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Theorem fislw 16115
Description: The sylow subgroups of a finite group are exactly the groups which have cardinality equal to the maximum power of  P dividing the group. (Contributed by Mario Carneiro, 16-Jan-2015.)
Hypothesis
Ref Expression
fislw.1  |-  X  =  ( Base `  G
)
Assertion
Ref Expression
fislw  |-  ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  ->  ( H  e.  ( P pSyl  G )  <->  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) ) )

Proof of Theorem fislw
Dummy variables  k  n  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 461 . . . 4  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  H  e.  ( P pSyl  G ) )  ->  H  e.  ( P pSyl  G ) )
2 slwsubg 16100 . . . 4  |-  ( H  e.  ( P pSyl  G
)  ->  H  e.  (SubGrp `  G ) )
31, 2syl 16 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  H  e.  ( P pSyl  G ) )  ->  H  e.  (SubGrp `  G )
)
4 fislw.1 . . . 4  |-  X  =  ( Base `  G
)
5 simpl2 992 . . . 4  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  H  e.  ( P pSyl  G ) )  ->  X  e.  Fin )
64, 5, 1slwhash 16114 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  H  e.  ( P pSyl  G ) )  ->  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) )
73, 6jca 532 . 2  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  H  e.  ( P pSyl  G ) )  ->  ( H  e.  (SubGrp `  G
)  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )
8 simpl3 993 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  P  e.  Prime )
9 simprl 755 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  H  e.  (SubGrp `  G ) )
10 simpl2 992 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  X  e.  Fin )
1110adantr 465 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  X  e.  Fin )
12 simprl 755 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  k  e.  (SubGrp `  G ) )
134subgss 15673 . . . . . . . . 9  |-  ( k  e.  (SubGrp `  G
)  ->  k  C_  X )
1412, 13syl 16 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  k  C_  X
)
15 ssfi 7525 . . . . . . . 8  |-  ( ( X  e.  Fin  /\  k  C_  X )  -> 
k  e.  Fin )
1611, 14, 15syl2anc 661 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  k  e.  Fin )
17 simprrl 763 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  H  C_  k
)
18 ssdomg 7347 . . . . . . . . 9  |-  ( k  e.  Fin  ->  ( H  C_  k  ->  H  ~<_  k ) )
1916, 17, 18sylc 60 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  H  ~<_  k )
20 simprrr 764 . . . . . . . . . . . . . . 15  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  P pGrp  ( Gs  k
) )
21 eqid 2438 . . . . . . . . . . . . . . . . . 18  |-  ( Gs  k )  =  ( Gs  k )
2221subggrp 15675 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  (SubGrp `  G
)  ->  ( Gs  k
)  e.  Grp )
2312, 22syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  ( Gs  k )  e.  Grp )
2421subgbas 15676 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  (SubGrp `  G
)  ->  k  =  ( Base `  ( Gs  k
) ) )
2512, 24syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  k  =  (
Base `  ( Gs  k
) ) )
2625, 16eqeltrrd 2513 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  ( Base `  ( Gs  k ) )  e. 
Fin )
27 eqid 2438 . . . . . . . . . . . . . . . . 17  |-  ( Base `  ( Gs  k ) )  =  ( Base `  ( Gs  k ) )
2827pgpfi 16095 . . . . . . . . . . . . . . . 16  |-  ( ( ( Gs  k )  e. 
Grp  /\  ( Base `  ( Gs  k ) )  e.  Fin )  -> 
( P pGrp  ( Gs  k
)  <->  ( P  e. 
Prime  /\  E. n  e. 
NN0  ( # `  ( Base `  ( Gs  k ) ) )  =  ( P ^ n ) ) ) )
2923, 26, 28syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  ( P pGrp  ( Gs  k )  <->  ( P  e.  Prime  /\  E. n  e.  NN0  ( # `  ( Base `  ( Gs  k ) ) )  =  ( P ^ n ) ) ) )
3020, 29mpbid 210 . . . . . . . . . . . . . 14  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  ( P  e. 
Prime  /\  E. n  e. 
