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Theorem fislw 16246
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 16231 . . . 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 16245 . . 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 15802 . . . . . . . . 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 7645 . . . . . . . 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 7466 . . . . . . . . 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 2454 . . . . . . . . . . . . . . . . . 18  |-  ( Gs  k )  =  ( Gs  k )
2221subggrp 15804 . . . . . . . . . . . . . . . . 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 15805 . . . . . . . . . . . . . . . . . 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 2543 . . . . . . . . . . . . . . . 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 2454 . . . . . . . . . . . . . . . . 17  |-  ( Base `  ( Gs  k ) )  =  ( Base `  ( Gs  k ) )
2827pgpfi 16226 . . . . . . . . . . . . . . . 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 13885 . . . . . . . . . . . . 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 10449 . . . . . . . . . . 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 10476 . . . . . . . . . . 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 2454 . . . . . . . . . . . . . . . . . 18  |-  ( 0g
`  G )  =  ( 0g `  G
)
3736subg0cl 15809 . . . . . . . . . . . . . . . . 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 3752 . . . . . . . . . . . . . . . 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 12252 . . . . . . . . . . . . . . . 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 14036 . . . . . . . . . . . . 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 10857 . . . . . . . . . . . 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 15687 . . . . . . . . . . . . . . . . 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 12252 . . . . . . . . . . . . . . . . 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 14036 . . . . . . . . . . . . . 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 10857 . . . . . . . . . . . 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 15863 . . . . . . . . . . . . . . 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 10858 . . . . . . . . . . . . . . 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 10858 . . . . . . . . . . . . . . 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 14064 . . . . . . . . . . . . . . 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 6208 . . . . . . . . . . . . . . 15  |-  ( p  =  P  ->  (
p  pCnt  ( # `  k
) )  =  ( P  pCnt  ( # `  k
) ) )
64 oveq1 6208 . . . . . . . . . . . . . . 15  |-  ( p  =  P  ->  (
p  pCnt  ( # `  X
) )  =  ( P  pCnt  ( # `  X
) ) )
6563, 64breq12d 4414 . . . . . . . . . . . . . 14  |-  ( p  =  P  ->  (
( p  pCnt  ( # `
 k ) )  <_  ( p  pCnt  (
# `  X )
)  <->  ( P  pCnt  (
# `  k )
)  <_  ( P  pCnt  ( # `  X
) ) ) )
6665rspcv 3175 . . . . . . . . . . . . 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 10979 . . . . . . . . . . . 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 12158 . . . . . . . . . 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 5804 . . . . . . . . . . . . . 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 2456 . . . . . . . . . . . . 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 2868 . . . . . . . . . . . 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 14068 . . . . . . . . . . . 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 4431 . . . . . . . . 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 15802 . . . . . . . . . . . . 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 7645 . . . . . . . . . . . 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 12261 . . . . . . . . . 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 7542 . . . . . . . 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 7636 . . . . . . 7  |-  ( ( k  e.  Fin  /\  H  C_  k  /\  H  ~~  k )  ->  H  =  k )
9216, 17, 90, 91syl3anc 1219 . . . . . 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 2454 . . . . . . . . . . . . 13  |-  ( Gs  H )  =  ( Gs  H )
9594subgbas 15805 . . . . . . . . . . . 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 5804 . . . . . . . . . 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 2497 . . . . . . . . 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 6209 . . . . . . . . . . 11  |-  ( n  =  ( P  pCnt  (
# `  X )
)  ->  ( P ^ n )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) )
101100eqeq2d 2468 . . . . . . . . . 10  |-  ( n  =  ( P  pCnt  (
# `  X )
)  ->  ( ( # `
 ( Base `  ( Gs  H ) ) )  =  ( P ^
n )  <->  ( # `  ( Base `  ( Gs  H ) ) )  =  ( P ^ ( P 
pCnt  ( # `  X
) ) ) ) )
102101rspcev 3179 . . . . . . . . 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 15804 . . . . . . . . . 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 2543 . . . . . . . . 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 2454 . . . . . . . . . 10  |-  ( Base `  ( Gs  H ) )  =  ( Base `  ( Gs  H ) )
108107pgpfi 16226 . . . . . . . . 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 6209 . . . . . . . 8  |-  ( H  =  k  ->  ( Gs  H )  =  ( Gs  k ) )
113112breq2d 4413 . . . . . . 7  |-  ( H  =  k  ->  ( P pGrp  ( Gs  H )  <->  P pGrp  ( Gs  k ) ) )
114 eqimss 3517 . . . . . . . 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 2830 . . 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 16229 . . 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 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   E.wrex 2800    C_ wss 3437   (/)c0 3746   class class class wbr 4401   ` cfv 5527  (class class class)co 6201    ~~ cen 7418    ~<_ cdom 7419   Fincfn 7421    <_ cle 9531   NNcn 10434   NN0cn0 10691   ZZcz 10758   ZZ>=cuz 10973   ^cexp 11983   #chash 12221    || cdivides 13654   Primecprime 13882    pCnt cpc 14022   Basecbs 14293   ↾s cress 14294   0gc0g 14498   Grpcgrp 15530  SubGrpcsubg 15795   pGrp cpgp 16152   pSyl cslw 16153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-inf2 7959  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471  ax-pre-sup 9472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-disj 4372  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-se 4789  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-2o 7032  df-oadd 7035  df-omul 7036  df-er 7212  df-ec 7214  df-qs 7218  df-map 7327  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-sup 7803  df-oi 7836  df-card 8221  df-acn 8224  df-cda 8449  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-div 10106  df-nn 10435  df-2 10492  df-3 10493  df-n0 10692  df-z 10759  df-uz 10974  df-q 11066  df-rp 11104  df-fz 11556  df-fzo 11667  df-fl 11760  df-mod 11827  df-seq 11925  df-exp 11984  df-fac 12170  df-bc 12197  df-hash 12222  df-cj 12707  df-re 12708  df-im 12709  df-sqr 12843  df-abs 12844  df-clim 13085  df-sum 13283  df-dvds 13655  df-gcd 13810  df-prm 13883  df-pc 14023  df-ndx 14296  df-slot 14297  df-base 14298  df-sets 14299  df-ress 14300  df-plusg 14371  df-0g 14500  df-mnd 15535  df-submnd 15585  df-grp 15665  df-minusg 15666  df-sbg 15667  df-mulg 15668  df-subg 15798  df-eqg 15800  df-ghm 15865  df-ga 15928  df-od 16154  df-pgp 16156  df-slw 16157
This theorem is referenced by:  sylow3lem1  16248
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