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Theorem slwhash 16121
Description: A sylow subgroup has cardinality equal to the maximum power of  P dividing the group. (Contributed by Mario Carneiro, 18-Jan-2015.)
Hypotheses
Ref Expression
fislw.1  |-  X  =  ( Base `  G
)
slwhash.3  |-  ( ph  ->  X  e.  Fin )
slwhash.4  |-  ( ph  ->  H  e.  ( P pSyl 
G ) )
Assertion
Ref Expression
slwhash  |-  ( ph  ->  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )

Proof of Theorem slwhash
Dummy variables  g 
k  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fislw.1 . . 3  |-  X  =  ( Base `  G
)
2 slwhash.4 . . . . 5  |-  ( ph  ->  H  e.  ( P pSyl 
G ) )
3 slwsubg 16107 . . . . 5  |-  ( H  e.  ( P pSyl  G
)  ->  H  e.  (SubGrp `  G ) )
42, 3syl 16 . . . 4  |-  ( ph  ->  H  e.  (SubGrp `  G ) )
5 subgrcl 15684 . . . 4  |-  ( H  e.  (SubGrp `  G
)  ->  G  e.  Grp )
64, 5syl 16 . . 3  |-  ( ph  ->  G  e.  Grp )
7 slwhash.3 . . 3  |-  ( ph  ->  X  e.  Fin )
8 slwprm 16106 . . . 4  |-  ( H  e.  ( P pSyl  G
)  ->  P  e.  Prime )
92, 8syl 16 . . 3  |-  ( ph  ->  P  e.  Prime )
101grpbn0 15565 . . . . . 6  |-  ( G  e.  Grp  ->  X  =/=  (/) )
116, 10syl 16 . . . . 5  |-  ( ph  ->  X  =/=  (/) )
12 hashnncl 12132 . . . . . 6  |-  ( X  e.  Fin  ->  (
( # `  X )  e.  NN  <->  X  =/=  (/) ) )
137, 12syl 16 . . . . 5  |-  ( ph  ->  ( ( # `  X
)  e.  NN  <->  X  =/=  (/) ) )
1411, 13mpbird 232 . . . 4  |-  ( ph  ->  ( # `  X
)  e.  NN )
159, 14pccld 13915 . . 3  |-  ( ph  ->  ( P  pCnt  ( # `
 X ) )  e.  NN0 )
16 pcdvds 13928 . . . 4  |-  ( ( P  e.  Prime  /\  ( # `
 X )  e.  NN )  ->  ( P ^ ( P  pCnt  (
# `  X )
) )  ||  ( # `
 X ) )
179, 14, 16syl2anc 661 . . 3  |-  ( ph  ->  ( P ^ ( P  pCnt  ( # `  X
) ) )  ||  ( # `  X ) )
181, 6, 7, 9, 15, 17sylow1 16100 . 2  |-  ( ph  ->  E. k  e.  (SubGrp `  G ) ( # `  k )  =  ( P ^ ( P 
pCnt  ( # `  X
) ) ) )
197adantr 465 . . . 4  |-  ( (
ph  /\  ( k  e.  (SubGrp `  G )  /\  ( # `  k
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  ->  X  e.  Fin )
204adantr 465 . . . 4  |-  ( (
ph  /\  ( k  e.  (SubGrp `  G )  /\  ( # `  k
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  ->  H  e.  (SubGrp `  G ) )
21 simprl 755 . . . 4  |-  ( (
ph  /\  ( k  e.  (SubGrp `  G )  /\  ( # `  k
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  ->  k  e.  (SubGrp `  G ) )
22 eqid 2441 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
23 eqid 2441 . . . . . . 7  |-  ( Gs  H )  =  ( Gs  H )
2423slwpgp 16110 . . . . . 6  |-  ( H  e.  ( P pSyl  G
)  ->  P pGrp  ( Gs  H ) )
252, 24syl 16 . . . . 5  |-  ( ph  ->  P pGrp  ( Gs  H ) )
2625adantr 465 . . . 4  |-  ( (
ph  /\  ( k  e.  (SubGrp `  G )  /\  ( # `  k
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  ->  P pGrp  ( Gs  H
) )
27 simprr 756 . . . 4  |-  ( (
ph  /\  ( k  e.  (SubGrp `  G )  /\  ( # `  k
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  ->  ( # `  k
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
28 eqid 2441 . . . 4  |-  ( -g `  G )  =  (
-g `  G )
291, 19, 20, 21, 22, 26, 27, 28sylow2b 16120 . . 3  |-  ( (
