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Theorem ablfacrplem 15578
Description: Lemma for ablfacrp2 15580. (Contributed by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
ablfacrp.b  |-  B  =  ( Base `  G
)
ablfacrp.o  |-  O  =  ( od `  G
)
ablfacrp.k  |-  K  =  { x  e.  B  |  ( O `  x )  ||  M }
ablfacrp.l  |-  L  =  { x  e.  B  |  ( O `  x )  ||  N }
ablfacrp.g  |-  ( ph  ->  G  e.  Abel )
ablfacrp.m  |-  ( ph  ->  M  e.  NN )
ablfacrp.n  |-  ( ph  ->  N  e.  NN )
ablfacrp.1  |-  ( ph  ->  ( M  gcd  N
)  =  1 )
ablfacrp.2  |-  ( ph  ->  ( # `  B
)  =  ( M  x.  N ) )
Assertion
Ref Expression
ablfacrplem  |-  ( ph  ->  ( ( # `  K
)  gcd  N )  =  1 )
Distinct variable groups:    x, B    x, G    x, O    x, M    x, N    ph, x
Allowed substitution hints:    K( x)    L( x)

Proof of Theorem ablfacrplem
Dummy variables  g  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nprmdvds1 13066 . . . . . . 7  |-  ( p  e.  Prime  ->  -.  p  ||  1 )
21adantl 453 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  -.  p  ||  1 )
3 ablfacrp.1 . . . . . . . 8  |-  ( ph  ->  ( M  gcd  N
)  =  1 )
43adantr 452 . . . . . . 7  |-  ( (
ph  /\  p  e.  Prime )  ->  ( M  gcd  N )  =  1 )
54breq2d 4184 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  ( p  ||  ( M  gcd  N
)  <->  p  ||  1 ) )
62, 5mtbird 293 . . . . 5  |-  ( (
ph  /\  p  e.  Prime )  ->  -.  p  ||  ( M  gcd  N
) )
7 ablfacrp.k . . . . . . . . . . . . . 14  |-  K  =  { x  e.  B  |  ( O `  x )  ||  M }
8 ablfacrp.g . . . . . . . . . . . . . . 15  |-  ( ph  ->  G  e.  Abel )
9 ablfacrp.m . . . . . . . . . . . . . . . 16  |-  ( ph  ->  M  e.  NN )
109nnzd 10330 . . . . . . . . . . . . . . 15  |-  ( ph  ->  M  e.  ZZ )
11 ablfacrp.o . . . . . . . . . . . . . . . 16  |-  O  =  ( od `  G
)
12 ablfacrp.b . . . . . . . . . . . . . . . 16  |-  B  =  ( Base `  G
)
1311, 12oddvdssubg 15425 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  Abel  /\  M  e.  ZZ )  ->  { x  e.  B  |  ( O `  x )  ||  M }  e.  (SubGrp `  G ) )
148, 10, 13syl2anc 643 . . . . . . . . . . . . . 14  |-  ( ph  ->  { x  e.  B  |  ( O `  x )  ||  M }  e.  (SubGrp `  G
) )
157, 14syl5eqel 2488 . . . . . . . . . . . . 13  |-  ( ph  ->  K  e.  (SubGrp `  G ) )
1615ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  K  e.  (SubGrp `  G )
)
17 eqid 2404 . . . . . . . . . . . . 13  |-  ( Gs  K )  =  ( Gs  K )
1817subggrp 14902 . . . . . . . . . . . 12  |-  ( K  e.  (SubGrp `  G
)  ->  ( Gs  K
)  e.  Grp )
1916, 18syl 16 . . . . . . . . . . 11  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  ( Gs  K )  e.  Grp )
2017subgbas 14903 . . . . . . . . . . . . 13  |-  ( K  e.  (SubGrp `  G
)  ->  K  =  ( Base `  ( Gs  K
) ) )
2116, 20syl 16 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  K  =  ( Base `  ( Gs  K ) ) )
22 ablfacrp.2 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( # `  B
)  =  ( M  x.  N ) )
239nnnn0d 10230 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  M  e.  NN0 )
24 ablfacrp.n . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  N  e.  NN )
2524nnnn0d 10230 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  N  e.  NN0 )
2623, 25nn0mulcld 10235 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( M  x.  N
)  e.  NN0 )
2722, 26eqeltrd 2478 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( # `  B
)  e.  NN0 )
28 fvex 5701 . . . . . . . . . . . . . . . . 17  |-  ( Base `  G )  e.  _V
2912, 28eqeltri 2474 . . . . . . . . . . . . . . . 16  |-  B  e. 
