Theorem zndvds 14634

Description: Express equality of equivalence classes in ZZ / n ZZ 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 e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( ( 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 2231	. 2 |- ( ( L ` A ) = ( L ` B ) <-> ( L ` B ) = ( L ` A ) )
2		eqid 2229	. . . . . 6 |- (RSpan ` ℤring ) = (RSpan ` ℤring )
3		eqid 2229	. . . . . 6 |- (ℤring ~QG ( (RSpan ` ℤring ) ` { N } ) ) = (ℤring ~QG ( (RSpan ` ℤring ) ` { N } ) )
4		zncyg.y	. . . . . 6 |- Y = (ℤ/nℤ ` N )
5		zndvds.2	. . . . . 6 |- L = ( ZRHom ` Y )
6	2, 3, 4, 5	znzrhval 14632	. . . . 5 |- ( ( N e. NN0 /\ B e. ZZ ) -> ( L ` B ) = [ B ] (ℤring ~QG ( (RSpan ` ℤring ) ` { N } ) ) )
7	6	3adant2 1040	. . . 4 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( L ` B ) = [ B ] (ℤring ~QG ( (RSpan ` ℤring ) ` { N } ) ) )
8	2, 3, 4, 5	znzrhval 14632	. . . . 5 |- ( ( N e. NN0 /\ A e. ZZ ) -> ( L ` A ) = [ A ] (ℤring ~QG ( (RSpan ` ℤring ) ` { N } ) ) )
9	8	3adant3 1041	. . . 4 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( L ` A ) = [ A ] (ℤring ~QG ( (RSpan ` ℤring ) ` { N } ) ) )
10	7, 9	eqeq12d 2244	. . 3 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( ( L ` B ) = ( L ` A ) <-> [ B ] (ℤring ~QG ( (RSpan ` ℤring ) ` { N } ) ) = [ A ] (ℤring ~QG ( (RSpan ` ℤring ) ` { N } ) ) ) )
11		zringring 14578	. . . . . 6 |- ℤring e. Ring
12		nn0z 9482	. . . . . . . . 9 |- ( N e. NN0 -> N e. ZZ )
13	12	3ad2ant1 1042	. . . . . . . 8 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> N e. ZZ )
14	13	snssd 3813	. . . . . . 7 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> { N } C_ ZZ )
15		zringbas 14581	. . . . . . . 8 |- ZZ = ( Base ` ℤring )
16		eqid 2229	. . . . . . . 8 |- (LIdeal ` ℤring ) = (LIdeal ` ℤring )
17	2, 15, 16	rspcl 14476	. . . . . . 7 |- ( (ℤring e. Ring /\ { N } C_ ZZ ) -> ( (RSpan ` ℤring ) ` { N } ) e. (LIdeal ` ℤring ) )
18	11, 14, 17	sylancr 414	. . . . . 6 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( (RSpan ` ℤring ) ` { N } ) e. (LIdeal ` ℤring ) )
19	16	lidlsubg 14471	. . . . . 6 |- ( (ℤring e. Ring /\ ( (RSpan ` ℤring ) ` { N } ) e. (LIdeal ` ℤring ) ) -> ( (RSpan ` ℤring ) ` { N } ) e. (SubGrp ` ℤring ) )
20	11, 18, 19	sylancr 414	. . . . 5 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( (RSpan ` ℤring ) ` { N } ) e. (SubGrp ` ℤring ) )
21	15, 3	eqger 13782	. . . . 5 |- ( ( (RSpan ` ℤring ) ` { N } ) e. (SubGrp ` ℤring ) -> (ℤring ~QG ( (RSpan ` ℤring ) ` { N } ) ) Er ZZ )
22	20, 21	syl 14	. . . 4 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> (ℤring ~QG ( (RSpan ` ℤring ) ` { N } ) ) Er ZZ )
23		simp3 1023	. . . 4 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> B e. ZZ )
24	22, 23	erth 6739	. . 3 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( B (ℤring ~QG ( (RSpan ` ℤring ) ` { N } ) ) A <-> [ B ] (ℤring ~QG ( (RSpan ` ℤring ) ` { N } ) ) = [ A ] (ℤring ~QG ( (RSpan ` ℤring ) ` { N } ) ) ) )
25		zringabl 14579	. . . . 5 |- ℤring e. Abel
26	15, 16	lidlss 14461	. . . . . 6 |- ( ( (RSpan ` ℤring ) ` { N } ) e. (LIdeal ` ℤring ) -> ( (RSpan ` ℤring ) ` { N } ) C_ ZZ )
