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Theorem zndvds 16785
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 2406 . 2  |-  ( ( L `  A )  =  ( L `  B )  <->  ( L `  B )  =  ( L `  A ) )
2 eqid 2404 . . . . . 6  |-  (flds  ZZ )  =  (flds  ZZ )
3 eqid 2404 . . . . . 6  |-  (RSpan `  (flds  ZZ ) )  =  (RSpan `  (flds  ZZ ) )
4 eqid 2404 . . . . . 6  |-  ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ ) ) `  { N } ) )  =  ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) )
5 zncyg.y . . . . . 6  |-  Y  =  (ℤ/n `  N )
6 zndvds.2 . . . . . 6  |-  L  =  ( ZRHom `  Y
)
72, 3, 4, 5, 6znzrhval 16782 . . . . 5  |-  ( ( N  e.  NN0  /\  B  e.  ZZ )  ->  ( L `  B
)  =  [ B ] ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) )
873adant2 976 . . . 4  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( L `  B )  =  [ B ] ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) )
92, 3, 4, 5, 6znzrhval 16782 . . . . 5  |-  ( ( N  e.  NN0  /\  A  e.  ZZ )  ->  ( L `  A
)  =  [ A ] ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) )
1093adant3 977 . . . 4  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( L `  A )  =  [ A ] ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) )
118, 10eqeq12d 2418 . . 3  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( L `  B
)  =  ( L `
 A )  <->  [ B ] ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) )  =  [ A ] ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) ) )
12 zsubrg 16707 . . . . . . 7  |-  ZZ  e.  (SubRing ` fld )
132subrgrng 15826 . . . . . . 7  |-  ( ZZ  e.  (SubRing ` fld )  ->  (flds  ZZ )  e.  Ring )
1412, 13ax-mp 8 . . . . . 6  |-  (flds  ZZ )  e.  Ring
15 nn0z 10260 . . . . . . . . 9  |-  ( N  e.  NN0  ->  N  e.  ZZ )
16153ad2ant1 978 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  N  e.  ZZ )
1716snssd 3903 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  { N }  C_  ZZ )
182subrgbas 15832 . . . . . . . . 9  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  =  ( Base `  (flds  ZZ ) ) )
1912, 18ax-mp 8 . . . . . . . 8  |-  ZZ  =  ( Base `  (flds  ZZ ) )
20 eqid 2404 . . . . . . . 8  |-  (LIdeal `  (flds  ZZ ) )  =  (LIdeal `  (flds  ZZ ) )
213, 19, 20rspcl 16248 . . . . . . 7  |-  ( ( (flds  ZZ )  e.  Ring  /\  { N }  C_  ZZ )  ->  ( (RSpan `  (flds  ZZ ) ) `  { N } )  e.  (LIdeal `  (flds  ZZ ) ) )
2214, 17, 21sylancr 645 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
(RSpan `  (flds  ZZ ) ) `  { N } )  e.  (LIdeal `  (flds  ZZ ) ) )
2320lidlsubg 16241 . . . . . 6  |-  ( ( (flds  ZZ )  e.  Ring  /\  (
(RSpan `  (flds  ZZ ) ) `  { N } )  e.  (LIdeal `  (flds  ZZ ) ) )  ->  ( (RSpan `  (flds  ZZ ) ) `  { N } )  e.  (SubGrp `  (flds  ZZ ) ) )
2414, 22, 23sylancr 645 . . . . 5  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
(RSpan `  (flds  ZZ ) ) `  { N } )  e.  (SubGrp `  (flds  ZZ ) ) )
2519, 4eqger 14945 . . . . 5  |-  ( ( (RSpan `  (flds  ZZ ) ) `  { N } )  e.  (SubGrp `  (flds  ZZ ) )  -> 
( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) )  Er  ZZ )
2624, 25syl 16 . . . 4  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
(flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ ) ) `  { N } ) )  Er  ZZ )
27 simp3 959 . . . 4  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  B  e.  ZZ )
2826, 27erth 6908 . . 3  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( B ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) A  <->  [ B ] ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ ) ) `  { N } ) )  =  [ A ]
( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) ) )
29 rngabl 15648 . . . . . 6  |-  ( (flds  ZZ )  e.  Ring  ->  (flds  ZZ )  e.  Abel )
3014, 29ax-mp 8 . . . . 5  |-  (flds  ZZ )  e.  Abel
3119, 20lidlss 16235 . . . . . 6  |-  ( ( (RSpan `  (flds  ZZ ) ) `  { N } )  e.  (LIdeal `  (flds  ZZ ) )  -> 
( (RSpan `  (flds  ZZ )
) `  { N } )  C_  ZZ )
3222, 31syl 16 . . . . 5  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
(RSpan `  (flds  ZZ ) ) `  { N } )  C_  ZZ )
33 eqid 2404 . . . . . 6  |-  ( -g `  (flds  ZZ ) )  =  (
-g `  (flds  ZZ ) )
3419, 33, 4eqgabl 15409 . . . . 5  |-  ( ( (flds  ZZ )  e.  Abel  /\  (
(RSpan `  (flds  ZZ ) ) `  { N } )  C_  ZZ )  ->  ( B ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) A  <-> 
( B  e.  ZZ  /\  A  e.  ZZ  /\  ( A ( -g `  (flds  ZZ )
) B )  e.  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) ) )
3530, 32, 34sylancr 645 . . . 4  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( B ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) A  <-> 
( B  e.  ZZ  /\  A  e.  ZZ  /\  ( A ( -g `  (flds  ZZ )