_V
30 hashclb 11596 . . . . . . . . . . . . . . . 16  |-  ( B  e.  _V  ->  ( B  e.  Fin  <->  ( # `  B
)  e.  NN0 )
)
3129, 30ax-mp 8 . . . . . . . . . . . . . . 15  |-  ( B  e.  Fin  <->  ( # `  B
)  e.  NN0 )
3227, 31sylibr 204 . . . . . . . . . . . . . 14  |-  ( ph  ->  B  e.  Fin )
33 ssrab2 3388 . . . . . . . . . . . . . . 15  |-  { x  e.  B  |  ( O `  x )  ||  M }  C_  B
347, 33eqsstri 3338 . . . . . . . . . . . . . 14  |-  K  C_  B
35 ssfi 7288 . . . . . . . . . . . . . 14  |-  ( ( B  e.  Fin  /\  K  C_  B )  ->  K  e.  Fin )
3632, 34, 35sylancl 644 . . . . . . . . . . . . 13  |-  ( ph  ->  K  e.  Fin )
3736ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  K  e.  Fin )
3821, 37eqeltrrd 2479 . . . . . . . . . . 11  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  ( Base `  ( Gs  K ) )  e.  Fin )
39 simplr 732 . . . . . . . . . . 11  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  p  e.  Prime )
40 simpr 448 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  p  ||  ( # `  K
) )
4121fveq2d 5691 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  ( # `
 K )  =  ( # `  ( Base `  ( Gs  K ) ) ) )
4240, 41breqtrd 4196 . . . . . . . . . . 11  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  p  ||  ( # `  ( Base `  ( Gs  K ) ) ) )
43 eqid 2404 . . . . . . . . . . . 12  |-  ( Base `  ( Gs  K ) )  =  ( Base `  ( Gs  K ) )
44 eqid 2404 . . . . . . . . . . . 12  |-  ( od
`  ( Gs  K ) )  =  ( od
`  ( Gs  K ) )
4543, 44odcau 15193 . . . . . . . . . . 11  |-  ( ( ( ( Gs  K )  e.  Grp  /\  ( Base `  ( Gs  K ) )  e.  Fin  /\  p  e.  Prime )  /\  p  ||  ( # `  ( Base `  ( Gs  K ) ) ) )  ->  E. g  e.  ( Base `  ( Gs  K ) ) ( ( od
`  ( Gs  K ) ) `  g )  =  p )
4619, 38, 39, 42, 45syl31anc 1187 . . . . . . . . . 10  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  E. g  e.  ( Base `  ( Gs  K ) ) ( ( od `  ( Gs  K ) ) `  g )  =  p )
4721rexeqdv 2871 . . . . . . . . . 10  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  ( E. g  e.  K  ( ( od `  ( Gs  K ) ) `  g )  =  p  <->  E. g  e.  ( Base `  ( Gs  K ) ) ( ( od
`  ( Gs  K ) ) `  g )  =  p ) )
4846, 47mpbird 224 . . . . . . . . 9  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  E. g  e.  K  ( ( od `  ( Gs  K ) ) `  g )  =  p )
4917, 11, 44subgod 15159 . . . . . . . . . . . . 13  |-  ( ( K  e.  (SubGrp `  G )  /\  g  e.  K )  ->  ( O `  g )  =  ( ( od
`  ( Gs  K ) ) `  g ) )
5016, 49sylan 458 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  /\  g  e.  K )  ->  ( O `  g )  =  ( ( od
`  ( Gs  K ) ) `  g ) )
51 fveq2 5687 . . . . . . . . . . . . . . . 16  |-  ( x  =  g  ->  ( O `  x )  =  ( O `  g ) )
5251breq1d 4182 . . . . . . . . . . . . . . 15  |-  ( x  =  g  ->  (
( O `  x
)  ||  M  <->  ( O `  g )  ||  M
) )