27	18, 26	syl 14	. . . . 5 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( (RSpan ` ℤring ) ` { N } ) C_ ZZ )
28		eqid 2229	. . . . . 6 |- ( -g ` ℤring ) = ( -g ` ℤring )
29	15, 28, 3	eqgabl 13888	. . . . 5 |- ( (ℤring e. Abel /\ ( (RSpan ` ℤring ) ` { N } ) C_ ZZ ) -> ( B (ℤring ~QG ( (RSpan ` ℤring ) ` { N } ) ) A <-> ( B e. ZZ /\ A e. ZZ /\ ( A ( -g ` ℤring ) B ) e. ( (RSpan ` ℤring ) ` { N } ) ) ) )
30	25, 27, 29	sylancr 414	. . . 4 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( B (ℤring ~QG ( (RSpan ` ℤring ) ` { N } ) ) A <-> ( B e. ZZ /\ A e. ZZ /\ ( A ( -g ` ℤring ) B ) e. ( (RSpan ` ℤring ) ` { N } ) ) ) )
31		simp2 1022	. . . . . . 7 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> A e. ZZ )
32	23, 31	jca 306	. . . . . 6 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( B e. ZZ /\ A e. ZZ ) )
33	32	biantrurd 305	. . . . 5 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( ( A ( -g ` ℤring ) B ) e. ( (RSpan ` ℤring ) ` { N } ) <-> ( ( B e. ZZ /\ A e. ZZ ) /\ ( A ( -g ` ℤring ) B ) e. ( (RSpan ` ℤring ) ` { N } ) ) ) )
34		df-3an 1004	. . . . 5 |- ( ( B e. ZZ /\ A e. ZZ /\ ( A ( -g ` ℤring ) B ) e. ( (RSpan ` ℤring ) ` { N } ) ) <-> ( ( B e. ZZ /\ A e. ZZ ) /\ ( A ( -g ` ℤring ) B ) e. ( (RSpan ` ℤring ) ` { N } ) ) )
35	33, 34	bitr4di 198	. . . 4 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( ( A ( -g ` ℤring ) B ) e. ( (RSpan ` ℤring ) ` { N } ) <-> ( B e. ZZ /\ A e. ZZ /\ ( A ( -g ` ℤring ) B ) e. ( (RSpan ` ℤring ) ` { N } ) ) ) )
36		zsubrg 14566	. . . . . . . . 9 |- ZZ e. (SubRing ` ℂfld )
37		subrgsubg 14212	. . . . . . . . 9 |- ( ZZ e. (SubRing ` ℂfld ) -> ZZ e. (SubGrp ` ℂfld ) )
38	36, 37	mp1i 10	. . . . . . . 8 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ZZ e. (SubGrp ` ℂfld ) )
39		cnfldsub 14560	. . . . . . . . 9 |- - = ( -g ` ℂfld )
40		df-zring 14576	. . . . . . . . 9 |- ℤring = (ℂfld ↾s ZZ )
41	39, 40, 28	subgsub 13744	. . . . . . . 8 |- ( ( ZZ e. (SubGrp ` ℂfld ) /\ A e. ZZ /\ B e. ZZ ) -> ( A - B ) = ( A ( -g ` ℤring ) B ) )
42	38, 41	syld3an1 1317	. . . . . . 7 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( A - B ) = ( A ( -g ` ℤring ) B ) )
43	42	eqcomd 2235	. . . . . 6 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( A ( -g ` ℤring ) B ) = ( A - B ) )
44		dvdsrzring 14588	. . . . . . . 8 |- || = ( ||r ` ℤring )
45	15, 2, 44	rspsn 14519	. . . . . . 7 |- ( (ℤring e. Ring /\ N e. ZZ ) -> ( (RSpan ` ℤring ) ` { N } ) = { x | N || x } )
46	11, 13, 45	sylancr 414	. . . . . 6 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( (RSpan ` ℤring ) ` { N } ) = { x | N || x } )
47	43, 46	eleq12d 2300	. . . . 5 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( ( A ( -g ` ℤring ) B ) e. ( (RSpan ` ℤring ) ` { N } ) <-> ( A - B ) e. { x | N || x } ) )
48	31, 23	zsubcld 9590	. . . . . 6 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( A - B ) e. ZZ )
49		breq2 4087	. . . . . . 7 |- ( x = ( A - B ) -> ( N || x <-> N || ( A - B ) ) )
50	49	elabg 2949	. . . . . 6 |- ( ( A - B ) e. ZZ -> ( ( A - B ) e. { x | N || x } <-> N || ( A - B ) ) )
