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Theorem zndvds 20757

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 2745	. 2 ⊢ ((𝐿‘𝐴) = (𝐿‘𝐵) ↔ (𝐿‘𝐵) = (𝐿‘𝐴))
2		eqid 2738	. . . . . 6 ⊢ (RSpan‘ℤ_ring) = (RSpan‘ℤ_ring)
3		eqid 2738	. . . . . 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 20754	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐵 ∈ ℤ) → (𝐿‘𝐵) = [𝐵](ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁})))
7	6	3adant2 1130	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐿‘𝐵) = [𝐵](ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁})))
8	2, 3, 4, 5	znzrhval 20754	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → (𝐿‘𝐴) = [𝐴](ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁})))
9	8	3adant3 1131	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐿‘𝐴) = [𝐴](ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁})))
10	7, 9	eqeq12d 2754	. . 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 20673	. . . . . 6 ⊢ ℤ_ring ∈ Ring
12		nn0z 12343	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
13	12	3ad2ant1 1132	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝑁 ∈ ℤ)
14	13	snssd 4742	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {𝑁} ⊆ ℤ)
15		zringbas 20676	. . . . . . . 8 ⊢ ℤ = (Base‘ℤ_ring)
16		eqid 2738	. . . . . . . 8 ⊢ (LIdeal‘ℤ_ring) = (LIdeal‘ℤ_ring)
17	2, 15, 16	rspcl 20493	. . . . . . 7 ⊢ ((ℤ_ring ∈ Ring ∧ {𝑁} ⊆ ℤ) → ((RSpan‘ℤ_ring)‘{𝑁}) ∈ (LIdeal‘ℤ_ring))
18	11, 14, 17	sylancr 587	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((RSpan‘ℤ_ring)‘{𝑁}) ∈ (LIdeal‘ℤ_ring))
19	16	lidlsubg 20486	. . . . . 6 ⊢ ((ℤ_ring ∈ Ring ∧ ((RSpan‘ℤ_ring)‘{𝑁}) ∈ (LIdeal‘ℤ_ring)) → ((RSpan‘ℤ_ring)‘{𝑁}) ∈ (SubGrp‘ℤ_ring))
20	11, 18, 19	sylancr 587	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((RSpan‘ℤ_ring)‘{𝑁}) ∈ (SubGrp‘ℤ_ring))
21	15, 3	eqger 18806	. . . . 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 1137	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℤ)
24	22, 23	erth 8547	. . 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 20674	. . . . 5 ⊢ ℤ_ring ∈ Abel
26	15, 16	lidlss 20481	. . . . . 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 2738	. . . . . 6 ⊢ (-_g‘ℤ_ring) = (-_g‘ℤ_ring)
29	15, 28, 3	eqgabl 19436	. . . . 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 587	. . . 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 1136	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∈ ℤ)
32	23, 31	jca 512	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ))
33	32	biantrurd 533	. . . . 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 1088	. . . . 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 20651	. . . . . . . . 9 ⊢ ℤ ∈ (SubRing‘ℂ_fld)
37		subrgsubg 20030	. . . . . . . . 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 20626	. . . . . . . . 9 ⊢ − = (-_g‘ℂ_fld)
40		df-zring 20671	. . . . . . . . 9 ⊢ ℤ_ring = (ℂ_fld ↾_s ℤ)
41	39, 40, 28	subgsub 18767	. . . . . . . 8 ⊢ ((ℤ ∈ (SubGrp‘ℂ_fld) ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 − 𝐵) = (𝐴(-_g‘ℤ_ring)𝐵))
42	38, 41	syld3an1 1409	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 − 𝐵) = (𝐴(-_g‘ℤ_ring)𝐵))
43	42	eqcomd 2744	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴(-_g‘ℤ_ring)𝐵) = (𝐴 − 𝐵))
44		dvdsrzring 20683	. . . . . . . 8 ⊢ ∥ = (∥_r‘ℤ_ring)
45	15, 2, 44	rspsn 20525	. . . . . . 7 ⊢ ((ℤ_ring ∈ Ring ∧ 𝑁 ∈ ℤ) → ((RSpan‘ℤ_ring)‘{𝑁}) = {𝑥 ∣ 𝑁 ∥ 𝑥})
46	11, 13, 45	sylancr 587	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((RSpan‘ℤ_ring)‘{𝑁}) = {𝑥 ∣ 𝑁 ∥ 𝑥})
47	43, 46	eleq12d 2833	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴(-_g‘ℤ_ring)𝐵) ∈ ((RSpan‘ℤ_ring)‘{𝑁}) ↔ (𝐴 − 𝐵) ∈ {𝑥 ∣ 𝑁 ∥ 𝑥}))
48		ovex 7308	. . . . . 6 ⊢ (𝐴 − 𝐵) ∈ V
49		breq2 5078	. . . . . 6 ⊢ (𝑥 = (𝐴 − 𝐵) → (𝑁 ∥ 𝑥 ↔ 𝑁 ∥ (𝐴 − 𝐵)))
50	48, 49	elab 3609	. . . . 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 282	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 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 {cab 2715 ⊆ wss 3887 {csn 4561 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 Er wer 8495 [cec 8496 − cmin 11205 ℕ₀cn0 12233 ℤcz 12319 ∥ cdvds 15963 -_gcsg 18579 SubGrpcsubg 18749 ~_QG cqg 18751 Abelcabl 19387 Ringcrg 19783 SubRingcsubrg 20020 LIdealclidl 20432 RSpancrsp 20433 ℂ_fldccnfld 20597 ℤ_ringczring 20670 ℤRHomczrh 20701 ℤ/nℤczn 20704

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8042 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-ec 8500 df-qs 8504 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-inf 9202 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-fz 13240 df-seq 13722 df-dvds 15964 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-0g 17152 df-imas 17219 df-qus 17220 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-mhm 18430 df-grp 18580 df-minusg 18581 df-sbg 18582 df-mulg 18701 df-subg 18752 df-nsg 18753 df-eqg 18754 df-ghm 18832 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-cring 19786 df-oppr 19862 df-dvdsr 19883 df-rnghom 19959 df-subrg 20022 df-lmod 20125 df-lss 20194 df-lsp 20234 df-sra 20434 df-rgmod 20435 df-lidl 20436 df-rsp 20437 df-2idl 20503 df-cnfld 20598 df-zring 20671 df-zrh 20705 df-zn 20708

This theorem is referenced by: zndvds0 20758 znf1o 20759 znunit 20771 cygznlem1 20774 lgsqrlem1 26494 lgsqrlem2 26495 lgsqrlem4 26497 lgsdchrval 26502 lgseisenlem3 26525 lgseisenlem4 26526 dchrisumlem1 26637 dirith 26677

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