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Theorem zndvds 18764
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 2463 . 2  |-  ( ( L `  A )  =  ( L `  B )  <->  ( L `  B )  =  ( L `  A ) )
2 eqid 2454 . . . . . 6  |-  (RSpan ` ring )  =  (RSpan ` ring )
3 eqid 2454 . . . . . 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 18761 . . . . 5  |-  ( ( N  e.  NN0  /\  B  e.  ZZ )  ->  ( L `  B
)  =  [ B ] (ring ~QG  (
(RSpan ` ring ) `  { N } ) ) )
763adant2 1013 . . . 4  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( L `  B )  =  [ B ] (ring ~QG  ( (RSpan ` ring ) `  { N } ) ) )
82, 3, 4, 5znzrhval 18761 . . . . 5  |-  ( ( N  e.  NN0  /\  A  e.  ZZ )  ->  ( L `  A
)  =  [ A ] (ring ~QG  (
(RSpan ` ring ) `  { N } ) ) )
983adant3 1014 . . . 4  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( L `  A )  =  [ A ] (ring ~QG  ( (RSpan ` ring ) `  { N } ) ) )
107, 9eqeq12d 2476 . . 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 zringring 18689 . . . . . 6  |-ring  e.  Ring
12 nn0z 10883 . . . . . . . . 9  |-  ( N  e.  NN0  ->  N  e.  ZZ )
13123ad2ant1 1015 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  N  e.  ZZ )
1413snssd 4161 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  { N }  C_  ZZ )
15 zringbas 18692 . . . . . . . 8  |-  ZZ  =  ( Base ` ring )
16 eqid 2454 . . . . . . . 8  |-  (LIdeal ` ring )  =  (LIdeal ` ring )
172, 15, 16rspcl 18068 . . . . . . 7  |-  ( (ring  e. 
Ring  /\  { N }  C_  ZZ )  ->  (
(RSpan ` ring ) `  { N } )  e.  (LIdeal ` ring ) )
1811, 14, 17sylancr 661 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
(RSpan ` ring ) `  { N } )  e.  (LIdeal ` ring ) )
1916lidlsubg 18060 . . . . . 6  |-  ( (ring  e. 
Ring  /\  ( (RSpan ` ring ) `  { N } )  e.  (LIdeal ` ring ) )  ->  (
(RSpan ` ring ) `  { N } )  e.  (SubGrp ` ring ) )
2011, 18, 19sylancr 661 . . . . 5  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
(RSpan ` ring ) `  { N } )  e.  (SubGrp ` ring ) )
2115, 3eqger 16453 . . . . 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 996 . . . 4  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  B  e.  ZZ )
2422, 23erth 7348 . . 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 18690 . . . . 5  |-ring  e.  Abel
2615, 16lidlss 18054 . . . . . 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 2454 . . . . . 6  |-  ( -g ` ring )  =  ( -g ` ring )
2915, 28, 3eqgabl 17045 . . . . 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 661 . . . 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 995 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  A  e.  ZZ )
3223, 31jca 530 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( B  e.  ZZ  /\  A  e.  ZZ ) )
3332biantrurd 506 . . . . 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 973 . . . . 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 18669 . . . . . . . . 9  |-  ZZ  e.  (SubRing ` fld )
37 subrgsubg 17633 . . . . . . . . 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 18644 . . . . . . . . 9  |-  -  =  ( -g ` fld )
40 df-zring 18687 . . . . . . . . 9  |-ring  =  (flds  ZZ )
4139, 40, 28subgsub 16415 . . . . . . . 8  |-  ( ( ZZ  e.  (SubGrp ` fld )  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  -  B )  =  ( A ( -g ` ring ) B ) )
4238, 41syld3an1 1272 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  -  B )  =  ( A (
-g ` ring ) B ) )
4342eqcomd 2462 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A ( -g ` ring ) B )  =  ( A  -  B
) )
44 dvdsrzring 18699 . . . . . . . 8  |-  ||  =  ( ||r `
ring )
4515, 2, 44rspsn 18100 . . . . . . 7  |-  ( (ring  e. 
Ring  /\  N  e.  ZZ )  ->  ( (RSpan ` ring ) `  { N } )  =  { x  |  N  ||  x }
)
4611, 13, 45sylancr 661 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
(RSpan ` ring ) `  { N } )  =  {
x  |  N  ||  x } )
4743, 46eleq12d 2536 . . . . 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 6298 . . . . . 6  |-  ( A  -  B )  e. 
_V
49 breq2 4443 . . . . . 6  |-  ( x  =  ( A  -  B )  ->  ( N  ||  x  <->  N  ||  ( A  -  B )
) )
5048, 49elab 3243 . . . . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   {cab 2439    C_ wss 3461   {csn 4016   class class class wbr 4439   ` cfv 5570  (class class class)co 6270    Er wer 7300   [cec 7301    - cmin 9796   NN0cn0 10791   ZZcz 10860    || cdvds 14073   -gcsg 16257  SubGrpcsubg 16397   ~QG cqg 16399   Abelcabl 17001   Ringcrg 17396  SubRingcsubrg 17623  LIdealclidl 18014  RSpancrsp 18015  ℂfldccnfld 18618  ℤringzring 18686   ZRHomczrh 18715  ℤ/nczn 18718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-tpos 6947  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-ec 7305  df-qs 7309  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-fz 11676  df-seq 12093  df-dvds 14074  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-mulr 14801  df-starv 14802  df-sca 14803  df-vsca 14804  df-ip 14805  df-tset 14806  df-ple 14807  df-ds 14809  df-unif 14810  df-0g 14934  df-imas 15000  df-qus 15001  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-mhm 16168  df-grp 16259  df-minusg 16260  df-sbg 16261  df-mulg 16262  df-subg 16400  df-nsg 16401  df-eqg 16402  df-ghm 16467  df-cmn 17002  df-abl 17003  df-mgp 17340  df-ur 17352  df-ring 17398  df-cring 17399  df-oppr 17470  df-dvdsr 17488  df-rnghom 17562  df-subrg 17625  df-lmod 17712  df-lss 17777  df-lsp 17816  df-sra 18016  df-rgmod 18017  df-lidl 18018  df-rsp 18019  df-2idl 18078  df-cnfld 18619  df-zring 18687  df-zrh 18719  df-zn 18722
This theorem is referenced by:  zndvds0  18765  znf1o  18766  znunit  18778  cygznlem1  18781  lgsqrlem1  23817  lgsqrlem2  23818  lgsqrlem4  23820  lgsdchrval  23823  lgseisenlem3  23827  lgseisenlem4  23828  dchrisumlem1  23875  dirith  23915
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