51	48, 50	syl 14	. . . . 5 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( ( A - B ) e. { x | N || x } <-> N || ( A - B ) ) )
52	47, 51	bitrd 188	. . . 4 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( ( A ( -g ` ℤring ) B ) e. ( (RSpan ` ℤring ) ` { N } ) <-> N || ( A - B ) ) )
53	30, 35, 52	3bitr2d 216	. . 3 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( B (ℤring ~QG ( (RSpan ` ℤring ) ` { N } ) ) A <-> N || ( A - B ) ) )
54	10, 24, 53	3bitr2d 216	. 2 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( ( L ` B ) = ( L ` A ) <-> N || ( A - B ) ) )
55	1, 54	bitrid 192	1 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( ( L ` A ) = ( L ` B ) <-> N || ( A - B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   {cab 2215   C_ wss 3197   {csn 3666   class class class wbr 4083   ` cfv 5321  (class class class)co 6010   Er wer 6690   [cec 6691   - cmin 8333   NN0cn0 9385   ZZcz 9462   || cdvds 12319   -gcsg 13556  SubGrpcsubg 13725   ~_QG cqg 13727   Abelcabl 13843   Ringcrg 13980  SubRingcsubrg 14202  LIdealclidl 14452  RSpancrsp 14453  ℂ_fldccnfld 14541  ℤ_ringczring 14575   ZRHomczrh 14596  ℤ/nℤczn 14598

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-mulrcl 8114  ax-addcom 8115  ax-mulcom 8116  ax-addass 8117  ax-mulass 8118  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-1rid 8122  ax-0id 8123  ax-rnegex 8124  ax-precex 8125  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-apti 8130  ax-pre-ltadd 8131  ax-pre-mulgt0 8132  ax-addf 8137  ax-mulf 8138

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-tp 3674  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4385  df-iord 4458  df-on 4460  df-ilim 4461  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-tpos 6402  df-recs 6462  df-frec 6548  df-er 6693  df-ec 6695  df-qs 6699  df-map 6810  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-reap 8738  df-inn 9127  df-2 9185  df-3 9186  df-4 9187  df-5 9188  df-6 9189  df-7 9190  df-8 9191  df-9 9192  df-n0 9386  df-z 9463  df-dec 9595  df-uz 9739  df-rp 9867  df-fz 10222  df-fzo 10356  df-seqfrec 10687  df-cj 11374  df-abs 11531  df-dvds 12320  df-struct 13055  df-ndx 13056  df-slot 13057  df-base 13059  df-sets 13060  df-iress 13061  df-plusg 13144  df-mulr 13145  df-starv 13146  df-sca 13147  df-vsca 13148  df-ip 13149  df-tset 13150  df-ple 13151  df-ds 13153  df-unif 13154  df-0g 13312  df-topgen 13314  df-iimas 13356  df-qus 13357  df-mgm 13410  df-sgrp 13456  df-mnd 13471  df-mhm 13513  df-grp 13557  df-minusg 13558  df-sbg 13559  df-mulg 13678  df-subg 13728  df-nsg 13729  df-eqg 13730  df-ghm 13799  df-cmn 13844  df-abl 13845  df-mgp 13905  df-rng 13917  df-ur 13944  df-srg 13948  df-ring 13982  df-cring 13983  df-oppr 14052  df-dvdsr 14073  df-rhm 14137  df-subrg 14204  df-lmod 14274  df-lssm 14338  df-lsp 14372  df-sra 14420  df-rgmod 14421  df-lidl 14454  df-rsp 14455  df-2idl 14485  df-bl 14531  df-mopn 14532  df-fg 14534  df-metu 14535  df-cnfld 14542  df-zring 14576  df-zrh 14599  df-zn 14601

This theorem is referenced by:  zndvds0  14635  znf1o  14636  znunit  14644  lgseisenlem3  15772  lgseisenlem4  15773