znle - Intuitionistic Logic Explorer

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Theorem znle 14370

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 2204	. . . 4 ⊢ ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) = ((𝐹 ∘ ≤ ) ∘ ^◡𝐹)
7	1, 2, 3, 4, 5, 6	znval 14369	. . 3 ⊢ (𝑁 ∈ ℕ₀ → 𝑌 = (𝑈 sSet ⟨(le‘ndx), ((𝐹 ∘ ≤ ) ∘ ^◡𝐹)⟩))
8	7	fveq2d 5579	. 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 14326	. . . . 5 ⊢ ℤ_ring ∈ Ring
12		rspex 14207	. . . . . . . . 9 ⊢ (ℤ_ring ∈ Ring → (RSpan‘ℤ_ring) ∈ V)
13	11, 12	ax-mp 5	. . . . . . . 8 ⊢ (RSpan‘ℤ_ring) ∈ V
14	1, 13	eqeltri 2277	. . . . . . 7 ⊢ 𝑆 ∈ V
15		snexg 4227	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → {𝑁} ∈ V)
16		fvexg 5594	. . . . . . 7 ⊢ ((𝑆 ∈ V ∧ {𝑁} ∈ V) → (𝑆‘{𝑁}) ∈ V)
17	14, 15, 16	sylancr 414	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → (𝑆‘{𝑁}) ∈ V)
18		eqgex 13528	. . . . . 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 13128	. . . . 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 2291	. . 3 ⊢ (𝑁 ∈ ℕ₀ → 𝑈 ∈ V)
23		eqid 2204	. . . . . . . 8 ⊢ (ℤRHom‘𝑈) = (ℤRHom‘𝑈)
24	23	zrhex 14354	. . . . . . 7 ⊢ (𝑈 ∈ V → (ℤRHom‘𝑈) ∈ V)
25		resexg 4998	. . . . . . 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 2291	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → 𝐹 ∈ V)
28		xrex 9977	. . . . . . 7 ⊢ ℝ^* ∈ V
29	28, 28	xpex 4789	. . . . . 6 ⊢ (ℝ^* × ℝ^*) ∈ V
30		lerelxr 8134	. . . . . 6 ⊢ ≤ ⊆ (ℝ^* × ℝ^*)
31	29, 30	ssexi 4181	. . . . 5 ⊢ ≤ ∈ V
32		coexg 5226	. . . . 5 ⊢ ((𝐹 ∈ V ∧ ≤ ∈ V) → (𝐹 ∘ ≤ ) ∈ V)
33	27, 31, 32	sylancl 413	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → (𝐹 ∘ ≤ ) ∈ V)
34		cnvexg 5219	. . . . 5 ⊢ (𝐹 ∈ V → ^◡𝐹 ∈ V)
35	27, 34	syl 14	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → ^◡𝐹 ∈ V)
36		coexg 5226	. . . 4 ⊢ (((𝐹 ∘ ≤ ) ∈ V ∧ ^◡𝐹 ∈ V) → ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) ∈ V)
37	33, 35, 36	syl2anc 411	. . 3 ⊢ (𝑁 ∈ ℕ₀ → ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) ∈ V)
38		pleslid 13005	. . . 4 ⊢ (le = Slot (le‘ndx) ∧ (le‘ndx) ∈ ℕ)
39	38	setsslid 12854	. . 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 2247	1 ⊢ (𝑁 ∈ ℕ₀ → ≤ = ((𝐹 ∘ ≤ ) ∘ ^◡𝐹))

Colors of variables: wff set class

Syntax hints: → wi 4 = wceq 1372 ∈ wcel 2175 Vcvv 2771 ifcif 3570 {csn 3632 ⟨cop 3635 × cxp 4672 ^◡ccnv 4673 ↾ cres 4676 ∘ ccom 4678 ‘cfv 5270 (class class class)co 5943 0cc0 7924 ℝ^*cxr 8105 ≤ cle 8107 ℕ₀cn0 9294 ℤcz 9371 ..^cfzo 10263 ndxcnx 12800 sSet csts 12801 lecple 12887 /_s cqus 13103 ~_QG cqg 13476 Ringcrg 13729 RSpancrsp 14201 ℤ_ringczring 14323 ℤRHomczrh 14344 ℤ/nℤczn 14346