NN0  ( # `  ( Base `  ( Gs  k ) ) )  =  ( P ^ n ) ) )
3130simpld 459 . . . . . . . . . . . . 13  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  P  e.  Prime )
32 prmnn 13758 . . . . . . . . . . . . 13  |-  ( P  e.  Prime  ->  P  e.  NN )
3331, 32syl 16 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  P  e.  NN )
3433nnred 10329 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  P  e.  RR )
3533nnge1d 10356 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  1  <_  P
)
36 eqid 2438 . . . . . . . . . . . . . . . . . 18  |-  ( 0g
`  G )  =  ( 0g `  G
)
3736subg0cl 15680 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  k )
3812, 37syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  ( 0g `  G )  e.  k )
39 ne0i 3638 . . . . . . . . . . . . . . . 16  |-  ( ( 0g `  G )  e.  k  ->  k  =/=  (/) )
4038, 39syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  k  =/=  (/) )
41 hashnncl 12126 . . . . . . . . . . . . . . . 16  |-  ( k  e.  Fin  ->  (
( # `  k )  e.  NN  <->  k  =/=  (/) ) )
4216, 41syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  ( ( # `  k )  e.  NN  <->  k  =/=  (/) ) )
4340, 42mpbird 232 . . . . . . . . . . . . . 14  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  ( # `  k
)  e.  NN )
4431, 43pccld 13909 . . . . . . . . . . . . 13  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  ( P  pCnt  (
# `  k )
)  e.  NN0 )
4544nn0zd 10737 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  ( P  pCnt  (
# `  k )
)  e.  ZZ )
46 simpl1 991 . . . . . . . . . . . . . . . . 17  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  G  e.  Grp )
474grpbn0 15558 . . . . . . . . . . . . . . . . 17  |-  ( G  e.  Grp  ->  X  =/=  (/) )
4846, 47syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  X  =/=  (/) )
49 hashnncl 12126 . . . . . . . . . . . . . . . . 17  |-  ( X  e.  Fin  ->  (
( # `  X )  e.  NN  <->  X  =/=  (/) ) )
5010, 49syl 16 . . . . . . . . . . . . . . . 16  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  ( ( # `
 X )  e.  NN  <->  X  =/=  (/) ) )
5148, 50mpbird 232 . . . . . . . . . . . . . . 15  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  ( # `  X
)  e.  NN )
528, 51pccld 13909 . . . . . . . . . . . . . 14  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  ( P  pCnt  ( # `  X
) )  e.  NN0 )
5352adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  ( P  pCnt  (
# `  X )
)  e.  NN0 )
5453nn0zd 10737 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  ( P  pCnt  (
# `  X )
)  e.  ZZ )
554lagsubg 15734 . . . . . . . . . . . . . . 15  |-  ( ( k  e.  (SubGrp `  G )  /\  X  e.  Fin )  ->  ( # `
 k )  ||  ( # `  X ) )
5612, 11, 55syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  ( # `  k
)  ||  ( # `  X
) )
5743nnzd 10738 . . . . . . . . . . . . . . 15  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  ( # `  k
)  e.  ZZ )
5851adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  ( # `  X
)  e.  NN )
5958nnzd 10738 . . . . . . . . . . . . . . 15  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  ( # `  X
)  e.  ZZ )
60 pc2dvds 13937 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  k
)  e.  ZZ  /\  ( # `  X )  e.  ZZ )  -> 
( ( # `  k
)  ||  ( # `  X
)  <->  A. p  e.  Prime  ( p  pCnt  ( # `  k
