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Theorem fislw 15251
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 448 . . . 4  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  H  e.  ( P pSyl  G ) )  ->  H  e.  ( P pSyl  G ) )
2 slwsubg 15236 . . . 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 961 . . . 4  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  H  e.  ( P pSyl  G ) )  ->  X  e.  Fin )
64, 5, 1slwhash 15250 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  H  e.  ( P pSyl  G ) )  ->  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) )
73, 6jca 519 . 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 962 . . 3  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  P  e.  Prime )
9 simprl 733 . . 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 961 . . . . . . . . 9  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  X  e.  Fin )
1110adantr 452 . . . . . . . 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 733 . . . . . . . . 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 14937 . . . . . . . . 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 7321 . . . . . . . 8  |-  ( ( X  e.  Fin  /\  k  C_  X )  -> 
k  e.  Fin )
1611, 14, 15syl2anc 643 . . . . . . 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 741 . . . . . . 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 7145 . . . . . . . . 9  |-  ( k  e.  Fin  ->  ( H  C_  k  ->  H  ~<_  k ) )
1916, 17, 18sylc 58 . . . . . . . 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 742 . . . . . . . . . . . . . . 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 2435 . . . . . . . . . . . . . . . . . 18  |-  ( Gs  k )  =  ( Gs  k )
2221subggrp 14939 . . . . . . . . . . . . . . . . 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 14940 . . . . . . . . . . . . . . . . . 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 2510 . . . . . . . . . . . . . . . 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 2435 . . . . . . . . . . . . . . . . 17  |-  ( Base `  ( Gs  k ) )  =  ( Base `  ( Gs  k ) )
2827pgpfi 15231 . . . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . . . 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 202 . . . . . . . . . . . . . 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 446 . . . . . . . . . . . . 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 13074 . . . . . . . . . . . . 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 10007 . . . . . . . . . . 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 10034 . . . . . . . . . . 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 2435 . . . . . . . . . . . . . . . . . 18  |-  ( 0g
`  G )  =  ( 0g `  G
)
3736subg0cl 14944 . . . . . . . . . . . . . . . . 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 3626 . . . . . . . . . . . . . . . 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 11637 . . . . . . . . . . . . . . . 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 224 . . . . . . . . . . . . . 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 13216 . . . . . . . . . . . . 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 10365 . . . . . . . . . . . 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 960 . . . . . . . . . . . . . . . . 17  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  G  e.  Grp )
474grpbn0 14826 . . . . . . . . . . . . . . . . 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 11637 . . . . . . . . . . . . . . . . 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 224 . . . . . . . . . . . . . . 15  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  ( # `  X
)  e.  NN )
528, 51pccld 13216 . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . 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 10365 . . . . . . . . . . . 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 14994 . . . . . . . . . . . . . . 15  |-  ( ( k  e.  (SubGrp `  G )  /\  X  e.  Fin )  ->  ( # `
 k )  ||  ( # `  X ) )
5612, 11, 55syl2anc 643 . . . . . . . . . . . . . 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 10366 . . . . . . . . . . . . . . 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 452 . . . . . . . . . . . . . . . 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 10366 . . . . . . . . . . . . . . 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 13244 . . . . . . . . . . . . . . 15  |-  ( ( ( # `  k
)  e.  ZZ  /\  ( # `  X )  e.  ZZ )  -> 
( ( # `  k
)  ||  ( # `  X
)  <->  A. p  e.  Prime  ( p  pCnt  ( # `  k
) )  <_  (
p  pCnt  ( # `  X
) ) ) )
6157, 59, 60syl2anc 643 . . . . . . . . . . . . . 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 202 . . . . . . . . . . . . 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 6080 . . . . . . . . . . . . . . 15  |-  ( p  =  P  ->  (
p  pCnt  ( # `  k
) )  =  ( P  pCnt  ( # `  k
) ) )
64 oveq1 6080 . . . . . . . . . . . . . . 15  |-  ( p  =  P  ->  (
p  pCnt  ( # `  X
) )  =  ( P  pCnt  ( # `  X
) ) )
6563, 64breq12d 4217 . . . . . . . . . . . . . 14  |-  ( p  =  P  ->  (
( p  pCnt  ( # `
 k ) )  <_  ( p  pCnt  (
# `  X )
)  <->  ( P  pCnt  (
# `  k )
)  <_  ( P  pCnt  ( # `  X
) ) ) )
6665rspcv 3040 . . . . . . . . . . . . 13  |-  ( P  e.  Prime  ->  ( A. p  e.  Prime  ( p 
pCnt  ( # `  k
) )  <_  (
p  pCnt  ( # `  X
) )  ->  ( P  pCnt  ( # `  k
) )  <_  ( P  pCnt  ( # `  X
) ) ) )
6731, 62, 66sylc 58 . . . . . . . . . . . 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 10486 . . . . . . . . . . . 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 1138 . . . . . . . . . . 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 11547 . . . . . . . . . 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 450 . . . . . . . . . . . 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 5724 . . . . . . . . . . . . . 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 2443 . . . . . . . . . . . . 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 2718 . . . . . . . . . . . 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 224 . . . . . . . . . . 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 13248 . . . . . . . . . . . 12  |-  ( ( P  e.  Prime  /\  ( # `
