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Theorem zndvds 17824
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 2435 . 2  |-  ( ( L `  A )  =  ( L `  B )  <->  ( L `  B )  =  ( L `  A ) )
2 eqid 2433 . . . . . 6  |-  (RSpan ` ring )  =  (RSpan ` ring )
3 eqid 2433 . . . . . 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 17821 . . . . 5  |-  ( ( N  e.  NN0  /\  B  e.  ZZ )  ->  ( L `  B
)  =  [ B ] (ring ~QG  (
(RSpan ` ring ) `  { N } ) ) )
763adant2 1000 . . . 4  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( L `  B )  =  [ B ] (ring ~QG  ( (RSpan ` ring ) `  { N } ) ) )
82, 3, 4, 5znzrhval 17821 . . . . 5  |-  ( ( N  e.  NN0  /\  A  e.  ZZ )  ->  ( L `  A
)  =  [ A ] (ring ~QG  (
(RSpan ` ring ) `  { N } ) ) )
983adant3 1001 . . . 4  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( L `  A )  =  [ A ] (ring ~QG  ( (RSpan ` ring ) `  { N } ) ) )
107, 9eqeq12d 2447 . . 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 17728 . . . . . 6  |-ring  e.  Ring
12 nn0z 10657 . . . . . . . . 9  |-  ( N  e.  NN0  ->  N  e.  ZZ )
13123ad2ant1 1002 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  N  e.  ZZ )
1413snssd 4006 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  { N }  C_  ZZ )
15 zringbas 17731 . . . . . . . 8  |-  ZZ  =  ( Base ` ring )
16 eqid 2433 . . . . . . . 8  |-  (LIdeal ` ring )  =  (LIdeal ` ring )
172, 15, 16rspcl 17226 . . . . . . 7  |-  ( (ring  e. 
Ring  /\  { N }  C_  ZZ )  ->  (
(RSpan ` ring ) `  { N } )  e.  (LIdeal ` ring ) )
1811, 14, 17sylancr 656 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
(RSpan ` ring ) `  { N } )  e.  (LIdeal ` ring ) )
1916lidlsubg 17219 . . . . . 6  |-  ( (ring  e. 
Ring  /\  ( (RSpan ` ring ) `  { N } )  e.  (LIdeal ` ring ) )  ->  (
(RSpan ` ring ) `  { N } )  e.  (SubGrp ` ring ) )
2011, 18, 19sylancr 656 . . . . 5  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
(RSpan ` ring ) `  { N } )  e.  (SubGrp ` ring ) )
2115, 3eqger 15711 . . . . 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 983 . . . 4  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  B  e.  ZZ )
2422, 23erth 7133 . . 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 17729 . . . . 5  |-ring  e.  Abel
2615, 16lidlss 17213 . . . . . 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 2433 . . . . . 6  |-  ( -g ` ring )  =  ( -g ` ring )
2915, 28, 3eqgabl 16299 . . . . 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 656 . . . 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 982 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  A  e.  ZZ )
3223, 31jca 529 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( B  e.  ZZ  /\  A  e.  ZZ ) )
3332biantrurd 505 . . . . 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 960 . . . . 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 17710 . . . . . . . . 9  |-  ZZ  e.  (SubRing ` fld )
37 subrgsubg 16795 . . . . . . . . 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 17688 . . . . . . . . 9  |-  -  =  ( -g ` fld )
40 df-zring 17726 . . . . . . . . 9  |-ring  =  (flds  ZZ )
4139, 40, 28subgsub 15673 . . . . . . . 8  |-  ( ( ZZ  e.  (SubGrp ` fld )  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  -  B )  =  ( A ( -g ` ring ) B ) )
4238, 41syld3an1 1257 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  -  B )  =  ( A (
-g ` ring ) B ) )
4342eqcomd 2438 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A ( -g ` ring ) B )  =  ( A  -  B
) )
44 dvdsrzring 17743 . . . . . . . 8  |-  ||  =  ( ||r `
ring )
4515, 2, 44rspsn 17258 . . . . . . 7  |-  ( (ring  e. 
Ring  /\  N  e.  ZZ )  ->  ( (RSpan ` ring ) `  { N } )  =  { x  |  N  ||  x }
)
4611, 13, 45sylancr 656 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
(RSpan ` ring ) `  { N } )  =  {
x  |  N  ||  x } )
4743, 46eleq12d 2501 . . . . 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 6105 . . . . . 6  |-  ( A  -  B )  e. 
_V
49 breq2 4284 . . . . . 6  |-  ( x  =  ( A  -  B )  ->  ( N  ||  x  <->  N  ||  ( A  -  B )
) )
5048, 49elab 3095 . . . . 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 958    = wceq 1362    e. wcel 1755   {cab 2419    C_ wss 3316   {csn 3865   class class class wbr 4280   ` cfv 5406  (class class class)co 6080    Er wer 7086   [cec 7087    - cmin 9583   NN0cn0 10567   ZZcz 10634    || cdivides 13518   -gcsg 15396  SubGrpcsubg 15655   ~QG cqg 15657   Abelcabel 16258   Ringcrg 16577  SubRingcsubrg 16785  LIdealclidl 17173  RSpancrsp 17174  ℂfldccnfld 17662  ℤringzring 17725   ZRHomczrh 17773  ℤ/nczn 17776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-addf 9349  ax-mulf 9350
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-tpos 6734  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-ec 7091  df-qs 7095  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-sup 7679  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-fz 11425  df-seq 11791  df-dvds 13519  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-starv 14236  df-sca 14237  df-vsca 14238  df-ip 14239  df-tset 14240  df-ple 14241  df-ds 14243  df-unif 14244  df-0g 14363  df-imas 14429  df-divs 14430  df-mnd 15398  df-mhm 15447  df-grp 15525  df-minusg 15526  df-sbg 15527  df-mulg 15528  df-subg 15658  df-nsg 15659  df-eqg 15660  df-ghm 15725  df-cmn 16259  df-abl 16260  df-mgp 16566  df-rng 16580  df-cring 16581  df-ur 16582  df-oppr 16649  df-dvdsr 16667  df-rnghom 16740  df-subrg 16787  df-lmod 16874  df-lss 16936  df-lsp 16975  df-sra 17175  df-rgmod 17176  df-lidl 17177  df-rsp 17178  df-2idl 17236  df-cnfld 17663  df-zring 17726  df-zrh 17777  df-zn 17780
This theorem is referenced by:  zndvds0  17825  znf1o  17826  znunit  17838  cygznlem1  17841  lgsqrlem1  22565  lgsqrlem2  22566  lgsqrlem4  22568  lgsdchrval  22571  lgseisenlem3  22575  lgseisenlem4  22576  dchrisumlem1  22623  dirith  22663
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