ph  /\  ( k  e.  (SubGrp `  G )  /\  ( # `  k
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  ->  E. g  e.  X  H  C_  ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) )
30 simprr 756 . . . . . 6  |-  ( ( ( ph  /\  (
k  e.  (SubGrp `  G )  /\  ( # `
 k )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  /\  ( g  e.  X  /\  H  C_ 
ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )  ->  H  C_  ran  ( x  e.  k  |->  ( ( g ( +g  `  G ) x ) ( -g `  G ) g ) ) )
312ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  (
k  e.  (SubGrp `  G )  /\  ( # `
 k )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  /\  ( g  e.  X  /\  H  C_ 
ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )  ->  H  e.  ( P pSyl  G )
)
3231, 8syl 16 . . . . . . 7  |-  ( ( ( ph  /\  (
k  e.  (SubGrp `  G )  /\  ( # `
 k )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  /\  ( g  e.  X  /\  H  C_ 
ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )  ->  P  e.  Prime )
3315ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  (
k  e.  (SubGrp `  G )  /\  ( # `
 k )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  /\  ( g  e.  X  /\  H  C_ 
ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )  ->  ( P  pCnt  ( # `  X
) )  e.  NN0 )
3421adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
k  e.  (SubGrp `  G )  /\  ( # `
 k )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  /\  ( g  e.  X  /\  H  C_ 
ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )  ->  k  e.  (SubGrp `  G ) )
35 simprl 755 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
k  e.  (SubGrp `  G )  /\  ( # `
 k )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  /\  ( g  e.  X  /\  H  C_ 
ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )  ->  g  e.  X )
36 eqid 2441 . . . . . . . . . . . . 13  |-  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) )  =  ( x  e.  k 
|->  ( ( g ( +g  `  G ) x ) ( -g `  G ) g ) )
371, 22, 28, 36conjsubg 15776 . . . . . . . . . . . 12  |-  ( ( k  e.  (SubGrp `  G )  /\  g  e.  X )  ->  ran  ( x  e.  k  |->  ( ( g ( +g  `  G ) x ) ( -g `  G ) g ) )  e.  (SubGrp `  G ) )
3834, 35, 37syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
k  e.  (SubGrp `  G )  /\  ( # `
 k )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  /\  ( g  e.  X  /\  H  C_ 
ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )  ->  ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) )  e.  (SubGrp `  G )
)
39 eqid 2441 . . . . . . . . . . . 12  |-  ( Gs  ran  ( x  e.  k 
|->  ( ( g ( +g  `  G ) x ) ( -g `  G ) g ) ) )  =  ( Gs 
ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) )
4039subgbas 15683 . . . . . . . . . . 11  |-  ( ran  ( x  e.  k 
|->  ( ( g ( +g  `  G ) x ) ( -g `  G ) g ) )  e.  (SubGrp `  G )  ->  ran  ( x  e.  k  |->  ( ( g ( +g  `  G ) x ) ( -g `  G ) g ) )  =  ( Base `  ( Gs  ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) ) )
4138, 40syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
k  e.  (SubGrp `  G )  /\  ( # `
 k )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  /\  ( g  e.  X  /\  H  C_ 
ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )  ->  ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) )  =  ( Base `  ( Gs  ran  ( x  e.  k 
|->  ( ( g ( +g  `  G ) x ) ( -g `  G ) g ) ) ) ) )
4241fveq2d 5693 . . . . . . . . 9  |-  ( ( ( ph  /\  (
k  e.  (SubGrp `  G )  /\  ( # `
 k )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  /\  ( g  e.  X  /\  H  C_ 
ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )  ->  ( # `  ran  ( x  e.  k  |->  ( ( g ( +g  `  G ) x ) ( -g `  G ) g ) ) )  =  (
# `  ( Base `  ( Gs  ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) ) ) )
431, 22, 28, 36conjsubgen 15777 . . . . . . . . . . . 12  |-  ( ( k  e.  (SubGrp `  G )  /\  g  e.  X )  ->  k  ~~  ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) )
4434, 35, 43syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
k  e.  (SubGrp `  G )  /\  ( # `
 k )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  /\  ( g  e.  X  /\  H  C_ 
ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )  ->  k  ~~  ran  ( x  e.  k 
|->  ( ( g ( +g  `  G ) x ) ( -g `  G ) g ) ) )
457ad2antrr 725 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
k  e.  (SubGrp `  G )  /\  ( # `
 k )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  /\  ( g  e.  X  /\  H  C_ 
ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )  ->  X  e.  Fin )
461subgss 15680 . . . . . . . . . . . . . 14  |-  ( k  e.  (SubGrp `  G
)  ->  k  C_  X )
4734, 46syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
k  e.  (SubGrp `  G )  /\  ( # `
 k )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  /\  ( g  e.  X  /\  H  C_ 
ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )  ->  k  C_  X )
48 ssfi 7531 . . . . . . . . . . . . 13  |-  ( ( X  e.  Fin  /\  k  C_  X )  -> 
k  e.  Fin )
4945, 47, 48syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
k  e.  (SubGrp `  G )  /\  ( # `
 k )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  /\  ( g  e.  X  /\  H  C_ 
ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )  ->  k  e.  Fin )
501subgss 15680 . . . . . . . . . . . . . 14  |-  ( ran  ( x  e.  k 
|->  ( ( g ( +g  `  G ) x ) ( -g `  G ) g ) )  e.  (SubGrp `  G )  ->  ran  ( x  e.  k  |->  ( ( g ( +g  `  G ) x ) ( -g `  G ) g ) )  C_  X )
5138, 50syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
k  e.  (SubGrp `  G )  /\  ( # `
 k )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  /\  ( g  e.  X  /\  H  C_ 
ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )  ->  ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) )  C_  X )
52 ssfi 7531 . . . . . . . . . . . . 13  |-  ( ( X  e.  Fin  /\  ran  ( x  e.  k 
|->  ( ( g ( +g  `  G ) x ) ( -g `  G ) g ) )  C_  X )  ->  ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) )  e. 
Fin )
5345, 51, 52syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
k  e.  (SubGrp `  G )  /\  ( # `
 k )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  /\  ( g  e.  X  /\  H  C_ 
ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )  ->  ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) )  e. 
