znle - Intuitionistic Logic Explorer

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Theorem znle 14399

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 2205	. . . 4 ⊢ ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) = ((𝐹 ∘ ≤ ) ∘ ^◡𝐹)
7	1, 2, 3, 4, 5, 6	znval 14398	. . 3 ⊢ (𝑁 ∈ ℕ₀ → 𝑌 = (𝑈 sSet ⟨(le‘ndx), ((𝐹 ∘ ≤ ) ∘ ^◡𝐹)⟩))
8	7	fveq2d 5580	. 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 14355	. . . . 5 ⊢ ℤ_ring ∈ Ring
12		rspex 14236	. . . . . . . . 9 ⊢ (ℤ_ring ∈ Ring → (RSpan‘ℤ_ring) ∈ V)
13	11, 12	ax-mp 5	. . . . . . . 8 ⊢ (RSpan‘ℤ_ring) ∈ V
14	1, 13	eqeltri 2278	. . . . . . 7 ⊢ 𝑆 ∈ V
15		snexg 4228	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → {𝑁} ∈ V)
16		fvexg 5595	. . . . . . 7 ⊢ ((𝑆 ∈ V ∧ {𝑁} ∈ V) → (𝑆‘{𝑁}) ∈ V)
17	14, 15, 16	sylancr 414	. . . . . 6 ⊢ (𝑁 ∈ ℕ₀ → (𝑆‘{𝑁}) ∈ V)
18		eqgex 13557	. . . . . 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 13157	. . . . 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 2292	. . 3 ⊢ (𝑁 ∈ ℕ₀ → 𝑈 ∈ V)
23		eqid 2205	. . . . . . . 8 ⊢ (ℤRHom‘𝑈) = (ℤRHom‘𝑈)
24	23	zrhex 14383	. . . . . . 7 ⊢ (𝑈 ∈ V → (ℤRHom‘𝑈) ∈ V)
25		resexg 4999	. . . . . . 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 2292	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → 𝐹 ∈ V)
28		xrex 9978	. . . . . . 7 ⊢ ℝ^* ∈ V
29	28, 28	xpex 4790	. . . . . 6 ⊢ (ℝ^* × ℝ^*) ∈ V
30		lerelxr 8135	. . . . . 6 ⊢ ≤ ⊆ (ℝ^* × ℝ^*)
31	29, 30	ssexi 4182	. . . . 5 ⊢ ≤ ∈ V
32		coexg 5227	. . . . 5 ⊢ ((𝐹 ∈ V ∧ ≤ ∈ V) → (𝐹 ∘ ≤ ) ∈ V)
33	27, 31, 32	sylancl 413	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → (𝐹 ∘ ≤ ) ∈ V)
34		cnvexg 5220	. . . . 5 ⊢ (𝐹 ∈ V → ^◡𝐹 ∈ V)
35	27, 34	syl 14	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → ^◡𝐹 ∈ V)
36		coexg 5227	. . . 4 ⊢ (((𝐹 ∘ ≤ ) ∈ V ∧ ^◡𝐹 ∈ V) → ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) ∈ V)
37	33, 35, 36	syl2anc 411	. . 3 ⊢ (𝑁 ∈ ℕ₀ → ((𝐹 ∘ ≤ ) ∘ ^◡𝐹) ∈ V)
38		pleslid 13034	. . . 4 ⊢ (le = Slot (le‘ndx) ∧ (le‘ndx) ∈ ℕ)
39	38	setsslid 12883	. . 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 2248	1 ⊢ (𝑁 ∈ ℕ₀ → ≤ = ((𝐹 ∘ ≤ ) ∘ ^◡𝐹))

Colors of variables: wff set class

Syntax hints: → wi 4 = wceq 1373 ∈ wcel 2176 Vcvv 2772 ifcif 3571 {csn 3633 ⟨cop 3636 × cxp 4673 ^◡ccnv 4674 ↾ cres 4677 ∘ ccom 4679 ‘cfv 5271 (class class class)co 5944 0cc0 7925 ℝ^*cxr 8106 ≤ cle 8108 ℕ₀cn0 9295 ℤcz 9372 ..^cfzo 10264 ndxcnx 12829 sSet csts 12830 lecple 12916 /_s cqus 13132 ~_QG cqg 13505 Ringcrg 13758 RSpancrsp 14230 ℤ_ringczring 14352 ℤRHomczrh 14373 ℤ/nℤczn 14375

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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-coll 4159 ax-sep 4162 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-cnex 8016 ax-resscn 8017 ax-1cn 8018 ax-1re 8019 ax-icn 8020 ax-addcl 8021 ax-addrcl 8022 ax-mulcl 8023 ax-mulrcl 8024 ax-addcom 8025 ax-mulcom 8026 ax-addass 8027 ax-mulass 8028 ax-distr 8029 ax-i2m1 8030 ax-0lt1 8031 ax-1rid 8032 ax-0id 8033 ax-rnegex 8034 ax-precex 8035 ax-cnre 8036 ax-pre-ltirr 8037 ax-pre-ltwlin 8038 ax-pre-lttrn 8039 ax-pre-apti 8040 ax-pre-ltadd 8041 ax-pre-mulgt0 8042 ax-addf 8047 ax-mulf 8048

This theorem depends on definitions: df-bi 117 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rmo 2492 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-if 3572 df-pw 3618 df-sn 3639 df-pr 3640 df-tp 3641 df-op 3642 df-uni 3851 df-int 3886 df-iun 3929 df-br 4045 df-opab 4106 df-mpt 4107 df-id 4340 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-ima 4688 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-f1 5276 df-fo 5277 df-f1o 5278 df-fv 5279 df-riota 5899 df-ov 5947 df-oprab 5948 df-mpo 5949 df-1st 6226 df-2nd 6227 df-ec 6622 df-map 6737 df-pnf 8109 df-mnf 8110 df-xr 8111 df-ltxr 8112 df-le 8113 df-sub 8245 df-neg 8246 df-reap 8648 df-inn 9037 df-2 9095 df-3 9096 df-4 9097 df-5 9098 df-6 9099 df-7 9100 df-8 9101 df-9 9102 df-n0 9296 df-z 9373 df-dec 9505 df-uz 9649 df-rp 9776 df-fz 10131 df-cj 11153 df-abs 11310 df-struct 12834 df-ndx 12835 df-slot 12836 df-base 12838 df-sets 12839 df-iress 12840 df-plusg 12922 df-mulr 12923 df-starv 12924 df-sca 12925 df-vsca 12926 df-ip 12927 df-tset 12928 df-ple 12929 df-ds 12931 df-unif 12932 df-0g 13090 df-topgen 13092 df-iimas 13134 df-qus 13135 df-mgm 13188 df-sgrp 13234 df-mnd 13249 df-grp 13335 df-minusg 13336 df-subg 13506 df-eqg 13508 df-cmn 13622 df-mgp 13683 df-ur 13722 df-ring 13760 df-cring 13761 df-rhm 13914 df-subrg 13981 df-lsp 14149 df-sra 14197 df-rgmod 14198 df-rsp 14232 df-bl 14308 df-mopn 14309 df-fg 14311 df-metu 14312 df-cnfld 14319 df-zring 14353 df-zrh 14376 df-zn 14378

This theorem is referenced by: znval2 14400 znle2 14414

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