) B )  e.  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) ) )
36 simp2 958 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  A  e.  ZZ )
3727, 36jca 519 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( B  e.  ZZ  /\  A  e.  ZZ ) )
3837biantrurd 495 . . . . 5  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( A ( -g `  (flds  ZZ ) ) B )  e.  ( (RSpan `  (flds  ZZ ) ) `  { N } )  <->  ( ( B  e.  ZZ  /\  A  e.  ZZ )  /\  ( A ( -g `  (flds  ZZ )
) B )  e.  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) ) )
39 df-3an 938 . . . . 5  |-  ( ( B  e.  ZZ  /\  A  e.  ZZ  /\  ( A ( -g `  (flds  ZZ )
) B )  e.  ( (RSpan `  (flds  ZZ )
) `  { N } ) )  <->  ( ( B  e.  ZZ  /\  A  e.  ZZ )  /\  ( A ( -g `  (flds  ZZ )
) B )  e.  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) )
4038, 39syl6bbr 255 . . . 4  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( A ( -g `  (flds  ZZ ) ) B )  e.  ( (RSpan `  (flds  ZZ ) ) `  { N } )  <->  ( B  e.  ZZ  /\  A  e.  ZZ  /\  ( A ( -g `  (flds  ZZ )
) B )  e.  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) ) )
41 subrgsubg 15829 . . . . . . . . 9  |-  ( ZZ  e.  (SubRing ` fld )  ->  ZZ  e.  (SubGrp ` fld ) )
4212, 41mp1i 12 . . . . . . . 8  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ZZ  e.  (SubGrp ` fld ) )
43 cnfldsub 16684 . . . . . . . . 9  |-  -  =  ( -g ` fld )
4443, 2, 33subgsub 14911 . . . . . . . 8  |-  ( ( ZZ  e.  (SubGrp ` fld )  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  -  B )  =  ( A ( -g `  (flds  ZZ )
) B ) )
4542, 44syld3an1 1230 . . . . . . 7  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  -  B )  =  ( A (
-g `  (flds  ZZ ) ) B ) )
4645eqcomd 2409 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A ( -g `  (flds  ZZ )
) B )  =  ( A  -  B
) )
472dvdsrz 16722 . . . . . . . 8  |-  ||  =  ( ||r `
 (flds  ZZ ) )
4819, 3, 47rspsn 16280 . . . . . . 7  |-  ( ( (flds  ZZ )  e.  Ring  /\  N  e.  ZZ )  ->  (
(RSpan `  (flds  ZZ ) ) `  { N } )  =  { x  |  N  ||  x } )
4914, 16, 48sylancr 645 . . . . . 6  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
(RSpan `  (flds  ZZ ) ) `  { N } )  =  { x  |  N  ||  x } )
5046, 49eleq12d 2472 . . . . 5  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( A ( -g `  (flds  ZZ ) ) B )  e.  ( (RSpan `  (flds  ZZ ) ) `  { N } )  <->  ( A  -  B )  e.  {
x  |  N  ||  x } ) )
51 ovex 6065 . . . . . 6  |-  ( A  -  B )  e. 
_V
52 breq2 4176 . . . . . 6  |-  ( x  =  ( A  -  B )  ->  ( N  ||  x  <->  N  ||  ( A  -  B )
) )
5351, 52elab 3042 . . . . 5  |-  ( ( A  -  B )  e.  { x  |  N  ||  x }  <->  N 
||  ( A  -  B ) )
5450, 53syl6bb 253 . . . 4  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( A ( -g `  (flds  ZZ ) ) B )  e.  ( (RSpan `  (flds  ZZ ) ) `  { N } )  <->  N  ||  ( A  -  B )
) )
5535, 40, 543bitr2d 273 . . 3  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( B ( (flds  ZZ ) ~QG  ( (RSpan `  (flds  ZZ )
) `  { N } ) ) A  <-> 
N  ||  ( A  -  B ) ) )
5611, 28, 553bitr2d 273 . 2  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( L `  B
)  =  ( L `
 A )  <->  N  ||  ( A  -  B )
) )
571, 56syl5bb 249 1  |-  ( ( N  e.  NN0  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( L `  A
)  =  ( L `
 B )  <->  N  ||  ( A  -  B )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   {cab 2390    C_ wss 3280   {csn 3774   class class class wbr 4172   ` cfv 5413  (class class class)co 6040    Er wer 6861   [cec 6862    - cmin 9247   NN0cn0 10177   ZZcz 10238    || cdivides 12807   Basecbs 13424   ↾s cress 13425   -gcsg 14643  SubGrpcsubg 14893   ~QG cqg 14895   Abelcabel 15368   Ringcrg 15615  SubRingcsubrg 15819  LIdealclidl 16197  RSpancrsp 16198  ℂfldccnfld 16658   ZRHomczrh 16733  ℤ/nczn 16736
This theorem is referenced by:  zndvds0  16786  znf1o  16787  znunit  16799  cygznlem1  16802  lgsqrlem1  21078  lgsqrlem2  21079  lgsqrlem4  21081  lgsdchrval  21084  lgseisenlem3  21088  lgseisenlem4  21089  dchrisumlem1  21136  dirith  21176
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-ec 6866  df-qs 6870  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-seq 11279  df-dvds 12808  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-0g 13682  df-imas 13689  df-divs 13690  df-mnd 14645  df-mhm 14693  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-nsg 14897  df-eqg 14898  df-ghm 14959  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-rnghom 15774  df-subrg 15821  df-lmod 15907  df-lss 15964  df-lsp 16003  df-sra 16199  df-rgmod 16200  df-lidl 16201  df-rsp 16202  df-2idl 16258  df-cnfld 16659  df-zrh 16737  df-zn 16740
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