5352, 7elrab2 3054 . . . . . . . . . . . . . 14  |-  ( g  e.  K  <->  ( g  e.  B  /\  ( O `  g )  ||  M ) )
5453simprbi 451 . . . . . . . . . . . . 13  |-  ( g  e.  K  ->  ( O `  g )  ||  M )
5554adantl 453 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  /\  g  e.  K )  ->  ( O `  g )  ||  M )
5650, 55eqbrtrrd 4194 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  /\  g  e.  K )  ->  (
( od `  ( Gs  K ) ) `  g )  ||  M
)
57 breq1 4175 . . . . . . . . . . 11  |-  ( ( ( od `  ( Gs  K ) ) `  g )  =  p  ->  ( ( ( od `  ( Gs  K ) ) `  g
)  ||  M  <->  p  ||  M
) )
5856, 57syl5ibcom 212 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  /\  g  e.  K )  ->  (
( ( od `  ( Gs  K ) ) `  g )  =  p  ->  p  ||  M
) )
5958rexlimdva 2790 . . . . . . . . 9  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  ( E. g  e.  K  ( ( od `  ( Gs  K ) ) `  g )  =  p  ->  p  ||  M
) )
6048, 59mpd 15 . . . . . . . 8  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  p  ||  M )
6160ex 424 . . . . . . 7  |-  ( (
ph  /\  p  e.  Prime )  ->  ( p  ||  ( # `  K
)  ->  p  ||  M
) )
6261anim1d 548 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  ( (
p  ||  ( # `  K
)  /\  p  ||  N
)  ->  ( p  ||  M  /\  p  ||  N ) ) )
63 prmz 13038 . . . . . . . 8  |-  ( p  e.  Prime  ->  p  e.  ZZ )
6463adantl 453 . . . . . . 7  |-  ( (
ph  /\  p  e.  Prime )  ->  p  e.  ZZ )
65 hashcl 11594 . . . . . . . . . 10  |-  ( K  e.  Fin  ->  ( # `
 K )  e. 
NN0 )
6636, 65syl 16 . . . . . . . . 9  |-  ( ph  ->  ( # `  K
)  e.  NN0 )
6766nn0zd 10329 . . . . . . . 8  |-  ( ph  ->  ( # `  K
)  e.  ZZ )
6867adantr 452 . . . . . . 7  |-  ( (
ph  /\  p  e.  Prime )  ->  ( # `  K
)  e.  ZZ )
6924nnzd 10330 . . . . . . . 8  |-  ( ph  ->  N  e.  ZZ )
7069adantr 452 . . . . . . 7  |-  ( (
ph  /\  p  e.  Prime )  ->  N  e.  ZZ )
71 dvdsgcdb 12999 . . . . . . 7  |-  ( ( p  e.  ZZ  /\  ( # `  K )  e.  ZZ  /\  N  e.  ZZ )  ->  (
( p  ||  ( # `
 K )  /\  p  ||  N )  <->  p  ||  (
( # `  K )  gcd  N ) ) )
7264, 68, 70, 71syl3anc 1184 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  ( (
p  ||  ( # `  K
)  /\  p  ||  N
)  <->  p  ||  ( (
# `  K )  gcd  N ) ) )
7310adantr 452 . . . . . . 7  |-  ( (
ph  /\  p  e.  Prime )  ->  M  e.  ZZ )
74 dvdsgcdb 12999 . . . . . . 7  |-  ( ( p  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( p  ||  M  /\  p  ||  N )  <-> 
p  ||  ( M  gcd  N ) ) )
7564, 73, 70, 74syl3anc 1184 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  ( (
p  ||  M  /\  p  ||  N )  <->  p  ||  ( M  gcd  N ) ) )
7662, 72, 753imtr3d 259 . . . . 5  |-  ( (
ph  /\  p  e.  Prime )  ->  ( p  ||  ( ( # `  K
)  gcd  N )  ->  p  ||  ( M  gcd  N ) ) )
776, 76mtod 170 . . . 4  |-  ( (
ph  /\  p  e.  Prime )  ->  -.  p  ||  ( ( # `  K
)  gcd  N )
)
7877nrexdv 2769 . . 3  |-  ( ph  ->  -.  E. p  e. 
