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Theorem fislw 14936
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 447 . . . 4  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  H  e.  ( P pSyl  G ) )  ->  H  e.  ( P pSyl  G ) )
2 slwsubg 14921 . . . 4  |-  ( H  e.  ( P pSyl  G
)  ->  H  e.  (SubGrp `  G ) )
31, 2syl 15 . . 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 959 . . . 4  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  H  e.  ( P pSyl  G ) )  ->  X  e.  Fin )
64, 5, 1slwhash 14935 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  H  e.  ( P pSyl  G ) )  ->  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) )
73, 6jca 518 . 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 960 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  P  e.  Prime )
9 simprl 732 . . 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 959 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  X  e.  Fin )
1110adantr 451 . . . . . . . 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 732 . . . . . . . . 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 14622 . . . . . . . . 9  |-  ( k  e.  (SubGrp `  G
)  ->  k  C_  X )
1412, 13syl 15 . . . . . . . 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 7083 . . . . . . . 8  |-  ( ( X  e.  Fin  /\  k  C_  X )  -> 
k  e.  Fin )
1611, 14, 15syl2anc 642 . . . . . . 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 740 . . . . . . 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 6907 . . . . . . . . 9  |-  ( k  e.  Fin  ->  ( H  C_  k  ->  H  ~<_  k ) )
1916, 17, 18sylc 56 . . . . . . . 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 741 . . . . . . . . . . . . . . 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 2283 . . . . . . . . . . . . . . . . . 18  |-  ( Gs  k )  =  ( Gs  k )
2221subggrp 14624 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  (SubGrp `  G
)  ->  ( Gs  k
)  e.  Grp )
2312, 22syl 15 . . . . . . . . . . . . . . . 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 14625 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  (SubGrp `  G
)  ->  k  =  ( Base `  ( Gs  k
) ) )
2512, 24syl 15 . . . . . . . . . . . . . . . . 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 2358 . . . . . . . . . . . . . . . 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 2283 . . . . . . . . . . . . . . . . 17  |-  ( Base `  ( Gs  k ) )  =  ( Base `  ( Gs  k ) )
2827pgpfi 14916 . . . . . . . . . . . . . . . 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 642 . . . . . . . . . . . . . . 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 201 . . . . . . . . . . . . . 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 445 . . . . . . . . . . . . 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 12761 . . . . . . . . . . . . 13  |-  ( P  e.  Prime  ->  P  e.  NN )
3331, 32syl 15 . . . . . . . . . . . 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 9761 . . . . . . . . . . 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 9788 . . . . . . . . . . 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 2283 . . . . . . . . . . . . . . . . . 18  |-  ( 0g
`  G )  =  ( 0g `  G
)
3736subg0cl 14629 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  k )
3812, 37syl 15 . . . . . . . . . . . . . . . 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 3461 . . . . . . . . . . . . . . . 16  |-  ( ( 0g `  G )  e.  k  ->  k  =/=  (/) )
4038, 39syl 15 . . . . . . . . . . . . . . 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 11354 . . . . . . . . . . . . . . . 16  |-  ( k  e.  Fin  ->  (
( # `  k )  e.  NN  <->  k  =/=  (/) ) )
4216, 41syl 15 . . . . . . . . . . . . . . 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 223 . . . . . . . . . . . . . 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 12903 . . . . . . . . . . . . 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 10115 . . . . . . . . . . . 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 958 . . . . . . . . . . . . . . . . 17  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  G  e.  Grp )
474grpbn0 14511 . . . . . . . . . . . . . . . . 17  |-  ( G  e.  Grp  ->  X  =/=  (/) )
