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Theorem zndvds 21529

Description: Express equality of equivalence classes in ℤ / 𝑛ℤ in terms of divisibility. (Contributed by Mario Carneiro, 15-Jun-2015.)

**Hypotheses**
Ref	Expression
zncyg.y	⊢ 𝑌 = (ℤ/nℤ‘𝑁)
zndvds.2	⊢ 𝐿 = (ℤRHom‘𝑌)

**Assertion**
Ref	Expression
zndvds	⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐿‘𝐴) = (𝐿‘𝐵) ↔ 𝑁 ∥ (𝐴 − 𝐵)))

Proof of Theorem zndvds Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqcom 2743	. 2 ⊢ ((𝐿‘𝐴) = (𝐿‘𝐵) ↔ (𝐿‘𝐵) = (𝐿‘𝐴))
2		eqid 2736	. . . . . 6 ⊢ (RSpan‘ℤ_ring) = (RSpan‘ℤ_ring)
3		eqid 2736	. . . . . 6 ⊢ (ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁})) = (ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁}))
4		zncyg.y	. . . . . 6 ⊢ 𝑌 = (ℤ/nℤ‘𝑁)
5		zndvds.2	. . . . . 6 ⊢ 𝐿 = (ℤRHom‘𝑌)
6	2, 3, 4, 5	znzrhval 21526	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐵 ∈ ℤ) → (𝐿‘𝐵) = [𝐵](ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁})))
7	6	3adant2 1132	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐿‘𝐵) = [𝐵](ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁})))
8	2, 3, 4, 5	znzrhval 21526	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → (𝐿‘𝐴) = [𝐴](ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁})))
9	8	3adant3 1133	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐿‘𝐴) = [𝐴](ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁})))
10	7, 9	eqeq12d 2752	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐿‘𝐵) = (𝐿‘𝐴) ↔ [𝐵](ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁})) = [𝐴](ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁}))))
11		zringring 21429	. . . . . 6 ⊢ ℤ_ring ∈ Ring
12		nn0z 12548	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
13	12	3ad2ant1 1134	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝑁 ∈ ℤ)
14	13	snssd 4730	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {𝑁} ⊆ ℤ)
15		zringbas 21433	. . . . . . . 8 ⊢ ℤ = (Base‘ℤ_ring)
16		eqid 2736	. . . . . . . 8 ⊢ (LIdeal‘ℤ_ring) = (LIdeal‘ℤ_ring)
17	2, 15, 16	rspcl 21233	. . . . . . 7 ⊢ ((ℤ_ring ∈ Ring ∧ {𝑁} ⊆ ℤ) → ((RSpan‘ℤ_ring)‘{𝑁}) ∈ (LIdeal‘ℤ_ring))
18	11, 14, 17	sylancr 588	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((RSpan‘ℤ_ring)‘{𝑁}) ∈ (LIdeal‘ℤ_ring))
19	16	lidlsubg 21221	. . . . . 6 ⊢ ((ℤ_ring ∈ Ring ∧ ((RSpan‘ℤ_ring)‘{𝑁}) ∈ (LIdeal‘ℤ_ring)) → ((RSpan‘ℤ_ring)‘{𝑁}) ∈ (SubGrp‘ℤ_ring))
20	11, 18, 19	sylancr 588	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((RSpan‘ℤ_ring)‘{𝑁}) ∈ (SubGrp‘ℤ_ring))
21	15, 3	eqger 19153	. . . . 5 ⊢ (((RSpan‘ℤ_ring)‘{𝑁}) ∈ (SubGrp‘ℤ_ring) → (ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁})) Er ℤ)
22	20, 21	syl 17	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁})) Er ℤ)
23		simp3 1139	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℤ)
24	22, 23	erth 8698	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵(ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁}))𝐴 ↔ [𝐵](ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁})) = [𝐴](ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁}))))
25		zringabl 21431	. . . . 5 ⊢ ℤ_ring ∈ Abel
26	15, 16	lidlss 21210	. . . . . 6 ⊢ (((RSpan‘ℤ_ring)‘{𝑁}) ∈ (LIdeal‘ℤ_ring) → ((RSpan‘ℤ_ring)‘{𝑁}) ⊆ ℤ)
27	18, 26	syl 17	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((RSpan‘ℤ_ring)‘{𝑁}) ⊆ ℤ)
28		eqid 2736	. . . . . 6 ⊢ (-_g‘ℤ_ring) = (-_g‘ℤ_ring)
