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Theorem ablfacrplem 15227
Description: Lemma for ablfacrp2 15229. (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
StepHypRef Expression
1 nprmdvds1 12717 . . . . . . 7  |-  ( p  e.  Prime  ->  -.  p  ||  1 )
21adantl 454 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  -.  p  ||  1 )
3 ablfacrp.1 . . . . . . . 8  |-  ( ph  ->  ( M  gcd  N
)  =  1 )
43adantr 453 . . . . . . 7  |-  ( (
ph  /\  p  e.  Prime )  ->  ( M  gcd  N )  =  1 )
54breq2d 3975 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  ( p  ||  ( M  gcd  N
)  <->  p  ||  1 ) )
62, 5mtbird 294 . . . . 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 10048 . . . . . . . . . . . . . . 15  |-  ( ph  ->  M  e.  ZZ )
11 ablfacrp.o . . . . . . . . . . . . . . . 16  |-  O  =  ( od `  G
)
12 ablfacrp.b . . . . . . . . . . . . . . . 16  |-  B  =  ( Base `  G
)
1311, 12oddvdssubg 15074 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  Abel  /\  M  e.  ZZ )  ->  { x  e.  B  |  ( O `  x )  ||  M }  e.  (SubGrp `  G ) )
148, 10, 13syl2anc 645 . . . . . . . . . . . . . 14  |-  ( ph  ->  { x  e.  B  |  ( O `  x )  ||  M }  e.  (SubGrp `  G
) )
157, 14syl5eqel 2340 . . . . . . . . . . . . 13  |-  ( ph  ->  K  e.  (SubGrp `  G ) )
1615ad2antrr 709 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  K  e.  (SubGrp `  G )
)
17 eqid 2256 . . . . . . . . . . . . 13  |-  ( Gs  K )  =  ( Gs  K )
1817subggrp 14551 . . . . . . . . . . . 12  |-  ( K  e.  (SubGrp `  G
)  ->  ( Gs  K
)  e.  Grp )
1916, 18syl 17 . . . . . . . . . . 11  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  ( Gs  K )  e.  Grp )
2017subgbas 14552 . . . . . . . . . . . . 13  |-  ( K  e.  (SubGrp `  G
)  ->  K  =  ( Base `  ( Gs  K
) ) )
2116, 20syl 17 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  K  =  ( Base `  ( Gs  K ) ) )
22 ablfacrp.2 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( # `  B
)  =  ( M  x.  N ) )
239nnnn0d 9950 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  M  e.  NN0 )
24 ablfacrp.n . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  N  e.  NN )
2524nnnn0d 9950 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  N  e.  NN0 )
2623, 25nn0mulcld 9955 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( M  x.  N
)  e.  NN0 )
2722, 26eqeltrd 2330 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( # `  B
)  e.  NN0 )
28 fvex 5437 . . . . . . . . . . . . . . . . 17  |-  ( Base `  G )  e.  _V
2912, 28eqeltri 2326 . . . . . . . . . . . . . . . 16  |-  B  e. 
_V
30 hashclb 11283 . . . . . . . . . . . . . . . 16  |-  ( B  e.  _V  ->  ( B  e.  Fin  <->  ( # `  B
)  e.  NN0 )
)
3129, 30ax-mp 10 . . . . . . . . . . . . . . 15  |-  ( B  e.  Fin  <->  ( # `  B
)  e.  NN0 )
3227, 31sylibr 205 . . . . . . . . . . . . . 14  |-  ( ph  ->  B  e.  Fin )
33 ssrab2 3200 . . . . . . . . . . . . . . 15  |-  { x  e.  B  |  ( O `  x )  ||  M }  C_  B
347, 33eqsstri 3150 . . . . . . . . . . . . . 14  |-  K  C_  B
35 ssfi 7016 . . . . . . . . . . . . . 14  |-  ( ( B  e.  Fin  /\  K  C_  B )  ->  K  e.  Fin )
3632, 34, 35sylancl 646 . . . . . . . . . . . . 13  |-  ( ph  ->  K  e.  Fin )
3736ad2antrr 709 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  K  e.  Fin )
3821, 37eqeltrrd 2331 . . . . . . . . . . 11  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  ( Base `  ( Gs  K ) )  e.  Fin )
39 simplr 734 . . . . . . . . . . 11  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  p  e.  Prime )
40 simpr 449 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  p  ||  ( # `  K
) )
4121fveq2d 5427 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  ( # `
 K )  =  ( # `  ( Base `  ( Gs  K ) ) ) )
4240, 41breqtrd 3987 . . . . . . . . . . 11  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  p  ||  ( # `  ( Base `  ( Gs  K ) ) ) )