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 615 ax-in2 616 ax-io 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-coll 4158 ax-sep 4161 ax-pow 4217 ax-pr 4252 ax-un 4479 ax-setind 4584 ax-cnex 8015 ax-resscn 8016 ax-1cn 8017 ax-1re 8018 ax-icn 8019 ax-addcl 8020 ax-addrcl 8021 ax-mulcl 8022 ax-mulrcl 8023 ax-addcom 8024 ax-mulcom 8025 ax-addass 8026 ax-mulass 8027 ax-distr 8028 ax-i2m1 8029 ax-0lt1 8030 ax-1rid 8031 ax-0id 8032 ax-rnegex 8033 ax-precex 8034 ax-cnre 8035 ax-pre-ltirr 8036 ax-pre-ltwlin 8037 ax-pre-lttrn 8038 ax-pre-apti 8039 ax-pre-ltadd 8040 ax-pre-mulgt0 8041 ax-addf 8046 ax-mulf 8047

This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1375 df-fal 1378 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ne 2376 df-nel 2471 df-ral 2488 df-rex 2489 df-reu 2490 df-rmo 2491 df-rab 2492 df-v 2773 df-sbc 2998 df-csb 3093 df-dif 3167 df-un 3169 df-in 3171 df-ss 3178 df-nul 3460 df-if 3571 df-pw 3617 df-sn 3638 df-pr 3639 df-tp 3640 df-op 3641 df-uni 3850 df-int 3885 df-iun 3928 df-br 4044 df-opab 4105 df-mpt 4106 df-id 4339 df-xp 4680 df-rel 4681 df-cnv 4682 df-co 4683 df-dm 4684 df-rn 4685 df-res 4686 df-ima 4687 df-iota 5231 df-fun 5272 df-fn 5273 df-f 5274 df-f1 5275 df-fo 5276 df-f1o 5277 df-fv 5278 df-riota 5898 df-ov 5946 df-oprab 5947 df-mpo 5948 df-1st 6225 df-2nd 6226 df-ec 6621 df-map 6736 df-pnf 8108 df-mnf 8109 df-xr 8110 df-ltxr 8111 df-le 8112 df-sub 8244 df-neg 8245 df-reap 8647 df-inn 9036 df-2 9094 df-3 9095 df-4 9096 df-5 9097 df-6 9098 df-7 9099 df-8 9100 df-9 9101 df-n0 9295 df-z 9372 df-dec 9504 df-uz 9648 df-rp 9775 df-fz 10130 df-cj 11124 df-abs 11281 df-struct 12805 df-ndx 12806 df-slot 12807 df-base 12809 df-sets 12810 df-iress 12811 df-plusg 12893 df-mulr 12894 df-starv 12895 df-sca 12896 df-vsca 12897 df-ip 12898 df-tset 12899 df-ple 12900 df-ds 12902 df-unif 12903 df-0g 13061 df-topgen 13063 df-iimas 13105 df-qus 13106 df-mgm 13159 df-sgrp 13205 df-mnd 13220 df-grp 13306 df-minusg 13307 df-subg 13477 df-eqg 13479 df-cmn 13593 df-mgp 13654 df-ur 13693 df-ring 13731 df-cring 13732 df-rhm 13885 df-subrg 13952 df-lsp 14120 df-sra 14168 df-rgmod 14169 df-rsp 14203 df-bl 14279 df-mopn 14280 df-fg 14282 df-metu 14283 df-cnfld 14290 df-zring 14324 df-zrh 14347 df-zn 14349

This theorem is referenced by: znval2 14371 znle2 14385

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