) )  <_  (
p  pCnt  ( # `  X
) ) ) )
6157, 59, 60syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  ( ( # `  k )  ||  ( # `
 X )  <->  A. p  e.  Prime  ( p  pCnt  (
# `  k )
)  <_  ( p  pCnt  ( # `  X
) ) ) )
6256, 61mpbid 210 . . . . . . . . . . . . 13  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  A. p  e.  Prime  ( p  pCnt  ( # `  k
) )  <_  (
p  pCnt  ( # `  X
) ) )
63 oveq1 6093 . . . . . . . . . . . . . . 15  |-  ( p  =  P  ->  (
p  pCnt  ( # `  k
) )  =  ( P  pCnt  ( # `  k
) ) )
64 oveq1 6093 . . . . . . . . . . . . . . 15  |-  ( p  =  P  ->  (
p  pCnt  ( # `  X
) )  =  ( P  pCnt  ( # `  X
) ) )
6563, 64breq12d 4300 . . . . . . . . . . . . . 14  |-  ( p  =  P  ->  (
( p  pCnt  ( # `
 k ) )  <_  ( p  pCnt  (
# `  X )
)  <->  ( P  pCnt  (
# `  k )
)  <_  ( P  pCnt  ( # `  X
) ) ) )
6665rspcv 3064 . . . . . . . . . . . . 13  |-  ( P  e.  Prime  ->  ( A. p  e.  Prime  ( p 
pCnt  ( # `  k
) )  <_  (
p  pCnt  ( # `  X
) )  ->  ( P  pCnt  ( # `  k
) )  <_  ( P  pCnt  ( # `  X
) ) ) )
6731, 62, 66sylc 60 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  ( P  pCnt  (
# `  k )
)  <_  ( P  pCnt  ( # `  X
) ) )
68 eluz2 10859 . . . . . . . . . . . 12  |-  ( ( P  pCnt  ( # `  X
) )  e.  (
ZZ>= `  ( P  pCnt  (
# `  k )
) )  <->  ( ( P  pCnt  ( # `  k
) )  e.  ZZ  /\  ( P  pCnt  ( # `
 X ) )  e.  ZZ  /\  ( P  pCnt  ( # `  k
) )  <_  ( P  pCnt  ( # `  X
) ) ) )
6945, 54, 67, 68syl3anbrc 1172 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  ( P  pCnt  (
# `  X )
)  e.  ( ZZ>= `  ( P  pCnt  ( # `  k ) ) ) )
7034, 35, 69leexp2ad 12032 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  ( P ^
( P  pCnt  ( # `
 k ) ) )  <_  ( P ^ ( P  pCnt  (
# `  X )
) ) )
7130simprd 463 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  E. n  e.  NN0  ( # `  ( Base `  ( Gs  k ) ) )  =  ( P ^ n ) )
7225fveq2d 5690 . . . . . . . . . . . . . 14  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  ( # `  k
)  =  ( # `  ( Base `  ( Gs  k ) ) ) )
7372eqeq1d 2446 . . . . . . . . . . . . 13  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  ( ( # `  k )  =  ( P ^ n )  <-> 
( # `  ( Base `  ( Gs  k ) ) )  =  ( P ^ n ) ) )
7473rexbidv 2731 . . . . . . . . . . . 12  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  ( E. n  e.  NN0  ( # `  k
)  =  ( P ^ n )  <->  E. n  e.  NN0  ( # `  ( Base `  ( Gs  k ) ) )  =  ( P ^ n ) ) )
7571, 74mpbird 232 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  E. n  e.  NN0  ( # `  k )  =  ( P ^
n ) )
76 pcprmpw 13941 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  ( # `
 k )  e.  NN )  ->  ( E. n  e.  NN0  ( # `  k )  =  ( P ^
n )  <->  ( # `  k
)  =  ( P ^ ( P  pCnt  (
# `  k )
) ) ) )
7731, 43, 76syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  ( E. n  e.  NN0  ( # `  k
)  =  ( P ^ n )  <->  ( # `  k
)  =  ( P ^ ( P  pCnt  (
# `  k )
) ) ) )
7875, 77mpbid 210 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  ( # `  k
)  =  ( P ^ ( P  pCnt  (
# `  k )
) ) )
79 simplrr 760 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
8070, 78, 793brtr4d 4317 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  ( # `  k
)  <_  ( # `  H
) )
814subgss 15673 . . . . . . . . . . . . 13  |-  ( H  e.  (SubGrp `  G
)  ->  H  C_  X
)
