MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fislw Unicode version

Theorem fislw 14952
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 14937 . . . 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 14951 . . 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 14638 . . . . . . . . 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 7099 . . . . . . . 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 6923 . . . . . . . . 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 2296 . . . . . . . . . . . . . . . . . 18  |-  ( Gs  k )  =  ( Gs  k )
2221subggrp 14640 . . . . . . . . . . . . . . . . 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 14641 . . . . . . . . . . . . . . . . . 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 2371 . . . . . . . . . . . . . . . 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 2296 . . . . . . . . . . . . . . . . 17  |-  ( Base `  ( Gs  k ) )  =  ( Base `  ( Gs  k ) )
2827pgpfi 14932 . . . . . . . . . . . . . . . 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 12777 . . . . . . . . . . . . 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 9777 . . . . . . . . . . 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 9804 . . . . . . . . . . 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 2296 . . . . . . . . . . . . . . . . . 18  |-  ( 0g
`  G )  =  ( 0g `  G
)
3736subg0cl 14645 . . . . . . . . . . . . . . . . 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 3474 . . . . . . . . . . . . . . . 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 11370 . . . . . . . . . . . . . . . 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 12919 . . . . . . . . . . . . 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 10131 . . . . . . . . . . . 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 14527 . . . . . . . . . . . . . . . . 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 11370 . . . . . . . . . . . . . . . . 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 12919 . . . . . . . . . . . . . 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 10131 . . . . . . . . . . . 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 14695 . . . . . . . . . . . . . . 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 10132 . . . . . . . . . . . . . . 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 10132 . . . . . . . . . . . . . . 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 12947 . . . . . . . . . . . . . . 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 5881 . . . . . . . . . . . . . . 15  |-  ( p  =  P  ->  (
p  pCnt  ( # `  k
) )  =  ( P  pCnt  ( # `  k
) ) )
64 oveq1 5881 . . . . . . . . . . . . . . 15  |-  ( p  =  P  ->  (
p  pCnt  ( # `  X
) )  =  ( P  pCnt  ( # `  X
) ) )
6563, 64breq12d 4052 . . . . . . . . . . . . . 14  |-  ( p  =  P  ->  (
( p  pCnt  ( # `
 k ) )  <_  ( p  pCnt  (
# `  X )
)  <->  ( P  pCnt  (
# `  k )
)  <_  ( P  pCnt  ( # `  X
) ) ) )
6665rspcv 2893 . . . . . . . . . . . . 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 10252 . . . . . . . . . . . 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 11293 . . . . . . . . . 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 5545 . . . . . . . . . . . . . 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 2304 . . . . . . . . . . . . 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 2577 . . . . . . . . . . . 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 12951 . . . . . . . . . . . 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 4069 . . . . . . . . 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 14638 . . . . . . . . . . . . 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 7099 . . . . . . . . . . . 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 11377 . . . . . . . . . 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 6997 . . . . . . . 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 7090 . . . . . . 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 2296 . . . . . . . . . . . . 13  |-  ( Gs  H )  =  ( Gs  H )
9594subgbas 14641 . . . . . . . . . . . 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 5545 . . . . . . . . . 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 2330 . . . . . . . . 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 5882 . . . . . . . . . . 11  |-  ( n  =  ( P  pCnt  (
# `  X )
)  ->  ( P ^ n )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) )
101100eqeq2d 2307 . . . . . . . . . 10  |-  ( n  =  ( P  pCnt  (
# `  X )
)  ->  ( ( # `
 ( Base `  ( Gs  H ) ) )  =  ( P ^
n )  <->  ( # `  ( Base `  ( Gs  H ) ) )  =  ( P ^ ( P 
pCnt  ( # `  X
) ) ) ) )
102101rspcev 2897 . . . . . . . . 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 14640 . . . . . . . . . 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 2371 . . . . . . . . 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 2296 . . . . . . . . . 10  |-  ( Base `  ( Gs  H ) )  =  ( Base `  ( Gs  H ) )
108107pgpfi 14932 . . . . . . . . 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 5882 . . . . . . . 8  |-  ( H  =  k  ->  ( Gs  H )  =  ( Gs  k ) )
113112breq2d 4051 . . . . . . 7  |-  ( H  =  k  ->  ( P pGrp  ( Gs  H )  <->  P pGrp  ( Gs  k ) ) )
114 eqimss 3243 . . . . . . . 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 2639 . . 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 14935 . . 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 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557    C_ wss 3165   (/)c0 3468   class class class wbr 4039   ` cfv 5271  (class class class)co 5874    ~~ cen 6876    ~<_ cdom 6877   Fincfn 6879    <_ cle 8884   NNcn 9762   NN0cn0 9981   ZZcz 10040   ZZ>=cuz 10246   ^cexp 11120   #chash 11353    || cdivides 12547   Primecprime 12774    pCnt cpc 12905   Basecbs 13164   ↾s cress 13165   0gc0g 13416   Grpcgrp 14378  SubGrpcsubg 14631   pGrp cpgp 14858   pSyl cslw 14859
This theorem is referenced by:  sylow3lem1  14954
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-disj 4010  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-omul 6500  df-er 6676  df-ec 6678  df-qs 6682  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-acn 7591  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-q 10333  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-sum 12175  df-dvds 12548  df-gcd 12702  df-prm 12775  df-pc 12906  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mulg 14508  df-subg 14634  df-eqg 14636  df-ghm 14697  df-ga 14760  df-od 14860  df-pgp 14862  df-slw 14863
  Copyright terms: Public domain W3C validator