29	15, 28, 3	eqgabl 19809	. . . . 5 ⊢ ((ℤ_ring ∈ Abel ∧ ((RSpan‘ℤ_ring)‘{𝑁}) ⊆ ℤ) → (𝐵(ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁}))𝐴 ↔ (𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ (𝐴(-_g‘ℤ_ring)𝐵) ∈ ((RSpan‘ℤ_ring)‘{𝑁}))))
30	25, 27, 29	sylancr 588	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵(ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁}))𝐴 ↔ (𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ (𝐴(-_g‘ℤ_ring)𝐵) ∈ ((RSpan‘ℤ_ring)‘{𝑁}))))
31		simp2 1138	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∈ ℤ)
32	23, 31	jca 511	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ))
33	32	biantrurd 532	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴(-_g‘ℤ_ring)𝐵) ∈ ((RSpan‘ℤ_ring)‘{𝑁}) ↔ ((𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ) ∧ (𝐴(-_g‘ℤ_ring)𝐵) ∈ ((RSpan‘ℤ_ring)‘{𝑁}))))
34		df-3an 1089	. . . . 5 ⊢ ((𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ (𝐴(-_g‘ℤ_ring)𝐵) ∈ ((RSpan‘ℤ_ring)‘{𝑁})) ↔ ((𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ) ∧ (𝐴(-_g‘ℤ_ring)𝐵) ∈ ((RSpan‘ℤ_ring)‘{𝑁})))
35	33, 34	bitr4di 289	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴(-_g‘ℤ_ring)𝐵) ∈ ((RSpan‘ℤ_ring)‘{𝑁}) ↔ (𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ (𝐴(-_g‘ℤ_ring)𝐵) ∈ ((RSpan‘ℤ_ring)‘{𝑁}))))
36		zsubrg 21400	. . . . . . . . 9 ⊢ ℤ ∈ (SubRing‘ℂ_fld)
37		subrgsubg 20554	. . . . . . . . 9 ⊢ (ℤ ∈ (SubRing‘ℂ_fld) → ℤ ∈ (SubGrp‘ℂ_fld))
38	36, 37	mp1i 13	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ℤ ∈ (SubGrp‘ℂ_fld))
39		cnfldsub 21380	. . . . . . . . 9 ⊢ − = (-_g‘ℂ_fld)
40		df-zring 21427	. . . . . . . . 9 ⊢ ℤ_ring = (ℂ_fld ↾_s ℤ)
41	39, 40, 28	subgsub 19114	. . . . . . . 8 ⊢ ((ℤ ∈ (SubGrp‘ℂ_fld) ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 − 𝐵) = (𝐴(-_g‘ℤ_ring)𝐵))
42	38, 41	syld3an1 1413	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 − 𝐵) = (𝐴(-_g‘ℤ_ring)𝐵))
43	42	eqcomd 2742	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴(-_g‘ℤ_ring)𝐵) = (𝐴 − 𝐵))
44		dvdsrzring 21441	. . . . . . . 8 ⊢ ∥ = (∥_r‘ℤ_ring)
45	15, 2, 44	rspsn 21331	. . . . . . 7 ⊢ ((ℤ_ring ∈ Ring ∧ 𝑁 ∈ ℤ) → ((RSpan‘ℤ_ring)‘{𝑁}) = {𝑥 ∣ 𝑁 ∥ 𝑥})
46	11, 13, 45	sylancr 588	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((RSpan‘ℤ_ring)‘{𝑁}) = {𝑥 ∣ 𝑁 ∥ 𝑥})
47	43, 46	eleq12d 2830	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴(-_g‘ℤ_ring)𝐵) ∈ ((RSpan‘ℤ_ring)‘{𝑁}) ↔ (𝐴 − 𝐵) ∈ {𝑥 ∣ 𝑁 ∥ 𝑥}))
48		ovex 7400	. . . . . 6 ⊢ (𝐴 − 𝐵) ∈ V
49		breq2 5089	. . . . . 6 ⊢ (𝑥 = (𝐴 − 𝐵) → (𝑁 ∥ 𝑥 ↔ 𝑁 ∥ (𝐴 − 𝐵)))
50	48, 49	elab 3622	. . . . 5 ⊢ ((𝐴 − 𝐵) ∈ {𝑥 ∣ 𝑁 ∥ 𝑥} ↔ 𝑁 ∥ (𝐴 − 𝐵))
51	47, 50	bitrdi 287	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴(-_g‘ℤ_ring)𝐵) ∈ ((RSpan‘ℤ_ring)‘{𝑁}) ↔ 𝑁 ∥ (𝐴 − 𝐵)))
52	30, 35, 51	3bitr2d 307	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵(ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁}))𝐴 ↔ 𝑁 ∥ (𝐴 − 𝐵)))
53	10, 24, 52	3bitr2d 307	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐿‘𝐵) = (𝐿‘𝐴) ↔ 𝑁 ∥ (𝐴 − 𝐵)))
54	1, 53	bitrid 283	1 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐿‘𝐴) = (𝐿‘𝐵) ↔ 𝑁 ∥ (𝐴 − 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 {cab 2714 ⊆ wss 3889 {csn 4567 class class class wbr 5085 ‘cfv 6498 (class class class)co 7367 Er wer 8640 [cec 8641 − cmin 11377 ℕ₀cn0 12437 ℤcz 12524 ∥ cdvds 16221 -_gcsg 18911 SubGrpcsubg 19096 ~_QG cqg 19098 Abelcabl 19756 Ringcrg 20214 SubRingcsubrg 20546 LIdealclidl 21204 RSpancrsp 21205 ℂ_fldccnfld 21352 ℤ_ringczring 21426 ℤRHomczrh 21479 ℤ/nℤczn 21482