4846, 47syl 15 . . . . . . . . . . . . . . . 16  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  X  =/=  (/) )
49 hashnncl 11354 . . . . . . . . . . . . . . . . 17  |-  ( X  e.  Fin  ->  (
( # `  X )  e.  NN  <->  X  =/=  (/) ) )
5010, 49syl 15 . . . . . . . . . . . . . . . 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 223 . . . . . . . . . . . . . . 15  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  ( # `  X
)  e.  NN )
528, 51pccld 12903 . . . . . . . . . . . . . 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 451 . . . . . . . . . . . . 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 10115 . . . . . . . . . . . 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 14679 . . . . . . . . . . . . . . 15  |-  ( ( k  e.  (SubGrp `  G )  /\  X  e.  Fin )  ->  ( # `
 k )  ||  ( # `  X ) )
5612, 11, 55syl2anc 642 . . . . . . . . . . . . . 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 10116 . . . . . . . . . . . . . . 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 451 . . . . . . . . . . . . . . . 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 10116 . . . . . . . . . . . . . . 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 12931 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  k
)  e.  ZZ  /\  ( # `  X )  e.  ZZ )  -> 
( ( # `  k
)  ||  ( # `  X
)  <->  A. p  e.  Prime  ( p  pCnt  ( # `  k
) )  <_  (
p  pCnt  ( # `  X
) ) ) )
6157, 59, 60syl2anc 642 . . . . . . . . . . . . . 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 201 . . . . . . . . . . . . 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 5865 . . . . . . . . . . . . . . 15  |-  ( p  =  P  ->  (
p  pCnt  ( # `  k
) )  =  ( P  pCnt  ( # `  k
) ) )
64 oveq1 5865 . . . . . . . . . . . . . . 15  |-  ( p  =  P  ->  (
p  pCnt  ( # `  X
) )  =  ( P  pCnt  ( # `  X
) ) )
6563, 64breq12d 4036 . . . . . . . . . . . . . 14  |-  ( p  =  P  ->  (
( p  pCnt  ( # `
 k ) )  <_  ( p  pCnt  (
# `  X )
)  <->  ( P  pCnt  (
# `  k )
)  <_  ( P  pCnt  ( # `  X
) ) ) )
6665rspcv 2880 . . . . . . . . . . . . 13  |-  ( P  e.  Prime  ->  ( A. p  e.  Prime  ( p 
pCnt  ( # `  k
) )  <_  (
p  pCnt  ( # `  X
) )  ->  ( P  pCnt  ( # `  k
) )  <_  ( P  pCnt  ( # `  X
) ) ) )
6731, 62, 66sylc 56 . . . . . . . . . . . 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 10236 . . . . . . . . . . . 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 1136 . . . . . . . . . . 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 11277 . . . . . . . . . 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 449 . . . . . . . . . . . 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 5529 . . . . . . . . . . . . . 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 2291 . . . . . . . . . . . . 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 2564 . . . . . . . . . . . 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 223 . . . . . . . . . . 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 12935 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  ( # `
 k )  e.  NN )  ->  ( E. n  e.  NN0  ( # `  k )  =  ( P ^
n )  <->  ( # `  k
)  =  ( P ^ ( P  pCnt  (
# `  k )
) ) ) )
7731, 43, 76syl2anc 642 . . . . . . . . . . 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 201 . . . . . . . . . 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 737 . . . . . . . . . 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 4053 . . . . . . . . 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 14622 . . . . . . . . . . . . 13  |-  ( H  e.  (SubGrp `  G
)  ->  H  C_  X
)
8281ad2antrl 708 . . . . . . . . . . . 12  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  H  C_  X
)
83 ssfi 7083 . . . . . . . . . . . 12  |-  ( ( X  e.  Fin  /\  H  C_  X )  ->  H  e.  Fin )
8410, 82, 83syl2anc 642 . . . . . . . . . . 11  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  H  e.  Fin )
8584adantr 451 . . . . . . . . . 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 11361 . . . . . . . . . 10  |-  ( ( k  e.  Fin  /\  H  e.  Fin )  ->  ( ( # `  k
)  <_  ( # `  H
)  <->  k  ~<_  H ) )
