zndvds - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > zndvds		Structured version Visualization version GIF version

Theorem zndvds 21484

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 2738	. 2 ⊢ ((𝐿‘𝐴) = (𝐿‘𝐵) ↔ (𝐿‘𝐵) = (𝐿‘𝐴))
2		eqid 2731	. . . . . 6 ⊢ (RSpan‘ℤ_ring) = (RSpan‘ℤ_ring)
3		eqid 2731	. . . . . 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 21481	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐵 ∈ ℤ) → (𝐿‘𝐵) = [𝐵](ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁})))
7	6	3adant2 1131	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐿‘𝐵) = [𝐵](ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁})))
8	2, 3, 4, 5	znzrhval 21481	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → (𝐿‘𝐴) = [𝐴](ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁})))
9	8	3adant3 1132	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐿‘𝐴) = [𝐴](ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁})))
10	7, 9	eqeq12d 2747	. . 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 21384	. . . . . 6 ⊢ ℤ_ring ∈ Ring
12		nn0z 12490	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
13	12	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝑁 ∈ ℤ)
14	13	snssd 4761	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {𝑁} ⊆ ℤ)
15		zringbas 21388	. . . . . . . 8 ⊢ ℤ = (Base‘ℤ_ring)
16		eqid 2731	. . . . . . . 8 ⊢ (LIdeal‘ℤ_ring) = (LIdeal‘ℤ_ring)
17	2, 15, 16	rspcl 21170	. . . . . . 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 21158	. . . . . 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 19088	. . . . 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 1138	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℤ)
24	22, 23	erth 8676	. . 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 21386	. . . . 5 ⊢ ℤ_ring ∈ Abel
26	15, 16	lidlss 21147	. . . . . 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 2731	. . . . . 6 ⊢ (-_g‘ℤ_ring) = (-_g‘ℤ_ring)
29	15, 28, 3	eqgabl 19744	. . . . 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 1137	. . . . . . 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 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 21355	. . . . . . . . 9 ⊢ ℤ ∈ (SubRing‘ℂ_fld)
37		subrgsubg 20490	. . . . . . . . 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 21332	. . . . . . . . 9 ⊢ − = (-_g‘ℂ_fld)
40		df-zring 21382	. . . . . . . . 9 ⊢ ℤ_ring = (ℂ_fld ↾_s ℤ)
41	39, 40, 28	subgsub 19048	. . . . . . . 8 ⊢ ((ℤ ∈ (SubGrp‘ℂ_fld) ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 − 𝐵) = (𝐴(-_g‘ℤ_ring)𝐵))
42	38, 41	syld3an1 1412	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 − 𝐵) = (𝐴(-_g‘ℤ_ring)𝐵))
43	42	eqcomd 2737	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴(-_g‘ℤ_ring)𝐵) = (𝐴 − 𝐵))
44		dvdsrzring 21396	. . . . . . . 8 ⊢ ∥ = (∥_r‘ℤ_ring)
45	15, 2, 44	rspsn 21268	. . . . . . 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 2825	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴(-_g‘ℤ_ring)𝐵) ∈ ((RSpan‘ℤ_ring)‘{𝑁}) ↔ (𝐴 − 𝐵) ∈ {𝑥 ∣ 𝑁 ∥ 𝑥}))
48		ovex 7379	. . . . . 6 ⊢ (𝐴 − 𝐵) ∈ V
49		breq2 5095	. . . . . 6 ⊢ (𝑥 = (𝐴 − 𝐵) → (𝑁 ∥ 𝑥 ↔ 𝑁 ∥ (𝐴 − 𝐵)))
50	48, 49	elab 3635	. . . . 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 1086 = wceq 1541 ∈ wcel 2111 {cab 2709 ⊆ wss 3902 {csn 4576 class class class wbr 5091 ‘cfv 6481 (class class class)co 7346 Er wer 8619 [cec 8620 − cmin 11341 ℕ₀cn0 12378 ℤcz 12465 ∥ cdvds 16160 -_gcsg 18845 SubGrpcsubg 19030 ~_QG cqg 19032 Abelcabl 19691 Ringcrg 20149 SubRingcsubrg 20482 LIdealclidl 21141 RSpancrsp 21142 ℂ_fldccnfld 21289 ℤ_ringczring 21381 ℤRHomczrh 21434 ℤ/nℤczn 21437

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11059 ax-resscn 11060 ax-1cn 11061 ax-icn 11062 ax-addcl 11063 ax-addrcl 11064 ax-mulcl 11065 ax-mulrcl 11066 ax-mulcom 11067 ax-addass 11068 ax-mulass 11069 ax-distr 11070 ax-i2m1 11071 ax-1ne0 11072 ax-1rid 11073 ax-rnegex 11074 ax-rrecex 11075 ax-cnre 11076 ax-pre-lttri 11077 ax-pre-lttrn 11078 ax-pre-ltadd 11079 ax-pre-mulgt0 11080 ax-addf 11082 ax-mulf 11083

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-tpos 8156 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-er 8622 df-ec 8624 df-qs 8628 df-map 8752 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-sup 9326 df-inf 9327 df-pnf 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149 df-sub 11343 df-neg 11344 df-nn 12123 df-2 12185 df-3 12186 df-4 12187 df-5 12188 df-6 12189 df-7 12190 df-8 12191 df-9 12192 df-n0 12379 df-z 12466 df-dec 12586 df-uz 12730 df-fz 13405 df-seq 13906 df-dvds 16161 df-struct 17055 df-sets 17072 df-slot 17090 df-ndx 17102 df-base 17118 df-ress 17139 df-plusg 17171 df-mulr 17172 df-starv 17173 df-sca 17174 df-vsca 17175 df-ip 17176 df-tset 17177 df-ple 17178 df-ds 17180 df-unif 17181 df-0g 17342 df-imas 17409 df-qus 17410 df-mgm 18545 df-sgrp 18624 df-mnd 18640 df-mhm 18688 df-grp 18846 df-minusg 18847 df-sbg 18848 df-mulg 18978 df-subg 19033 df-nsg 19034 df-eqg 19035 df-ghm 19123 df-cmn 19692 df-abl 19693 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-cring 20152 df-oppr 20253 df-dvdsr 20273 df-rhm 20388 df-subrng 20459 df-subrg 20483 df-lmod 20793 df-lss 20863 df-lsp 20903 df-sra 21105 df-rgmod 21106 df-lidl 21143 df-rsp 21144 df-2idl 21185 df-cnfld 21290 df-zring 21382 df-zrh 21438 df-zn 21441

This theorem is referenced by: zndvds0 21485 znf1o 21486 znunit 21498 cygznlem1 21501 lgsqrlem1 27282 lgsqrlem2 27283 lgsqrlem4 27285 lgsdchrval 27290 lgseisenlem3 27313 lgseisenlem4 27314 dchrisumlem1 27425 dirith 27465 hashscontpow1 42153 aks6d1c2 42162

W3C validator