8281ad2antrl 727 . . . . . . . . . . . 12  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  H  C_  X
)
83 ssfi 7525 . . . . . . . . . . . 12  |-  ( ( X  e.  Fin  /\  H  C_  X )  ->  H  e.  Fin )
8410, 82, 83syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  H  e.  Fin )
8584adantr 465 . . . . . . . . . 10  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  H  e.  Fin )
86 hashdom 12134 . . . . . . . . . 10  |-  ( ( k  e.  Fin  /\  H  e.  Fin )  ->  ( ( # `  k
)  <_  ( # `  H
)  <->  k  ~<_  H ) )
8716, 85, 86syl2anc 661 . . . . . . . . 9  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  ( ( # `  k )  <_  ( # `
 H )  <->  k  ~<_  H ) )
8880, 87mpbid 210 . . . . . . . 8  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  k  ~<_  H )
89 sbth 7423 . . . . . . . 8  |-  ( ( H  ~<_  k  /\  k  ~<_  H )  ->  H  ~~  k )
9019, 88, 89syl2anc 661 . . . . . . 7  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  H  ~~  k
)
91 fisseneq 7516 . . . . . . 7  |-  ( ( k  e.  Fin  /\  H  C_  k  /\  H  ~~  k )  ->  H  =  k )
9216, 17, 90, 91syl3anc 1218 . . . . . 6  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  ( k  e.  (SubGrp `  G )  /\  ( H  C_  k  /\  P pGrp  ( Gs  k
) ) ) )  ->  H  =  k )
9392expr 615 . . . . 5  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  k  e.  (SubGrp `  G ) )  -> 
( ( H  C_  k  /\  P pGrp  ( Gs  k ) )  ->  H  =  k ) )
94 eqid 2438 . . . . . . . . . . . . 13  |-  ( Gs  H )  =  ( Gs  H )
9594subgbas 15676 . . . . . . . . . . . 12  |-  ( H  e.  (SubGrp `  G
)  ->  H  =  ( Base `  ( Gs  H
) ) )
9695ad2antrl 727 . . . . . . . . . . 11  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  H  =  ( Base `  ( Gs  H
) ) )
9796fveq2d 5690 . . . . . . . . . 10  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  ( # `  H
)  =  ( # `  ( Base `  ( Gs  H ) ) ) )
98 simprr 756 . . . . . . . . . 10  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
9997, 98eqtr3d 2472 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  ( # `  ( Base `  ( Gs  H ) ) )  =  ( P ^ ( P 
pCnt  ( # `  X
) ) ) )
100 oveq2 6094 . . . . . . . . . . 11  |-  ( n  =  ( P  pCnt  (
# `  X )
)  ->  ( P ^ n )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) )
101100eqeq2d 2449 . . . . . . . . . 10  |-  ( n  =  ( P  pCnt  (
# `  X )
)  ->  ( ( # `
 ( Base `  ( Gs  H ) ) )  =  ( P ^
n )  <->  ( # `  ( Base `  ( Gs  H ) ) )  =  ( P ^ ( P 
pCnt  ( # `  X
) ) ) ) )
102101rspcev 3068 . . . . . . . . 9  |-  ( ( ( P  pCnt  ( # `
 X ) )  e.  NN0  /\  ( # `
 ( Base `  ( Gs  H ) ) )  =  ( P ^
( P  pCnt  ( # `
 X ) ) ) )  ->  E. n  e.  NN0  ( # `  ( Base `  ( Gs  H ) ) )  =  ( P ^ n ) )
10352, 99, 102syl2anc 661 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  E. n  e.  NN0  ( # `  ( Base `  ( Gs  H ) ) )  =  ( P ^ n ) )
10494subggrp 15675 . . . . . . . . . 10  |-  ( H  e.  (SubGrp `  G
)  ->  ( Gs  H
)  e.  Grp )
105104ad2antrl 727 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  ( Gs  H
)  e.  Grp )
10696, 84eqeltrrd 2513 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  ( Base `  ( Gs  H ) )  e. 
Fin )
107 eqid 2438 . . . . . . . . . 10  |-  ( Base `  ( Gs  H ) )  =  ( Base `  ( Gs  H ) )
108107pgpfi 16095 . . . . . . . . 9  |-  ( ( ( Gs  H )  e.  Grp  /\  ( Base `  ( Gs  H ) )  e. 
