Theorem zndvds 19898

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 2629	. 2 |- ( ( L ` A ) = ( L ` B ) <-> ( L ` B ) = ( L ` A ) )
2		eqid 2622	. . . . . 6 |- (RSpan ` ℤring ) = (RSpan ` ℤring )
3		eqid 2622	. . . . . 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 19895	. . . . 5 |- ( ( N e. NN0 /\ B e. ZZ ) -> ( L ` B ) = [ B ] (ℤring ~QG ( (RSpan ` ℤring ) ` { N } ) ) )
7	6	3adant2 1080	. . . 4 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( L ` B ) = [ B ] (ℤring ~QG ( (RSpan ` ℤring ) ` { N } ) ) )
8	2, 3, 4, 5	znzrhval 19895	. . . . 5 |- ( ( N e. NN0 /\ A e. ZZ ) -> ( L ` A ) = [ A ] (ℤring ~QG ( (RSpan ` ℤring ) ` { N } ) ) )
9	8	3adant3 1081	. . . 4 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( L ` A ) = [ A ] (ℤring ~QG ( (RSpan ` ℤring ) ` { N } ) ) )
10	7, 9	eqeq12d 2637	. . 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 19821	. . . . . 6 |- ℤring e. Ring
12		nn0z 11400	. . . . . . . . 9 |- ( N e. NN0 -> N e. ZZ )
13	12	3ad2ant1 1082	. . . . . . . 8 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> N e. ZZ )
14	13	snssd 4340	. . . . . . 7 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> { N } C_ ZZ )
15		zringbas 19824	. . . . . . . 8 |- ZZ = ( Base ` ℤring )
16		eqid 2622	. . . . . . . 8 |- (LIdeal ` ℤring ) = (LIdeal ` ℤring )
17	2, 15, 16	rspcl 19222	. . . . . . 7 |- ( (ℤring e. Ring /\ { N } C_ ZZ ) -> ( (RSpan ` ℤring ) ` { N } ) e. (LIdeal ` ℤring ) )
18	11, 14, 17	sylancr 695	. . . . . 6 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( (RSpan ` ℤring ) ` { N } ) e. (LIdeal ` ℤring ) )
19	16	lidlsubg 19215	. . . . . 6 |- ( (ℤring e. Ring /\ ( (RSpan ` ℤring ) ` { N } ) e. (LIdeal ` ℤring ) ) -> ( (RSpan ` ℤring ) ` { N } ) e. (SubGrp ` ℤring ) )
20	11, 18, 19	sylancr 695	. . . . 5 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( (RSpan ` ℤring ) ` { N } ) e. (SubGrp ` ℤring ) )
21	15, 3	eqger 17644	. . . . 5 |- ( ( (RSpan ` ℤring ) ` { N } ) e. (SubGrp ` ℤring ) -> (ℤring ~QG ( (RSpan ` ℤring ) ` { N } ) ) Er ZZ )
22	20, 21	syl 17	. . . 4 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> (ℤring ~QG ( (RSpan ` ℤring ) ` { N } ) ) Er ZZ )
23		simp3 1063	. . . 4 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> B e. ZZ )
24	22, 23	erth 7791	. . 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 19822	. . . . 5 |- ℤring e. Abel
26	15, 16	lidlss 19210	. . . . . 6 |- ( ( (RSpan ` ℤring ) ` { N } ) e. (LIdeal ` ℤring ) -> ( (RSpan ` ℤring ) ` { N } ) C_ ZZ )
27	18, 26	syl 17	. . . . 5 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( (RSpan ` ℤring ) ` { N } ) C_ ZZ )
28		eqid 2622	. . . . . 6 |- ( -g ` ℤring ) = ( -g ` ℤring )
29	15, 28, 3	eqgabl 18240	. . . . 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 695	. . . 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 1062	. . . . . . 7 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> A e. ZZ )
32	23, 31	jca 554	. . . . . 6 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( B e. ZZ /\ A e. ZZ ) )
33	32	biantrurd 529	. . . . 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 1039	. . . . 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	syl6bbr 278	. . . 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 19799	. . . . . . . . 9 |- ZZ e. (SubRing ` ℂfld )
