znle - Intuitionistic Logic Explorer

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Theorem znle 14616

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 2229	. . . 4 ⊢ ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) = ((𝐹 ∘ ≤ ) ∘ ^◡𝐹)
7	1, 2, 3, 4, 5, 6	znval 14615	. . 3 ⊢ (𝑁 ∈ ℕ₀ → 𝑌 = (𝑈 sSet ⟨(le‘ndx), ((𝐹 ∘ ≤ ) ∘ ^◡𝐹)⟩))
8	7	fveq2d 5633	. 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 14572	. . . . 5 ⊢ ℤ_ring ∈ Ring
12		rspex 14453	. . . . . . . . 9 ⊢ (ℤ_ring ∈ Ring → (RSpan‘ℤ_ring) ∈ V)
13	11, 12	ax-mp 5	. . . . . . . 8 ⊢ (RSpan‘ℤ_ring) ∈ V
14	1, 13	eqeltri 2302	. . . . . . 7 ⊢ 𝑆 ∈ V
15		snexg 4268	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → {𝑁} ∈ V)
16		fvexg 5648	. . . . . . 7 ⊢ ((𝑆 ∈ V ∧ {𝑁} ∈ V) → (𝑆‘{𝑁}) ∈ V)
17	14, 15, 16	sylancr 414	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → (𝑆‘{𝑁}) ∈ V)
18		eqgex 13773	. . . . . 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 13373	. . . . 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 2316	. . 3 ⊢ (𝑁 ∈ ℕ₀ → 𝑈 ∈ V)
23		eqid 2229	. . . . . . . 8 ⊢ (ℤRHom‘𝑈) = (ℤRHom‘𝑈)
24	23	zrhex 14600	. . . . . . 7 ⊢ (𝑈 ∈ V → (ℤRHom‘𝑈) ∈ V)
25		resexg 5045	. . . . . . 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 2316	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → 𝐹 ∈ V)
28		xrex 10064	. . . . . . 7 ⊢ ℝ^* ∈ V
29	28, 28	xpex 4834	. . . . . 6 ⊢ (ℝ^* × ℝ^*) ∈ V
30		lerelxr 8220	. . . . . 6 ⊢ ≤ ⊆ (ℝ^* × ℝ^*)
31	29, 30	ssexi 4222	. . . . 5 ⊢ ≤ ∈ V
32		coexg 5273	. . . . 5 ⊢ ((𝐹 ∈ V ∧ ≤ ∈ V) → (𝐹 ∘ ≤ ) ∈ V)
33	27, 31, 32	sylancl 413	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → (𝐹 ∘ ≤ ) ∈ V)
34		cnvexg 5266	. . . . 5 ⊢ (𝐹 ∈ V → ^◡𝐹 ∈ V)
35	27, 34	syl 14	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → ^◡𝐹 ∈ V)
36		coexg 5273	. . . 4 ⊢ (((𝐹 ∘ ≤ ) ∈ V ∧ ^◡𝐹 ∈ V) → ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) ∈ V)
37	33, 35, 36	syl2anc 411	. . 3 ⊢ (𝑁 ∈ ℕ₀ → ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) ∈ V)
38		pleslid 13250	. . . 4 ⊢ (le = Slot (le‘ndx) ∧ (le‘ndx) ∈ ℕ)
39	38	setsslid 13098	. . 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 2272	1 ⊢ (𝑁 ∈ ℕ₀ → ≤ = ((𝐹 ∘ ≤ ) ∘ ^◡𝐹))

Colors of variables: wff set class

Syntax hints: → wi 4 = wceq 1395 ∈ wcel 2200 Vcvv 2799 ifcif 3602 {csn 3666 ⟨cop 3669 × cxp 4717 ^◡ccnv 4718 ↾ cres 4721 ∘ ccom 4723 ‘cfv 5318 (class class class)co 6007 0cc0 8010 ℝ^*cxr 8191 ≤ cle 8193 ℕ₀cn0 9380 ℤcz 9457 ..^cfzo 10350 ndxcnx 13044 sSet csts 13045 lecple 13132 /_s cqus 13348 ~_QG cqg 13721 Ringcrg 13974 RSpancrsp 14447 ℤ_ringczring 14569 ℤRHomczrh 14590 ℤ/nℤczn 14592

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-cnex 8101 ax-resscn 8102 ax-1cn 8103 ax-1re 8104 ax-icn 8105 ax-addcl 8106 ax-addrcl 8107 ax-mulcl 8108 ax-mulrcl 8109 ax-addcom 8110 ax-mulcom 8111 ax-addass 8112 ax-mulass 8113 ax-distr 8114 ax-i2m1 8115 ax-0lt1 8116 ax-1rid 8117 ax-0id 8118 ax-rnegex 8119 ax-precex 8120 ax-cnre 8121 ax-pre-ltirr 8122 ax-pre-ltwlin 8123 ax-pre-lttrn 8124 ax-pre-apti 8125 ax-pre-ltadd 8126 ax-pre-mulgt0 8127 ax-addf 8132 ax-mulf 8133

This theorem depends on definitions: df-bi 117 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-tp 3674 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-id 4384 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-1st 6292 df-2nd 6293 df-ec 6690 df-map 6805 df-pnf 8194 df-mnf 8195 df-xr 8196 df-ltxr 8197 df-le 8198 df-sub 8330 df-neg 8331 df-reap 8733 df-inn 9122 df-2 9180 df-3 9181 df-4 9182 df-5 9183 df-6 9184 df-7 9185 df-8 9186 df-9 9187 df-n0 9381 df-z 9458 df-dec 9590 df-uz 9734 df-rp 9862 df-fz 10217 df-cj 11368 df-abs 11525 df-struct 13049 df-ndx 13050 df-slot 13051 df-base 13053 df-sets 13054 df-iress 13055 df-plusg 13138 df-mulr 13139 df-starv 13140 df-sca 13141 df-vsca 13142 df-ip 13143 df-tset 13144 df-ple 13145 df-ds 13147 df-unif 13148 df-0g 13306 df-topgen 13308 df-iimas 13350 df-qus 13351 df-mgm 13404 df-sgrp 13450 df-mnd 13465 df-grp 13551 df-minusg 13552 df-subg 13722 df-eqg 13724 df-cmn 13838 df-mgp 13899 df-ur 13938 df-ring 13976 df-cring 13977 df-rhm 14131 df-subrg 14198 df-lsp 14366 df-sra 14414 df-rgmod 14415 df-rsp 14449 df-bl 14525 df-mopn 14526 df-fg 14528 df-metu 14529 df-cnfld 14536 df-zring 14570 df-zrh 14593 df-zn 14595

This theorem is referenced by: znval2 14617 znle2 14631

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