Fin )  ->  ( P pGrp  ( Gs  H )  <->  ( P  e.  Prime  /\  E. n  e.  NN0  ( # `  ( Base `  ( Gs  H ) ) )  =  ( P ^ n ) ) ) )
109105, 106, 108syl2anc 661 . . . . . . . 8  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  ( P pGrp  ( Gs  H )  <->  ( P  e.  Prime  /\  E. n  e.  NN0  ( # `  ( Base `  ( Gs  H ) ) )  =  ( P ^ n ) ) ) )
1108, 103, 109mpbir2and 913 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  P pGrp  ( Gs  H ) )
111110adantr 465 . . . . . 6  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  k  e.  (SubGrp `  G ) )  ->  P pGrp  ( Gs  H ) )
112 oveq2 6094 . . . . . . . 8  |-  ( H  =  k  ->  ( Gs  H )  =  ( Gs  k ) )
113112breq2d 4299 . . . . . . 7  |-  ( H  =  k  ->  ( P pGrp  ( Gs  H )  <->  P pGrp  ( Gs  k ) ) )
114 eqimss 3403 . . . . . . . 8  |-  ( H  =  k  ->  H  C_  k )
115114biantrurd 508 . . . . . . 7  |-  ( H  =  k  ->  ( P pGrp  ( Gs  k )  <->  ( H  C_  k  /\  P pGrp  ( Gs  k ) ) ) )
116113, 115bitrd 253 . . . . . 6  |-  ( H  =  k  ->  ( P pGrp  ( Gs  H )  <->  ( H  C_  k  /\  P pGrp  ( Gs  k ) ) ) )
117111, 116syl5ibcom 220 . . . . 5  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  k  e.  (SubGrp `  G ) )  -> 
( H  =  k  ->  ( H  C_  k  /\  P pGrp  ( Gs  k ) ) ) )
11893, 117impbid 191 . . . 4  |-  ( ( ( ( G  e. 
Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  /\  k  e.  (SubGrp `  G ) )  -> 
( ( H  C_  k  /\  P pGrp  ( Gs  k ) )  <->  H  =  k ) )
119118ralrimiva 2794 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  A. k  e.  (SubGrp `  G )
( ( H  C_  k  /\  P pGrp  ( Gs  k ) )  <->  H  =  k ) )
120 isslw 16098 . . 3  |-  ( H  e.  ( P pSyl  G
)  <->  ( P  e. 
Prime  /\  H  e.  (SubGrp `  G )  /\  A. k  e.  (SubGrp `  G
) ( ( H 
C_  k  /\  P pGrp  ( Gs  k ) )  <-> 
H  =  k ) ) )
1218, 9, 119, 120syl3anbrc 1172 . 2  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  H  e.  ( P pSyl  G )
)
1227, 121impbida 828 1  |-  ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  ->  ( H  e.  ( P pSyl  G )  <->  ( H  e.  (SubGrp `  G )  /\  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601   A.wral 2710   E.wrex 2711    C_ wss 3323   (/)c0 3632   class class class wbr 4287   ` cfv 5413  (class class class)co 6086    ~~ cen 7299    ~<_ cdom 7300   Fincfn 7302    <_ cle 9411   NNcn 10314   NN0cn0 10571   ZZcz 10638   ZZ>=cuz 10853   ^cexp 11857   #chash 12095    || cdivides 13527   Primecprime 13755    pCnt cpc 13895   Basecbs 14166   ↾s cress 14167   0gc0g 14370   Grpcgrp 15402  SubGrpcsubg 15666   pGrp cpgp 16021   pSyl cslw 16022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-disj 4258  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  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-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-omul 6917  df-er 7093  df-ec 7095  df-qs 7099  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-oi 7716  df-card 8101  df-acn 8104  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-q 10946  df-rp 10984  df-fz 11430  df-fzo 11541  df-fl 11634  df-mod 11701  df-seq 11799  df-exp 11858  df-fac 12044  df-bc 12071  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-sum 13156  df-dvds 13528  df-gcd 13683  df-prm 13756  df-pc 13896  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-0g 14372  df-mnd 15407  df-submnd 15457  df-grp 15536  df-minusg 15537  df-sbg 15538  df-mulg 15539  df-subg 15669  df-eqg 15671  df-ghm 15736  df-ga 15799  df-od 16023  df-pgp 16025  df-slw 16026
This theorem is referenced by:  sylow3lem1  16117
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