37		subrgsubg 18786	. . . . . . . . 9 |- ( ZZ e. (SubRing ` ℂfld ) -> ZZ e. (SubGrp ` ℂfld ) )
38	36, 37	mp1i 13	. . . . . . . 8 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ZZ e. (SubGrp ` ℂfld ) )
39		cnfldsub 19774	. . . . . . . . 9 |- - = ( -g ` ℂfld )
40		df-zring 19819	. . . . . . . . 9 |- ℤring = (ℂfld ↾s ZZ )
41	39, 40, 28	subgsub 17606	. . . . . . . 8 |- ( ( ZZ e. (SubGrp ` ℂfld ) /\ A e. ZZ /\ B e. ZZ ) -> ( A - B ) = ( A ( -g ` ℤring ) B ) )
42	38, 41	syld3an1 1372	. . . . . . 7 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( A - B ) = ( A ( -g ` ℤring ) B ) )
43	42	eqcomd 2628	. . . . . 6 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( A ( -g ` ℤring ) B ) = ( A - B ) )
44		dvdsrzring 19831	. . . . . . . 8 |- || = ( ||r ` ℤring )
45	15, 2, 44	rspsn 19254	. . . . . . 7 |- ( (ℤring e. Ring /\ N e. ZZ ) -> ( (RSpan ` ℤring ) ` { N } ) = { x | N || x } )
46	11, 13, 45	sylancr 695	. . . . . 6 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( (RSpan ` ℤring ) ` { N } ) = { x | N || x } )
47	43, 46	eleq12d 2695	. . . . 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		ovex 6678	. . . . . 6 |- ( A - B ) e. _V
49		breq2 4657	. . . . . 6 |- ( x = ( A - B ) -> ( N || x <-> N || ( A - B ) ) )
50	48, 49	elab 3350	. . . . 5 |- ( ( A - B ) e. { x | N || x } <-> N || ( A - B ) )
51	47, 50	syl6bb 276	. . . 4 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( ( A ( -g ` ℤring ) B ) e. ( (RSpan ` ℤring ) ` { N } ) <-> N || ( A - B ) ) )
52	30, 35, 51	3bitr2d 296	. . 3 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( B (ℤring ~QG ( (RSpan ` ℤring ) ` { N } ) ) A <-> N || ( A - B ) ) )
53	10, 24, 52	3bitr2d 296	. 2 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( ( L ` B ) = ( L ` A ) <-> N || ( A - B ) ) )
54	1, 53	syl5bb 272	1 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( ( L ` A ) = ( L ` B ) <-> N || ( A - B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   C_ wss 3574   {csn 4177   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Er wer 7739   [cec 7740   - cmin 10266   NN0cn0 11292   ZZcz 11377   || cdvds 14983   -gcsg 17424  SubGrpcsubg 17588   ~_QG cqg 17590   Abelcabl 18194   Ringcrg 18547  SubRingcsubrg 18776  LIdealclidl 19170  RSpancrsp 19171  ℂ_fldccnfld 19746  ℤ_ringzring 19818   ZRHomczrh 19848  ℤ/nℤczn 19851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-seq 12802  df-dvds 14984  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-0g 16102  df-imas 16168  df-qus 16169  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-nsg 17592  df-eqg 17593  df-ghm 17658  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-oppr 18623  df-dvdsr 18641  df-rnghom 18715  df-subrg 18778  df-lmod 18865  df-lss 18933  df-lsp 18972  df-sra 19172  df-rgmod 19173  df-lidl 19174  df-rsp 19175  df-2idl 19232  df-cnfld 19747  df-zring 19819  df-zrh 19852  df-zn 19855

This theorem is referenced by:  zndvds0  19899  znf1o  19900  znunit  19912  cygznlem1  19915  lgsqrlem1  25071  lgsqrlem2  25072  lgsqrlem4  25074  lgsdchrval  25079  lgseisenlem3  25102  lgseisenlem4  25103  dchrisumlem1  25178  dirith  25218