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Theorem zndvds 21885

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 2770	. 2 ⊢ ((𝐿‘𝐴) = (𝐿‘𝐵) ↔ (𝐿‘𝐵) = (𝐿‘𝐴))
2		eqid 2763	. . . . . 6 ⊢ (RSpan‘ℤ_ring) = (RSpan‘ℤ_ring)
3		eqid 2763	. . . . . 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 21882	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐵 ∈ ℤ) → (𝐿‘𝐵) = [𝐵](ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁})))
7	6	3adant2 1149	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐿‘𝐵) = [𝐵](ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁})))
8	2, 3, 4, 5	znzrhval 21882	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ) → (𝐿‘𝐴) = [𝐴](ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁})))
9	8	3adant3 1150	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐿‘𝐴) = [𝐴](ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁})))
10	7, 9	eqeq12d 2779	. . 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 21785	. . . . . 6 ⊢ ℤ_ring ∈ Ring
12		nn0z 12744	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℤ)
13	12	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝑁 ∈ ℤ)
14	13	snssd 4747	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {𝑁} ⊆ ℤ)
15		zringbas 21789	. . . . . . . 8 ⊢ ℤ = (Base‘ℤ_ring)
16		eqid 2763	. . . . . . . 8 ⊢ (LIdeal‘ℤ_ring) = (LIdeal‘ℤ_ring)
17	2, 15, 16	rspcl 21548	. . . . . . 7 ⊢ ((ℤ_ring ∈ Ring ∧ {𝑁} ⊆ ℤ) → ((RSpan‘ℤ_ring)‘{𝑁}) ∈ (LIdeal‘ℤ_ring))
18	11, 14, 17	sylancr 599	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((RSpan‘ℤ_ring)‘{𝑁}) ∈ (LIdeal‘ℤ_ring))
19	16	lidlsubg 21532	. . . . . 6 ⊢ ((ℤ_ring ∈ Ring ∧ ((RSpan‘ℤ_ring)‘{𝑁}) ∈ (LIdeal‘ℤ_ring)) → ((RSpan‘ℤ_ring)‘{𝑁}) ∈ (SubGrp‘ℤ_ring))
20	11, 18, 19	sylancr 599	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((RSpan‘ℤ_ring)‘{𝑁}) ∈ (SubGrp‘ℤ_ring))
21	15, 3	eqger 19419	. . . . 5 ⊢ (((RSpan‘ℤ_ring)‘{𝑁}) ∈ (SubGrp‘ℤ_ring) → (ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁})) Er ℤ)
22	20, 21	syl 18	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁})) Er ℤ)
23		simp3 1156	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∈ ℤ)
24	22, 23	erth 8800	. . 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 21787	. . . . 5 ⊢ ℤ_ring ∈ Abel
26	15, 16	lidlss 21520	. . . . . 6 ⊢ (((RSpan‘ℤ_ring)‘{𝑁}) ∈ (LIdeal‘ℤ_ring) → ((RSpan‘ℤ_ring)‘{𝑁}) ⊆ ℤ)
27	18, 26	syl 18	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((RSpan‘ℤ_ring)‘{𝑁}) ⊆ ℤ)
28		eqid 2763	. . . . . 6 ⊢ (-_g‘ℤ_ring) = (-_g‘ℤ_ring)
29	15, 28, 3	eqgabl 20078	. . . . 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 599	. . . 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 1155	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐴 ∈ ℤ)
32	23, 31	jca 521	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ))
33	32	biantrurd 542	. . . . 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 1105	. . . . 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 292	. . . 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 21756	. . . . . . . . 9 ⊢ ℤ ∈ (SubRing‘ℂ_fld)
37		subrgsubg 20859	. . . . . . . . 9 ⊢ (ℤ ∈ (SubRing‘ℂ_fld) → ℤ ∈ (SubGrp‘ℂ_fld))
38	36, 37	mp1i 14	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ℤ ∈ (SubGrp‘ℂ_fld))
39		cnfldsub 21736	. . . . . . . . 9 ⊢ − = (-_g‘ℂ_fld)
40		df-zring 21783	. . . . . . . . 9 ⊢ ℤ_ring = (ℂ_fld ↾_s ℤ)
41	39, 40, 28	subgsub 19378	. . . . . . . 8 ⊢ ((ℤ ∈ (SubGrp‘ℂ_fld) ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 − 𝐵) = (𝐴(-_g‘ℤ_ring)𝐵))
42	38, 41	syld3an1 1438	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 − 𝐵) = (𝐴(-_g‘ℤ_ring)𝐵))
43	42	eqcomd 2769	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴(-_g‘ℤ_ring)𝐵) = (𝐴 − 𝐵))
44		dvdsrzring 21797	. . . . . . . 8 ⊢ ∥ = (∥_r‘ℤ_ring)
45	15, 2, 44	rspsn 21687	. . . . . . 7 ⊢ ((ℤ_ring ∈ Ring ∧ 𝑁 ∈ ℤ) → ((RSpan‘ℤ_ring)‘{𝑁}) = {𝑥 ∣ 𝑁 ∥ 𝑥})
46	11, 13, 45	sylancr 599	. . . . . 6 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((RSpan‘ℤ_ring)‘{𝑁}) = {𝑥 ∣ 𝑁 ∥ 𝑥})
47	43, 46	eleq12d 2858	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴(-_g‘ℤ_ring)𝐵) ∈ ((RSpan‘ℤ_ring)‘{𝑁}) ↔ (𝐴 − 𝐵) ∈ {𝑥 ∣ 𝑁 ∥ 𝑥}))
48		ovex 7476	. . . . . 6 ⊢ (𝐴 − 𝐵) ∈ V
49		breq2 5110	. . . . . 6 ⊢ (𝑥 = (𝐴 − 𝐵) → (𝑁 ∥ 𝑥 ↔ 𝑁 ∥ (𝐴 − 𝐵)))
50	48, 49	elab 3636	. . . . 5 ⊢ ((𝐴 − 𝐵) ∈ {𝑥 ∣ 𝑁 ∥ 𝑥} ↔ 𝑁 ∥ (𝐴 − 𝐵))
51	47, 50	bitrdi 290	. . . 4 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴(-_g‘ℤ_ring)𝐵) ∈ ((RSpan‘ℤ_ring)‘{𝑁}) ↔ 𝑁 ∥ (𝐴 − 𝐵)))
52	30, 35, 51	3bitr2d 310	. . 3 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵(ℤ_ring ~_QG ((RSpan‘ℤ_ring)‘{𝑁}))𝐴 ↔ 𝑁 ∥ (𝐴 − 𝐵)))
53	10, 24, 52	3bitr2d 310	. 2 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐿‘𝐵) = (𝐿‘𝐴) ↔ 𝑁 ∥ (𝐴 − 𝐵)))
54	1, 53	bitrid 286	1 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐿‘𝐴) = (𝐿‘𝐵) ↔ 𝑁 ∥ (𝐴 − 𝐵)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 {cab 2741 ⊆ wss 3900 {csn 4588 class class class wbr 5106 ‘cfv 6558 (class class class)co 7443 Er wer 8742 [cec 8743 − cmin 11568 ℕ₀cn0 12633 ℤcz 12720 ∥ cdvds 16451 -_gcsg 19175 SubGrpcsubg 19359 ~_QG cqg 19361 Abelcabl 20025 Ringcrg 20489 SubRingcsubrg 20851 LIdealclidl 21514 RSpancrsp 21515 ℂ_fldccnfld 21708 ℤ_ringczring 21782 ℤRHomczrh 21835 ℤ/nℤczn 21838

