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Theorem zndvds 18091
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 2460 . 2  |-  ( ( L `  A )  =  ( L `  B )  <->  ( L `  B )  =  ( L `  A ) )
2 eqid 2451 . . . . . 6  |-  (RSpan ` ring )  =  (RSpan ` ring )
3 eqid 2451 . . . . . 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 18088 . . . . 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 18088 . . . . 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 2473 . . 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 17995 . . . . . 6  |-ring  e.  Ring
12 nn0z 10770 . . . . . . . . 9  |-  ( N  e.  NN0  ->  N  e.  ZZ )
13123ad2ant1 1009 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  N  e.  ZZ )
1413snssd 4116 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  { N }  C_  ZZ )
15 zringbas 17998 . . . . . . . 8  |-  ZZ  =  ( Base ` ring )
16 eqid 2451 . . . . . . . 8  |-  (LIdeal ` ring )  =  (LIdeal ` ring )
172, 15, 16rspcl 17410 . . . . . . 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 17403 . . . . . 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 15833 . . . . 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 7245 . . 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 17996 . . . . 5  |-ring  e.  Abel
2615, 16lidlss 17397 . . . . . 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 2451 . . . . . 6  |-  ( -g ` ring )  =  ( -g ` ring )
2915, 28, 3eqgabl 16423 . . . . 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 17975 . . . . . . . . 9  |-  ZZ  e.  (SubRing ` fld )
37 subrgsubg 16977 . . . . . . . . 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 17953 . . . . . . . . 9  |-  -  =  ( -g ` fld )
40 df-zring 17993 . . . . . . . . 9  |-ring  =  (flds  ZZ )
4139, 40, 28subgsub 15795 . . . . . . . 8  |-  ( ( ZZ  e.  (SubGrp ` fld )  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  -  B )  =  ( A ( -g ` ring ) B ) )
4238, 41syld3an1 1265 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  -  B )  =  ( A (
-g ` ring ) B ) )
4342eqcomd 2459 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A ( -g ` ring ) B )  =  ( A  -  B
) )
44 dvdsrzring 18010 . . . . . . . 8  |-  ||  =  ( ||r `
ring )
4515, 2, 44rspsn 17442 . . . . . . 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 2533 . . . . 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 6215 . . . . . 6  |-  ( A  -  B )  e. 
_V
49 breq2 4394 . . . . . 6  |-  ( x  =  ( A  -  B )  ->  ( N  ||  x  <->  N  ||  ( A  -  B )
) )
5048, 49elab 3203 . . . . 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 1370    e. wcel 1758   {cab 2436    C_ wss 3426   {csn 3975   class class class wbr 4390   ` cfv 5516  (class class class)co 6190    Er wer 7198   [cec 7199    - cmin 9696   NN0cn0 10680   ZZcz 10747    || cdivides 13637   -gcsg 15515  SubGrpcsubg 15777   ~QG cqg 15779   Abelcabel 16382   Ringcrg 16751  SubRingcsubrg 16967  LIdealclidl 17357  RSpancrsp 17358  ℂfldccnfld 17927  ℤringzring 17992   ZRHomczrh 18040  ℤ/nczn 18043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-addf 9462  ax-mulf 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-om 6577  df-1st 6677  df-2nd 6678  df-tpos 6845  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-er 7201  df-ec 7203  df-qs 7207  df-map 7316  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-sup 7792  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-3 10482  df-4 10483  df-5 10484  df-6 10485  df-7 10486  df-8 10487  df-9 10488  df-10 10489  df-n0 10681  df-z 10748  df-dec 10857  df-uz 10963  df-fz 11539  df-seq 11908  df-dvds 13638  df-struct 14278  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-ress 14283  df-plusg 14353  df-mulr 14354  df-starv 14355  df-sca 14356  df-vsca 14357  df-ip 14358  df-tset 14359  df-ple 14360  df-ds 14362  df-unif 14363  df-0g 14482  df-imas 14548  df-divs 14549  df-mnd 15517  df-mhm 15566  df-grp 15647  df-minusg 15648  df-sbg 15649  df-mulg 15650  df-subg 15780  df-nsg 15781  df-eqg 15782  df-ghm 15847  df-cmn 16383  df-abl 16384  df-mgp 16697  df-ur 16709  df-rng 16753  df-cring 16754  df-oppr 16821  df-dvdsr 16839  df-rnghom 16912  df-subrg 16969  df-lmod 17056  df-lss 17120  df-lsp 17159  df-sra 17359  df-rgmod 17360  df-lidl 17361  df-rsp 17362  df-2idl 17420  df-cnfld 17928  df-zring 17993  df-zrh 18044  df-zn 18047
This theorem is referenced by:  zndvds0  18092  znf1o  18093  znunit  18105  cygznlem1  18108  lgsqrlem1  22796  lgsqrlem2  22797  lgsqrlem4  22799  lgsdchrval  22802  lgseisenlem3  22806  lgseisenlem4  22807  dchrisumlem1  22854  dirith  22894
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