MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  zndvds Structured version   Unicode version

Theorem zndvds 17962
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/n `  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.
StepHypRef Expression
1 eqcom 2440 . 2  |-  ( ( L `  A )  =  ( L `  B )  <->  ( L `  B )  =  ( L `  A ) )
2 eqid 2438 . . . . . 6  |-  (RSpan ` ring )  =  (RSpan ` ring )
3 eqid 2438 . . . . . 6  |-  (ring ~QG  ( (RSpan ` ring ) `  { N } ) )  =  (ring ~QG  (
(RSpan ` ring ) `  { N } ) )
4 zncyg.y . . . . . 6  |-  Y  =  (ℤ/n `  N )
5 zndvds.2 . . . . . 6  |-  L  =  ( ZRHom `  Y
)
62, 3, 4, 5znzrhval 17959 . . . . 5  |-  ( ( N  e.  NN0  /\  B  e.  ZZ )  ->  ( L `  B
)  =  [ B ] (ring ~QG  (
(RSpan ` ring ) `  { N } ) ) )
763adant2 1007 . . . 4  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( L `  B )  =  [ B ] (ring ~QG  ( (RSpan ` ring ) `  { N } ) ) )
82, 3, 4, 5znzrhval 17959 . . . . 5  |-  ( ( N  e.  NN0  /\  A  e.  ZZ )  ->  ( L `  A
)  =  [ A ] (ring ~QG  (
(RSpan ` ring ) `  { N } ) ) )
983adant3 1008 . . . 4  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( L `  A )  =  [ A ] (ring ~QG  ( (RSpan ` ring ) `  { N } ) ) )
107, 9eqeq12d 2452 . . 3  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( L `  B
)  =  ( L `
 A )  <->  [ B ] (ring ~QG  (
(RSpan ` ring ) `  { N } ) )  =  [ A ] (ring ~QG  ( (RSpan ` ring ) `  { N } ) ) ) )
11 zringrng 17866 . . . . . 6  |-ring  e.  Ring
12 nn0z 10661 . . . . . . . . 9  |-  ( N  e.  NN0  ->  N  e.  ZZ )
13123ad2ant1 1009 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  N  e.  ZZ )
1413snssd 4013 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  { N }  C_  ZZ )
15 zringbas 17869 . . . . . . . 8  |-  ZZ  =  ( Base ` ring )
16 eqid 2438 . . . . . . . 8  |-  (LIdeal ` ring )  =  (LIdeal ` ring )
172, 15, 16rspcl 17284 . . . . . . 7  |-  ( (ring  e. 
Ring  /\  { N }  C_  ZZ )  ->  (
(RSpan ` ring ) `  { N } )  e.  (LIdeal ` ring ) )
1811, 14, 17sylancr 663 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
(RSpan ` ring ) `  { N } )  e.  (LIdeal ` ring ) )
1916lidlsubg 17277 . . . . . 6  |-  ( (ring  e. 
Ring  /\  ( (RSpan ` ring ) `  { N } )  e.  (LIdeal ` ring ) )  ->  (
(RSpan ` ring ) `  { N } )  e.  (SubGrp ` ring ) )
2011, 18, 19sylancr 663 . . . . 5  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
(RSpan ` ring ) `  { N } )  e.  (SubGrp ` ring ) )
2115, 3eqger 15722 . . . . 5  |-  ( ( (RSpan ` ring ) `  { N } )  e.  (SubGrp ` ring )  ->  (ring ~QG  (
(RSpan ` ring ) `  { N } ) )  Er  ZZ )
2220, 21syl 16 . . . 4  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (ring ~QG  ( (RSpan ` ring ) `  { N } ) )  Er  ZZ )
23 simp3 990 . . . 4  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  B  e.  ZZ )
2422, 23erth 7137 . . 3  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( B (ring ~QG  (
(RSpan ` ring ) `  { N } ) ) A  <->  [ B ] (ring ~QG  ( (RSpan ` ring ) `  { N } ) )  =  [ A ] (ring ~QG  ( (RSpan ` ring ) `  { N } ) ) ) )
25 zringabl 17867 . . . . 5  |-ring  e.  Abel
2615, 16lidlss 17271 . . . . . 6  |-  ( ( (RSpan ` ring ) `  { N } )  e.  (LIdeal ` ring )  ->  ( (RSpan ` ring ) `  { N } ) 
C_  ZZ )
2718, 26syl 16 . . . . 5  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
(RSpan ` ring ) `  { N } )  C_  ZZ )
28 eqid 2438 . . . . . 6  |-  ( -g ` ring )  =  ( -g ` ring )
2915, 28, 3eqgabl 16310 . . . . 5  |-  ( (ring  e. 