43 eqid 2256 . . . . . . . . . . . 12  |-  ( Base `  ( Gs  K ) )  =  ( Base `  ( Gs  K ) )
44 eqid 2256 . . . . . . . . . . . 12  |-  ( od
`  ( Gs  K ) )  =  ( od
`  ( Gs  K ) )
4543, 44odcau 14842 . . . . . . . . . . 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 1190 . . . . . . . . . 10  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  E. g  e.  ( Base `  ( Gs  K ) ) ( ( od `  ( Gs  K ) ) `  g )  =  p )
4721rexeqdv 2704 . . . . . . . . . 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 225 . . . . . . . . 9  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  E. g  e.  K  ( ( od `  ( Gs  K ) ) `  g )  =  p )
4917, 11, 44subgod 14808 . . . . . . . . . . . . 13  |-  ( ( K  e.  (SubGrp `  G )  /\  g  e.  K )  ->  ( O `  g )  =  ( ( od
`  ( Gs  K ) ) `  g ) )
5016, 49sylan 459 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  /\  g  e.  K )  ->  ( O `  g )  =  ( ( od
`  ( Gs  K ) ) `  g ) )
51 fveq2 5423 . . . . . . . . . . . . . . . 16  |-  ( x  =  g  ->  ( O `  x )  =  ( O `  g ) )
5251breq1d 3973 . . . . . . . . . . . . . . 15  |-  ( x  =  g  ->  (
( O `  x
)  ||  M  <->  ( O `  g )  ||  M
) )
5352, 7elrab2 2876 . . . . . . . . . . . . . 14  |-  ( g  e.  K  <->  ( g  e.  B  /\  ( O `  g )  ||  M ) )
5453simprbi 452 . . . . . . . . . . . . 13  |-  ( g  e.  K  ->  ( O `  g )  ||  M )
5554adantl 454 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  /\  g  e.  K )  ->  ( O `  g )  ||  M )
5650, 55eqbrtrrd 3985 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  /\  g  e.  K )  ->  (
( od `  ( Gs  K ) ) `  g )  ||  M
)
57 breq1 3966 . . . . . . . . . . 11  |-  ( ( ( od `  ( Gs  K ) ) `  g )  =  p  ->  ( ( ( od `  ( Gs  K ) ) `  g
)  ||  M  <->  p  ||  M
) )
5856, 57syl5ibcom 213 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  /\  g  e.  K )  ->  (
( ( od `  ( Gs  K ) ) `  g )  =  p  ->  p  ||  M
) )
5958rexlimdva 2638 . . . . . . . . 9  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  ( E. g  e.  K  ( ( od `  ( Gs  K ) ) `  g )  =  p  ->  p  ||  M
) )
6048, 59mpd 16 . . . . . . . 8  |-  ( ( ( ph  /\  p  e.  Prime )  /\  p  ||  ( # `  K
) )  ->  p  ||  M )
6160ex 425 . . . . . . 7  |-  ( (
ph  /\  p  e.  Prime )  ->  ( p  ||  ( # `  K
)  ->  p  ||  M
) )
6261anim1d 549 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  ( (
p  ||  ( # `  K
)  /\  p  ||  N
)  ->  ( p  ||  M  /\  p  ||  N ) ) )
63 prmz 12688 . . . . . . . 8  |-  ( p  e.  Prime  ->  p  e.  ZZ )
6463adantl 454 . . . . . . 7  |-  ( (
ph  /\  p  e.  Prime )  ->  p  e.  ZZ )
65 hashcl 11281 . . . . . . . . . 10  |-  ( K  e.  Fin  ->  ( # `
 K )  e. 
NN0 )
6636, 65syl 17 . . . . . . . . 9  |-  ( ph  ->  ( # `  K
)  e.  NN0 )
6766nn0zd 10047 . . . . . . . 8  |-  ( ph  ->  ( # `  K
)  e.  ZZ )
6867adantr 453 . . . . . . 7  |-  ( (
ph  /\  p  e.  Prime )  ->  ( # `  K
)  e.  ZZ )
6924nnzd 10048 . . . . . . . 8  |-  ( ph  ->  N  e.  ZZ )
7069adantr 453 . . . . . . 7  |-  ( (
ph  /\  p  e.  Prime )  ->  N  e.  ZZ )
71 dvdsgcdb 12650 . . . . . . 7  |-  ( ( p  e.  ZZ  /\  ( # `  K )  e.  ZZ  /\  N  e.  ZZ )  ->  (
( p  ||  ( # `
 K )  /\  p  ||  N )  <->  p  ||  (
( # `  K )  gcd  N ) ) )
7264, 68, 70, 71syl3anc 1187 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  ( (
p  ||  ( # `  K
)  /\  p  ||  N
)  <->  p  ||  ( (
# `  K )  gcd  N ) ) )
7310adantr 453 . . . . . . 7  |-  ( (
ph  /\  p  e.  Prime )  ->  M  e.  ZZ )
74 dvdsgcdb 12650 . . . . . . 7  |-  ( ( p  e.  ZZ  /\  M  e.  ZZ  /\  N  e.  ZZ )  ->  (
( p  ||  M  /\  p  ||  N )  <-> 
p  ||  ( M  gcd  N ) ) )
7564, 73, 70, 74syl3anc 1187 . . . . . 6  |-  ( (
ph  /\  p  e.  Prime )  ->  ( (
p  ||  M  /\  p  ||  N )  <->  p  ||  ( M  gcd  N ) ) )
7662, 72, 753imtr3d 260 . . . . 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 2617 . . 3  |-  ( ph  ->  -.  E. p  e. 