8716, 85, 86syl2anc 642 . . . . . . . . 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 201 . . . . . . . 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 6981 . . . . . . . 8  |-  ( ( H  ~<_  k  /\  k  ~<_  H )  ->  H  ~~  k )
9019, 88, 89syl2anc 642 . . . . . . 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 7074 . . . . . . 7  |-  ( ( k  e.  Fin  /\  H  C_  k  /\  H  ~~  k )  ->  H  =  k )
9216, 17, 90, 91syl3anc 1182 . . . . . 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 598 . . . . 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 2283 . . . . . . . . . . . . 13  |-  ( Gs  H )  =  ( Gs  H )
9594subgbas 14625 . . . . . . . . . . . 12  |-  ( H  e.  (SubGrp `  G
)  ->  H  =  ( Base `  ( Gs  H
) ) )
9695ad2antrl 708 . . . . . . . . . . 11  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  H  =  ( Base `  ( Gs  H
) ) )
9796fveq2d 5529 . . . . . . . . . 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 733 . . . . . . . . . 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 2317 . . . . . . . . 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 5866 . . . . . . . . . . 11  |-  ( n  =  ( P  pCnt  (
# `  X )
)  ->  ( P ^ n )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) )
101100eqeq2d 2294 . . . . . . . . . 10  |-  ( n  =  ( P  pCnt  (
# `  X )
)  ->  ( ( # `
 ( Base `  ( Gs  H ) ) )  =  ( P ^
n )  <->  ( # `  ( Base `  ( Gs  H ) ) )  =  ( P ^ ( P 
pCnt  ( # `  X
) ) ) ) )
102101rspcev 2884 . . . . . . . . 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 642 . . . . . . . 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 14624 . . . . . . . . . 10  |-  ( H  e.  (SubGrp `  G
)  ->  ( Gs  H
)  e.  Grp )
105104ad2antrl 708 . . . . . . . . 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 2358 . . . . . . . . 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 2283 . . . . . . . . . 10  |-  ( Base `  ( Gs  H ) )  =  ( Base `  ( Gs  H ) )
108107pgpfi 14916 . . . . . . . . 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 642 . . . . . . . 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 888 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  P pGrp  ( Gs  H ) )
111110adantr 451 . . . . . 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 5866 . . . . . . . 8  |-  ( H  =  k  ->  ( Gs  H )  =  ( Gs  k ) )
113112breq2d 4035 . . . . . . 7  |-  ( H  =  k  ->  ( P pGrp  ( Gs  H )  <->  P pGrp  ( Gs  k ) ) )
114 eqimss 3230 . . . . . . . 8  |-  ( H  =  k  ->  H  C_  k )
115114biantrurd 494 . . . . . . 7  |-  ( H  =  k  ->  ( P pGrp  ( Gs  k )  <->  ( H  C_  k  /\  P pGrp  ( Gs  k ) ) ) )
116113, 115bitrd 244 . . . . . 6  |-  ( H  =  k  ->  ( P pGrp  ( Gs  H )  <->  ( H  C_  k  /\  P pGrp  ( Gs  k ) ) ) )
117111, 116syl5ibcom 211 . . . . 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 183 . . . 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 2626 . . 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 14919 . . 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 1136 . 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 805 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544    C_ wss 3152   (/)c0 3455   class class class wbr 4023   ` cfv 5255  (class class class)co 5858    ~~ cen 6860    ~<_ cdom 6861   Fincfn 6863    <_ cle 8868   NNcn 9746   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   ^cexp 11104   #chash 11337    || cdivides 12531   Primecprime 12758    pCnt cpc 12889   Basecbs 13148   ↾s cress 13149   0gc0g 13400   Grpcgrp 14362  SubGrpcsubg 14615   pGrp cpgp 14842   pSyl cslw 14843
This theorem is referenced by:  sylow3lem1  14938
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-disj 3994  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-er 6660  df-ec 6662  df-qs 6666  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-acn 7575  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-mod 10974  df-seq 11047  df-exp 11105  df-fac 11289  df-bc 11316  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-dvds 12532  df-gcd 12686  df-prm 12759  df-pc 12890  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mulg 14492  df-subg 14618  df-eqg 14620  df-ghm 14681  df-ga 14744  df-od 14844  df-pgp 14846  df-slw 14847
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