Theorem zndvds 15092

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 15090	. . . . 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 15090	. . . . 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 15036	. . . . . 6 |- ℤring e. Ring
12		nn0z 9673	. . . . . . . . 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 15039	. . . . . . . 8 |- ZZ = ( Base ` ℤring )
16		eqid 2238	. . . . . . . 8 |- (LIdeal ` ℤring ) = (LIdeal ` ℤring )
17	2, 15, 16	rspcl 14936	. . . . . . 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 14931	. . . . . 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 14085	. . . . 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 6858	. . 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 15037	. . . . 5 |- ℤring e. Abel
26	15, 16	lidlss 14921	. . . . . 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 14223	. . . . 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 15026	. . . . . . . . 9 |- ZZ e. (SubRing ` ℂfld )
37		subrgsubg 14643	. . . . . . . . 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 15020	. . . . . . . . 9 |- - = ( -g ` ℂfld )
40		df-zring 15034	. . . . . . . . 9 |- ℤring = (ℂfld ↾s ZZ )
41	39, 40, 28	subgsub 14047	. . . . . . . 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 15046	. . . . . . . 8 |- || = ( ||r ` ℤring )
45	15, 2, 44	rspsn 14979	. . . . . . 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 9783	. . . . . 6 |- ( ( N e. NN0 /\ A e. ZZ /\ B e. ZZ ) -> ( A - B ) e. ZZ )
49		breq2 4135	. . . . . . 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 4131   ` cfv 5380  (class class class)co 6088   Er wer 6809   [cec 6810   - cmin 8504   NN0cn0 9574   ZZcz 9653   || cdvds 12578   -gcsg 13865  SubGrpcsubg 14028   ~_QG cqg 14030   Abelcabl 14177   Ringcrg 14406  SubRingcsubrg 14633  LIdealclidl 14912  RSpancrsp 14913  ℂ_fldccnfld 15001  ℤ_ringczring 15033   ZRHomczrh 15054  ℤ/nℤczn 15056

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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-addf 8307  ax-mulf 8308

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 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-tpos 6521  df-recs 6581  df-frec 6667  df-er 6812  df-ec 6814  df-qs 6818  df-map 6929  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-5 9375  df-6 9376  df-7 9377  df-8 9378  df-9 9379  df-n0 9575  df-z 9654  df-dec 9788  df-uz 9937  df-rp 10071  df-fz 10428  df-fzo 10566  df-seqfrec 10905  df-cj 11628  df-abs 11786  df-dvds 12579  df-struct 13411  df-ndx 13412  df-slot 13413  df-base 13415  df-sets 13416  df-iress 13417  df-plusg 13502  df-mulr 13503  df-starv 13504  df-sca 13505  df-vsca 13506  df-ip 13507  df-tset 13508  df-ple 13509  df-ds 13511  df-unif 13512  df-0g 13670  df-topgen 13672  df-iimas 13682  df-qus 13683  df-mgm 13734  df-sgrp 13775  df-mnd 13788  df-mhm 13824  df-grp 13866  df-minusg 13867  df-sbg 13868  df-mulg 13981  df-subg 14031  df-nsg 14032  df-eqg 14033  df-ghm 14102  df-cmn 14178  df-abl 14179  df-mgp 14324  df-rng 14338  df-ur 14369  df-srg 14374  df-ring 14408  df-cring 14409  df-oppr 14481  df-dvdsr 14503  df-rhm 14567  df-subrg 14635  df-lmod 14733  df-lssm 14798  df-lsp 14832  df-sra 14880  df-rgmod 14881  df-lidl 14914  df-rsp 14915  df-2idl 14945  df-bl 14991  df-mopn 14992  df-fg 14994  df-metu 14995  df-cnfld 15002  df-zring 15034  df-zrh 15057  df-zn 15059

This theorem is used by:  zndvds0  15093  znf1o  15094  znunit  15102  lgseisenlem3  16413  lgseisenlem4  16414