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-addf 11117 ax-mulf 11118

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-tpos 8176 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-ec 8645 df-qs 8649 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-sup 9355 df-inf 9356 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-fz 13462 df-seq 13964 df-dvds 16222 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-0g 17404 df-imas 17472 df-qus 17473 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-mhm 18751 df-grp 18912 df-minusg 18913 df-sbg 18914 df-mulg 19044 df-subg 19099 df-nsg 19100 df-eqg 19101 df-ghm 19188 df-cmn 19757 df-abl 19758 df-mgp 20122 df-rng 20134 df-ur 20163 df-ring 20216 df-cring 20217 df-oppr 20317 df-dvdsr 20337 df-rhm 20452 df-subrng 20523 df-subrg 20547 df-lmod 20857 df-lss 20927 df-lsp 20967 df-sra 21168 df-rgmod 21169 df-lidl 21206 df-rsp 21207 df-2idl 21248 df-cnfld 21353 df-zring 21427 df-zrh 21483 df-zn 21486

This theorem is referenced by: zndvds0 21530 znf1o 21531 znunit 21543 cygznlem1 21546 lgsqrlem1 27309 lgsqrlem2 27310 lgsqrlem4 27312 lgsdchrval 27317 lgseisenlem3 27340 lgseisenlem4 27341 dchrisumlem1 27452 dirith 27492 hashscontpow1 42560 aks6d1c2 42569

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