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304 ax-addf 11306 ax-mulf 11307

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-om 7902 df-1st 8025 df-2nd 8026 df-tpos 8271 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-1o 8504 df-er 8745 df-ec 8747 df-qs 8751 df-map 8877 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-sup 9461 df-inf 9462 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-nn 12363 df-2 12432 df-3 12433 df-4 12434 df-5 12435 df-6 12436 df-7 12437 df-8 12438 df-9 12439 df-n0 12634 df-z 12721 df-dec 12842 df-uz 12993 df-fz 13667 df-seq 14172 df-dvds 16452 df-struct 17354 df-sets 17371 df-slot 17389 df-ndx 17401 df-base 17417 df-ress 17438 df-plusg 17470 df-mulr 17471 df-starv 17472 df-sca 17473 df-vsca 17474 df-ip 17475 df-tset 17476 df-ple 17477 df-ds 17479 df-unif 17480 df-0g 17641 df-imas 17709 df-qus 17710 df-mgm 18845 df-sgrp 18937 df-mnd 18953 df-mhm 19007 df-grp 19176 df-minusg 19177 df-sbg 19178 df-mulg 19307 df-subg 19362 df-nsg 19363 df-eqg 19364 df-ghm 19457 df-cmn 20026 df-abl 20027 df-mgp 20391 df-rng 20405 df-ur 20438 df-ring 20491 df-cring 20492 df-oppr 20597 df-dvdsr 20617 df-rhm 20732 df-subrng 20828 df-subrg 20852 df-lmod 21167 df-lss 21237 df-lsp 21277 df-sra 21478 df-rgmod 21479 df-lidl 21516 df-rsp 21517 df-2idl 21573 df-cnfld 21709 df-zring 21783 df-zrh 21839 df-zn 21842

This theorem is used by: zndvds0 21886 znf1o 21887 znunit 21899 cygznlem1 21902 lgsqrlem1 27703 lgsqrlem2 27704 lgsqrlem4 27706 lgsdchrval 27711 lgseisenlem3 27734 lgseisenlem4 27735 dchrisumlem1 27846 dirith 27886 hashscontpow1 43363 aks6d1c2 43372

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