 k )  e.  NN )  ->  ( E. n  e.  NN0  ( # `  k )  =  ( P ^
n )  <->  ( # `  k
)  =  ( P ^ ( P  pCnt  (
# `  k )
) ) ) )
7731, 43, 76syl2anc 643 . . . . . . . . . . 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 202 . . . . . . . . . 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 738 . . . . . . . . . 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 4234 . . . . . . . . 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 14937 . . . . . . . . . . . . 13  |-  ( H  e.  (SubGrp `  G
)  ->  H  C_  X
)
8281ad2antrl 709 . . . . . . . . . . . 12  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  H  C_  X
)
83 ssfi 7321 . . . . . . . . . . . 12  |-  ( ( X  e.  Fin  /\  H  C_  X )  ->  H  e.  Fin )
8410, 82, 83syl2anc 643 . . . . . . . . . . 11  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  H  e.  Fin )
8584adantr 452 . . . . . . . . . 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 11645 . . . . . . . . . 10  |-  ( ( k  e.  Fin  /\  H  e.  Fin )  ->  ( ( # `  k
)  <_  ( # `  H
)  <->  k  ~<_  H ) )
8716, 85, 86syl2anc 643 . . . . . . . . 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 202 . . . . . . . 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 7219 . . . . . . . 8  |-  ( ( H  ~<_  k  /\  k  ~<_  H )  ->  H  ~~  k )
9019, 88, 89syl2anc 643 . . . . . . 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 7312 . . . . . . 7  |-  ( ( k  e.  Fin  /\  H  C_  k  /\  H  ~~  k )  ->  H  =  k )
9216, 17, 90, 91syl3anc 1184 . . . . . 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 599 . . . . 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 2435 . . . . . . . . . . . . 13  |-  ( Gs  H )  =  ( Gs  H )
9594subgbas 14940 . . . . . . . . . . . 12  |-  ( H  e.  (SubGrp `  G
)  ->  H  =  ( Base `  ( Gs  H
) ) )
9695ad2antrl 709 . . . . . . . . . . 11  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  H  =  ( Base `  ( Gs  H
) ) )
9796fveq2d 5724 . . . . . . . . . 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 734 . . . . . . . . . 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 2469 . . . . . . . . 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 6081 . . . . . . . . . . 11  |-  ( n  =  ( P  pCnt  (
# `  X )
)  ->  ( P ^ n )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) )
101100eqeq2d 2446 . . . . . . . . . 10  |-  ( n  =  ( P  pCnt  (
# `  X )
)  ->  ( ( # `
 ( Base `  ( Gs  H ) ) )  =  ( P ^
n )  <->  ( # `  ( Base `  ( Gs  H ) ) )  =  ( P ^ ( P 
pCnt  ( # `  X
) ) ) ) )
102101rspcev 3044 . . . . . . . . 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 643 . . . . . . . 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 14939 . . . . . . . . . 10  |-  ( H  e.  (SubGrp `  G
)  ->  ( Gs  H
)  e.  Grp )
105104ad2antrl 709 . . . . . . . . 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 2510 . . . . . . . . 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 2435 . . . . . . . . . 10  |-  ( Base `  ( Gs  H ) )  =  ( Base `  ( Gs  H ) )
108107pgpfi 15231 . . . . . . . . 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 643 . . . . . . . 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 889 . . . . . . 7  |-  ( ( ( G  e.  Grp  /\  X  e.  Fin  /\  P  e.  Prime )  /\  ( H  e.  (SubGrp `  G )  /\  ( # `
 H )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  ->  P pGrp  ( Gs  H ) )
111110adantr 452 . . . . . 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 6081 . . . . . . . 8  |-  ( H  =  k  ->  ( Gs  H )  =  ( Gs  k ) )
113112breq2d 4216 . . . . . . 7  |-  ( H  =  k  ->  ( P pGrp  ( Gs  H )  <->  P pGrp  ( Gs  k ) ) )
114 eqimss 3392 . . . . . . . 8  |-  ( H  =  k  ->  H  C_  k )
115114biantrurd 495 . . . . . . 7  |-  ( H  =  k  ->  ( P pGrp  ( Gs  k )  <->  ( H  C_  k  /\  P pGrp  ( Gs  k ) ) ) )
116113, 115bitrd 245 . . . . . 6  |-  ( H  =  k  ->  ( P pGrp  ( Gs  H )  <->  ( H  C_  k  /\  P pGrp  ( Gs  k ) ) ) )
117111, 116syl5ibcom 212 . . . . 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 184 . . . 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 2781 . . 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 15234 . . 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 1138 . 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 806 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698    C_ wss 3312   (/)c0 3620   class class class wbr 4204   ` cfv 5446  (class class class)co 6073    ~~ cen 7098    ~<_ cdom 7099   Fincfn 7101    <_ cle 9113   NNcn 9992   NN0cn0 10213   ZZcz 10274   ZZ>=cuz 10480   ^cexp 11374   #chash 11610    || cdivides 12844   Primecprime 13071    pCnt cpc 13202   Basecbs 13461   ↾s cress 13462   0gc0g 13715   Grpcgrp 14677  SubGrpcsubg 14930   pGrp cpgp 15157   pSyl cslw 15158
This theorem is referenced by:  sylow3lem1  15253
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-disj 4175  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-omul 6721  df-er 6897  df-ec 6899  df-qs 6903  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-acn 7821  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-q 10567  df-rp 10605  df-fz 11036  df-fzo 11128  df-fl 11194  df-mod 11243  df-seq 11316  df-exp 11375  df-fac 11559  df-bc 11586  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-sum 12472  df-dvds 12845  df-gcd 12999  df-prm 13072  df-pc 13203  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-0g 13719  df-mnd 14682  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-mulg 14807  df-subg 14933  df-eqg 14935  df-ghm 14996  df-ga 15059  df-od 15159  df-pgp 15161  df-slw 15162
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