Theorem zndvds 15068

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 2240	. 2 |- ( ( L ` A ) = ( L ` B ) <-> ( L ` B ) = ( L ` A ) )
2		eqid 2238	. . . . . 6 |- (RSpan ` ℤring ) = (RSpan ` ℤring )
3		eqid 2238	. . . . . 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 15066	. . . . 5 |- ( ( N e. NN0 /\ B e. ZZ ) -> ( L ` B ) = [ B ] (ℤring ~QG ( (RSpan ` ℤring ) ` { N } ) ) )
7	6	3adant2 1047	. . . 4 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( L ` B ) = [ B ] (ℤring ~QG ( (RSpan ` ℤring ) ` { N } ) ) )
8	2, 3, 4, 5	znzrhval 15066	. . . . 5 |- ( ( N e. NN0 /\ A e. ZZ ) -> ( L ` A ) = [ A ] (ℤring ~QG ( (RSpan ` ℤring ) ` { N } ) ) )
9	8	3adant3 1048	. . . 4 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( L ` A ) = [ A ] (ℤring ~QG ( (RSpan ` ℤring ) ` { N } ) ) )
10	7, 9	eqeq12d 2253	. . 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 15012	. . . . . 6 |- ℤring e. Ring
12		nn0z 9669	. . . . . . . . 9 |- ( N e. NN0 -> N e. ZZ )
13	12	3ad2ant1 1049	. . . . . . . 8 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> N e. ZZ )
14	13	snssd 3860	. . . . . . 7 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> { N } C_ ZZ )
15		zringbas 15015	. . . . . . . 8 |- ZZ = ( Base ` ℤring )
16		eqid 2238	. . . . . . . 8 |- (LIdeal ` ℤring ) = (LIdeal ` ℤring )
17	2, 15, 16	rspcl 14912	. . . . . . 7 |- ( (ℤring e. Ring /\ { N } C_ ZZ ) -> ( (RSpan ` ℤring ) ` { N } ) e. (LIdeal ` ℤring ) )
18	11, 14, 17	sylancr 418	. . . . . 6 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( (RSpan ` ℤring ) ` { N } ) e. (LIdeal ` ℤring ) )
19	16	lidlsubg 14907	. . . . . 6 |- ( (ℤring e. Ring /\ ( (RSpan ` ℤring ) ` { N } ) e. (LIdeal ` ℤring ) ) -> ( (RSpan ` ℤring ) ` { N } ) e. (SubGrp ` ℤring ) )
20	11, 18, 19	sylancr 418	. . . . 5 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( (RSpan ` ℤring ) ` { N } ) e. (SubGrp ` ℤring ) )
21	15, 3	eqger 14080	. . . . 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 1030	. . . 4 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> B e. ZZ )
24	22, 23	erth 6853	. . 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 15013	. . . . 5 |- ℤring e. Abel
26	15, 16	lidlss 14897	. . . . . 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 2238	. . . . . 6 |- ( -g ` ℤring ) = ( -g ` ℤring )
29	15, 28, 3	eqgabl 14218	. . . . 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 418	. . . 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 1029	. . . . . . 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 1011	. . . . 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 15002	. . . . . . . . 9 |- ZZ e. (SubRing ` ℂfld )
37		subrgsubg 14619	. . . . . . . . 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 14996	. . . . . . . . 9 |- - = ( -g ` ℂfld )
40		df-zring 15010	. . . . . . . . 9 |- ℤring = (ℂfld ↾s ZZ )
41	39, 40, 28	subgsub 14042	. . . . . . . 8 |- ( ( ZZ e. (SubGrp ` ℂfld ) /\ A e. ZZ /\ B e. ZZ ) -> ( A - B ) = ( A ( -g ` ℤring ) B ) )
42	38, 41	syld3an1 1324	. . . . . . 7 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( A - B ) = ( A ( -g ` ℤring ) B ) )
43	42	eqcomd 2244	. . . . . 6 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( A ( -g ` ℤring ) B ) = ( A - B ) )
44		dvdsrzring 15022	. . . . . . . 8 |- || = ( ||r ` ℤring )
45	15, 2, 44	rspsn 14955	. . . . . . 7 |- ( (ℤring e. Ring /\ N e. ZZ ) -> ( (RSpan ` ℤring ) ` { N } ) = { x | N || x } )
46	11, 13, 45	sylancr 418	. . . . . 6 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( (RSpan ` ℤring ) ` { N } ) = { x | N || x } )
47	43, 46	eleq12d 2309	. . . . 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 9778	. . . . . 6 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( A - B ) e. ZZ )
49		breq2 4134	. . . . . . 7 |- ( x = ( A - B ) -> ( N || x <-> N || ( A - B ) ) )
50	49	elabg 2972	. . . . . 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 ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   {cab 2224   C_ wss 3220   {csn 3709   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   Er wer 6804   [cec 6805   - cmin 8499   NN0cn0 9568   ZZcz 9649   || cdvds 12573   -gcsg 13860  SubGrpcsubg 14023   ~_QG cqg 14025   Abelcabl 14172   Ringcrg 14384  SubRingcsubrg 14609  LIdealclidl 14888  RSpancrsp 14889  ℂ_fldccnfld 14977  ℤ_ringczring 15009   ZRHomczrh 15030  ℤ/nℤczn 15032

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-mulrcl 8279  ax-addcom 8280  ax-mulcom 8281  ax-addass 8282  ax-mulass 8283  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-1rid 8287  ax-0id 8288  ax-rnegex 8289  ax-precex 8290  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-apti 8295  ax-pre-ltadd 8296  ax-pre-mulgt0 8297  ax-addf 8302  ax-mulf 8303

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-tp 3717  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-tpos 6516  df-recs 6576  df-frec 6662  df-er 6807  df-ec 6809  df-qs 6813  df-map 6924  df-pnf 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-reap 8906  df-inn 9308  df-2 9366  df-3 9367  df-4 9368  df-5 9369  df-6 9370  df-7 9371  df-8 9372  df-9 9373  df-n0 9569  df-z 9650  df-dec 9783  df-uz 9932  df-rp 10066  df-fz 10423  df-fzo 10561  df-seqfrec 10900  df-cj 11623  df-abs 11781  df-dvds 12574  df-struct 13406  df-ndx 13407  df-slot 13408  df-base 13410  df-sets 13411  df-iress 13412  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-ip 13502  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-0g 13665  df-topgen 13667  df-iimas 13677  df-qus 13678  df-mgm 13729  df-sgrp 13770  df-mnd 13783  df-mhm 13819  df-grp 13861  df-minusg 13862  df-sbg 13863  df-mulg 13976  df-subg 14026  df-nsg 14027  df-eqg 14028  df-ghm 14097  df-cmn 14173  df-abl 14174  df-mgp 14302  df-rng 14316  df-ur 14347  df-srg 14352  df-ring 14386  df-cring 14387  df-oppr 14457  df-dvdsr 14479  df-rhm 14543  df-subrg 14611  df-lmod 14709  df-lssm 14774  df-lsp 14808  df-sra 14856  df-rgmod 14857  df-lidl 14890  df-rsp 14891  df-2idl 14921  df-bl 14967  df-mopn 14968  df-fg 14970  df-metu 14971  df-cnfld 14978  df-zring 15010  df-zrh 15033  df-zn 15035

This theorem is used by:  zndvds0  15069  znf1o  15070  znunit  15078  lgseisenlem3  16362  lgseisenlem4  16363