020 df-oprab 6021 df-mpo 6022 df-1st 6302 df-2nd 6303 df-ec 6703 df-map 6818 df-pnf 8215 df-mnf 8216 df-xr 8217 df-ltxr 8218 df-le 8219 df-sub 8351 df-neg 8352 df-reap 8754 df-inn 9143 df-2 9201 df-3 9202 df-4 9203 df-5 9204 df-6 9205 df-7 9206 df-8 9207 df-9 9208 df-n0 9402 df-z 9479 df-dec 9611 df-uz 9755 df-rp 9888 df-fz 10243 df-cj 11402 df-abs 11559 df-struct 13083 df-ndx 13084 df-slot 13085 df-base 13087 df-sets 13088 df-iress 13089 df-plusg 13172 df-mulr 13173 df-starv 13174 df-sca 13175 df-vsca 13176 df-ip 13177 df-tset 13178 df-ple 13179 df-ds 13181 df-unif 13182 df-0g 13340 df-topgen 13342 df-iimas 13384 df-qus 13385 df-mgm 13438 df-sgrp 13484 df-mnd 13499 df-grp 13585 df-minusg 13586 df-subg 13756 df-eqg 13758 df-cmn 13872 df-mgp 13933 df-ur 13972 df-ring 14010 df-cring 14011 df-rhm 14165 df-subrg 14232 df-lsp 14400 df-sra 14448 df-rgmod 14449 df-rsp 14483 df-bl 14559 df-mopn 14560 df-fg 14562 df-metu 14563 df-cnfld 14570 df-zring 14604 df-zrh 14627 df-zn 14629

This theorem is referenced by: znval2 14651 znle2 14665

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