Prime  p  ||  ( (
# `  K )  gcd  N ) )
79 exprmfct 13065 . . 3  |-  ( ( ( # `  K
)  gcd  N )  e.  ( ZZ>= `  2 )  ->  E. p  e.  Prime  p 
||  ( ( # `  K )  gcd  N
) )
8078, 79nsyl 115 . 2  |-  ( ph  ->  -.  ( ( # `  K )  gcd  N
)  e.  ( ZZ>= ` 
2 ) )
8124nnne0d 10000 . . . . . 6  |-  ( ph  ->  N  =/=  0 )
82 simpr 448 . . . . . . 7  |-  ( ( ( # `  K
)  =  0  /\  N  =  0 )  ->  N  =  0 )
8382necon3ai 2607 . . . . . 6  |-  ( N  =/=  0  ->  -.  ( ( # `  K
)  =  0  /\  N  =  0 ) )
8481, 83syl 16 . . . . 5  |-  ( ph  ->  -.  ( ( # `  K )  =  0  /\  N  =  0 ) )
85 gcdn0cl 12969 . . . . 5  |-  ( ( ( ( # `  K
)  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( ( # `  K )  =  0  /\  N  =  0 ) )  ->  (
( # `  K )  gcd  N )  e.  NN )
8667, 69, 84, 85syl21anc 1183 . . . 4  |-  ( ph  ->  ( ( # `  K
)  gcd  N )  e.  NN )
87 elnn1uz2 10508 . . . 4  |-  ( ( ( # `  K
)  gcd  N )  e.  NN  <->  ( ( (
# `  K )  gcd  N )  =  1  \/  ( ( # `  K )  gcd  N
)  e.  ( ZZ>= ` 
2 ) ) )
8886, 87sylib 189 . . 3  |-  ( ph  ->  ( ( ( # `  K )  gcd  N
)  =  1  \/  ( ( # `  K
)  gcd  N )  e.  ( ZZ>= `  2 )
) )
8988ord 367 . 2  |-  ( ph  ->  ( -.  ( (
# `  K )  gcd  N )  =  1  ->  ( ( # `  K )  gcd  N
)  e.  ( ZZ>= ` 
2 ) ) )
9080, 89mt3d 119 1  |-  ( ph  ->  ( ( # `  K
)  gcd  N )  =  1 )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667   {crab 2670   _Vcvv 2916    C_ wss 3280   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Fincfn 7068   0cc0 8946   1c1 8947    x. cmul 8951   NNcn 9956   2c2 10005   NN0cn0 10177   ZZcz 10238   ZZ>=cuz 10444   #chash 11573    || cdivides 12807    gcd cgcd 12961   Primecprime 13034   Basecbs 13424   ↾s cress 13425   Grpcgrp 14640  SubGrpcsubg 14893   odcod 15118   Abelcabel 15368
This theorem is referenced by:  ablfacrp2  15580
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-pre-sup 9024
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-disj 4143  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-se 4502  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-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-omul 6688  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-oi 7435  df-card 7782  df-acn 7785  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435  df-dvds 12808  df-gcd 12962  df-prm 13035  df-pc 13166  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-0g 13682  df-mnd 14645  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-eqg 14898  df-ga 15022  df-od 15122  df-cmn 15369  df-abl 15370
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