Prime  p  ||  ( (
# `  K )  gcd  N ) )
79 exprmfct 12716 . . 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 9723 . . . . . 6  |-  ( ph  ->  N  =/=  0 )
82 simpr 449 . . . . . . 7  |-  ( ( ( # `  K
)  =  0  /\  N  =  0 )  ->  N  =  0 )
8382necon3ai 2459 . . . . . 6  |-  ( N  =/=  0  ->  -.  ( ( # `  K
)  =  0  /\  N  =  0 ) )
8481, 83syl 17 . . . . 5  |-  ( ph  ->  -.  ( ( # `  K )  =  0  /\  N  =  0 ) )
85 gcdn0cl 12620 . . . . 5  |-  ( ( ( ( # `  K
)  e.  ZZ  /\  N  e.  ZZ )  /\  -.  ( ( # `  K )  =  0  /\  N  =  0 ) )  ->  (
( # `  K )  gcd  N )  e.  NN )
8667, 69, 84, 85syl21anc 1186 . . . 4  |-  ( ph  ->  ( ( # `  K
)  gcd  N )  e.  NN )
87 elnn1uz2 10226 . . . 4  |-  ( ( ( # `  K
)  gcd  N )  e.  NN  <->  ( ( (
# `  K )  gcd  N )  =  1  \/  ( ( # `  K )  gcd  N
)  e.  ( ZZ>= ` 
2 ) ) )
8886, 87sylib 190 . . 3  |-  ( ph  ->  ( ( ( # `  K )  gcd  N
)  =  1  \/  ( ( # `  K
)  gcd  N )  e.  ( ZZ>= `  2 )
) )
8988ord 368 . 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 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2419   E.wrex 2517   {crab 2519   _Vcvv 2740    C_ wss 3094   class class class wbr 3963   ` cfv 4638  (class class class)co 5757   Fincfn 6796   0cc0 8670   1c1 8671    x. cmul 8675   NNcn 9679   2c2 9728   NN0cn0 9897   ZZcz 9956   ZZ>=cuz 10162   #chash 11268    || cdivides 12458    gcd cgcd 12612   Primecprime 12685   Basecbs 13075   ↾s cress 13076   Grpcgrp 14289  SubGrpcsubg 14542   odcod 14767   Abelcabel 15017
This theorem is referenced by:  ablfacrp2  15229
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449  ax-inf2 7275  ax-cnex 8726  ax-resscn 8727  ax-1cn 8728  ax-icn 8729  ax-addcl 8730  ax-addrcl 8731  ax-mulcl 8732  ax-mulrcl 8733  ax-mulcom 8734  ax-addass 8735  ax-mulass 8736  ax-distr 8737  ax-i2m1 8738  ax-1ne0 8739  ax-1rid 8740  ax-rnegex 8741  ax-rrecex 8742  ax-cnre 8743  ax-pre-lttri 8744  ax-pre-lttrn 8745  ax-pre-ltadd 8746  ax-pre-mulgt0 8747  ax-pre-sup 8748
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-disj 3935  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-se 4290  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-isom 4655  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-iota 6190  df-riota 6237  df-recs 6321  df-rdg 6356  df-1o 6412  df-2o 6413  df-oadd 6416  df-omul 6417  df-er 6593  df-ec 6595  df-qs 6599  df-map 6707  df-en 6797  df-dom 6798  df-sdom 6799  df-fin 6800  df-sup 7127  df-oi 7158  df-card 7505  df-acn 7508  df-cda 7727  df-pnf 8802  df-mnf 8803  df-xr 8804  df-ltxr 8805  df-le 8806  df-sub 8972  df-neg 8973  df-div 9357  df-n 9680  df-2 9737  df-3 9738  df-n0 9898  df-z 9957  df-uz 10163  df-q 10249  df-rp 10287  df-fz 10714  df-fzo 10802  df-fl 10856  df-mod 10905  df-seq 10978  df-exp 11036  df-fac 11220  df-bc 11247  df-hash 11269  df-cj 11514  df-re 11515  df-im 11516  df-sqr 11650  df-abs 11651  df-clim 11892  df-sum 12089  df-divides 12459  df-gcd 12613  df-prime 12686  df-pc 12817  df-ndx 13078  df-slot 13079  df-base 13080  df-sets 13081  df-ress 13082  df-plusg 13148  df-0g 13331  df-mnd 14294  df-submnd 14343  df-grp 14416  df-minusg 14417  df-sbg 14418  df-mulg 14419  df-subg 14545  df-eqg 14547  df-ga 14671  df-od 14771  df-cmn 15018  df-abl 15019
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