Abel  /\  ( (RSpan ` ring ) `  { N } ) 
C_  ZZ )  -> 
( B (ring ~QG  ( (RSpan ` ring ) `  { N } ) ) A  <-> 
( B  e.  ZZ  /\  A  e.  ZZ  /\  ( A ( -g ` ring ) B )  e.  ( (RSpan ` ring ) `  { N } ) ) ) )
3025, 27, 29sylancr 663 . . . 4  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( B (ring ~QG  (
(RSpan ` ring ) `  { N } ) ) A  <-> 
( B  e.  ZZ  /\  A  e.  ZZ  /\  ( A ( -g ` ring ) B )  e.  ( (RSpan ` ring ) `  { N } ) ) ) )
31 simp2 989 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  A  e.  ZZ )
3223, 31jca 532 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( B  e.  ZZ  /\  A  e.  ZZ ) )
3332biantrurd 508 . . . . 5  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( A ( -g ` ring ) B )  e.  ( (RSpan ` ring ) `  { N } )  <->  ( ( B  e.  ZZ  /\  A  e.  ZZ )  /\  ( A ( -g ` ring ) B )  e.  ( (RSpan ` ring ) `  { N } ) ) ) )
34 df-3an 967 . . . . 5  |-  ( ( B  e.  ZZ  /\  A  e.  ZZ  /\  ( A ( -g ` ring ) B )  e.  ( (RSpan ` ring ) `  { N } ) )  <->  ( ( B  e.  ZZ  /\  A  e.  ZZ )  /\  ( A ( -g ` ring ) B )  e.  ( (RSpan ` ring ) `  { N } ) ) )
3533, 34syl6bbr 263 . . . 4  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( A ( -g ` ring ) B )  e.  ( (RSpan ` ring ) `  { N } )  <->  ( B  e.  ZZ  /\  A  e.  ZZ  /\  ( A ( -g ` ring ) B )  e.  ( (RSpan ` ring ) `  { N } ) ) ) )
36 zsubrg 17846 . . . . . . . . 9  |-  ZZ  e.  (SubRing ` fld )
37 subrgsubg 16851 . . . . . . . . 9  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  e.  (SubGrp ` fld ) )
3836, 37mp1i 12 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ZZ  e.  (SubGrp ` fld ) )
39 cnfldsub 17824 . . . . . . . . 9  |-  -  =  ( -g ` fld )
40 df-zring 17864 . . . . . . . . 9  |-ring  =  (flds  ZZ )
4139, 40, 28subgsub 15684 . . . . . . . 8  |-  ( ( ZZ  e.  (SubGrp ` fld )  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  -  B )  =  ( A ( -g ` ring ) B ) )
4238, 41syld3an1 1264 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  -  B )  =  ( A (
-g ` ring ) B ) )
4342eqcomd 2443 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A ( -g ` ring ) B )  =  ( A  -  B
) )
44 dvdsrzring 17881 . . . . . . . 8  |-  ||  =  ( ||r `
ring )
4515, 2, 44rspsn 17316 . . . . . . 7  |-  ( (ring  e. 
Ring  /\  N  e.  ZZ )  ->  ( (RSpan ` ring ) `  { N } )  =  { x  |  N  ||  x }
)
4611, 13, 45sylancr 663 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
(RSpan ` ring ) `  { N } )  =  {
x  |  N  ||  x } )
4743, 46eleq12d 2506 . . . . 5  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( A ( -g ` ring ) B )  e.  ( (RSpan ` ring ) `  { N } )  <->  ( A  -  B )  e.  {
x  |  N  ||  x } ) )
48 ovex 6111 . . . . . 6  |-  ( A  -  B )  e. 
_V
49 breq2 4291 . . . . . 6  |-  ( x  =  ( A  -  B )  ->  ( N  ||  x  <->  N  ||  ( A  -  B )
) )
5048, 49elab 3101 . . . . 5  |-  ( ( A  -  B )  e.  { x  |  N  ||  x }  <->  N 
||  ( A  -  B ) )
5147, 50syl6bb 261 . . . 4  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( A ( -g ` ring ) B )  e.  ( (RSpan ` ring ) `  { N } )  <->  N  ||  ( A  -  B )
) )
5230, 35, 513bitr2d 281 . . 3  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( B (ring ~QG  (
(RSpan ` ring ) `  { N } ) ) A  <-> 
N  ||  ( A  -  B ) ) )
5310, 24, 523bitr2d 281 . 2  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( L `  B
)  =  ( L `
 A )  <->  N  ||  ( A  -  B )
) )
541, 53syl5bb 257 1  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( L `  A
)  =  ( L `
 B )  <->  N  ||  ( A  -  B )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   {cab 2424    C_ wss 3323   {csn 3872   class class class wbr 4287   ` cfv 5413  (class class class)co 6086    Er wer 7090   [cec 7091    - cmin 9587   NN0cn0 10571   ZZcz 10638    || cdivides 13527   -gcsg 15405  SubGrpcsubg 15666   ~QG cqg 15668   Abelcabel 16269   Ringcrg 16635  SubRingcsubrg 16841  LIdealclidl 17231  RSpancrsp 17232  ℂfldccnfld 17798  ℤringzring 17863   ZRHomczrh 17911  ℤ/nczn 17914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-addf 9353  ax-mulf 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-tpos 6740  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-ec 7095  df-qs 7099  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-fz 11430  df-seq 11799  df-dvds 13528  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-0g 14372  df-imas 14438  df-divs 14439  df-mnd 15407  df-mhm 15456  df-grp 15536  df-minusg 15537  df-sbg 15538  df-mulg 15539  df-subg 15669  df-nsg 15670  df-eqg 15671  df-ghm 15736  df-cmn 16270  df-abl 16271  df-mgp 16582  df-ur 16594  df-rng 16637  df-cring 16638  df-oppr 16705  df-dvdsr 16723  df-rnghom 16796  df-subrg 16843  df-lmod 16930  df-lss 16994  df-lsp 17033  df-sra 17233  df-rgmod 17234  df-lidl 17235  df-rsp 17236  df-2idl 17294  df-cnfld 17799  df-zring 17864  df-zrh 17915  df-zn 17918
This theorem is referenced by:  zndvds0  17963  znf1o  17964  znunit  17976  cygznlem1  17979  lgsqrlem1  22660  lgsqrlem2  22661  lgsqrlem4  22663  lgsdchrval  22666  lgseisenlem3  22670  lgseisenlem4  22671  dchrisumlem1  22718  dirith  22758
  Copyright terms: Public domain W3C validator