Fin )
54 hashen 12116 . . . . . . . . . . . 12  |-  ( ( k  e.  Fin  /\  ran  ( x  e.  k 
|->  ( ( g ( +g  `  G ) x ) ( -g `  G ) g ) )  e.  Fin )  ->  ( ( # `  k
)  =  ( # `  ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) )  <-> 
k  ~~  ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )
5549, 53, 54syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
k  e.  (SubGrp `  G )  /\  ( # `
 k )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  /\  ( g  e.  X  /\  H  C_ 
ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )  ->  ( ( # `
 k )  =  ( # `  ran  ( x  e.  k  |->  ( ( g ( +g  `  G ) x ) ( -g `  G ) g ) ) )  <->  k  ~~  ran  ( x  e.  k 
|->  ( ( g ( +g  `  G ) x ) ( -g `  G ) g ) ) ) )
5644, 55mpbird 232 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
k  e.  (SubGrp `  G )  /\  ( # `
 k )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  /\  ( g  e.  X  /\  H  C_ 
ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )  ->  ( # `  k
)  =  ( # `  ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )
57 simplrr 760 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
k  e.  (SubGrp `  G )  /\  ( # `
 k )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  /\  ( g  e.  X  /\  H  C_ 
ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )  ->  ( # `  k
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
5856, 57eqtr3d 2475 . . . . . . . . 9  |-  ( ( ( ph  /\  (
k  e.  (SubGrp `  G )  /\  ( # `
 k )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  /\  ( g  e.  X  /\  H  C_ 
ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )  ->  ( # `  ran  ( x  e.  k  |->  ( ( g ( +g  `  G ) x ) ( -g `  G ) g ) ) )  =  ( P ^ ( P 
pCnt  ( # `  X
) ) ) )
5942, 58eqtr3d 2475 . . . . . . . 8  |-  ( ( ( ph  /\  (
k  e.  (SubGrp `  G )  /\  ( # `
 k )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  /\  ( g  e.  X  /\  H  C_ 
ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )  ->  ( # `  ( Base `  ( Gs  ran  (
x  e.  k  |->  ( ( g ( +g  `  G ) x ) ( -g `  G
) g ) ) ) ) )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) )
60 oveq2 6097 . . . . . . . . . 10  |-  ( n  =  ( P  pCnt  (
# `  X )
)  ->  ( P ^ n )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) )
6160eqeq2d 2452 . . . . . . . . 9  |-  ( n  =  ( P  pCnt  (
# `  X )
)  ->  ( ( # `
 ( Base `  ( Gs  ran  ( x  e.  k 
|->  ( ( g ( +g  `  G ) x ) ( -g `  G ) g ) ) ) ) )  =  ( P ^
n )  <->  ( # `  ( Base `  ( Gs  ran  (
x  e.  k  |->  ( ( g ( +g  `  G ) x ) ( -g `  G
) g ) ) ) ) )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )
6261rspcev 3071 . . . . . . . 8  |-  ( ( ( P  pCnt  ( # `
 X ) )  e.  NN0  /\  ( # `
 ( Base `  ( Gs  ran  ( x  e.  k 
|->  ( ( g ( +g  `  G ) x ) ( -g `  G ) g ) ) ) ) )  =  ( P ^
( P  pCnt  ( # `
 X ) ) ) )  ->  E. n  e.  NN0  ( # `  ( Base `  ( Gs  ran  (
x  e.  k  |->  ( ( g ( +g  `  G ) x ) ( -g `  G
) g ) ) ) ) )  =  ( P ^ n
) )
6333, 59, 62syl2anc 661 . . . . . . 7  |-  ( ( ( ph  /\  (
k  e.  (SubGrp `  G )  /\  ( # `
 k )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  /\  ( g  e.  X  /\  H  C_ 
ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )  ->  E. n  e.  NN0  ( # `  ( Base `  ( Gs  ran  (
x  e.  k  |->  ( ( g ( +g  `  G ) x ) ( -g `  G
) g ) ) ) ) )  =  ( P ^ n
) )
6439subggrp 15682 . . . . . . . . 9  |-  ( ran  ( x  e.  k 
|->  ( ( g ( +g  `  G ) x ) ( -g `  G ) g ) )  e.  (SubGrp `  G )  ->  ( Gs  ran  ( x  e.  k 
|->  ( ( g ( +g  `  G ) x ) ( -g `  G ) g ) ) )  e.  Grp )
6538, 64syl 16 . . . . . . . 8  |-  ( ( ( ph  /\  (
k  e.  (SubGrp `  G )  /\  ( # `
 k )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  /\  ( g  e.  X  /\  H  C_ 
ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )  ->  ( Gs  ran  ( x  e.  k  |->  ( ( g ( +g  `  G ) x ) ( -g `  G ) g ) ) )  e.  Grp )
6641, 53eqeltrrd 2516 . . . . . . . 8  |-  ( ( ( ph  /\  (
k  e.  (SubGrp `  G )  /\  ( # `
 k )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  /\  ( g  e.  X  /\  H  C_ 
ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )  ->  ( Base `  ( Gs  ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )  e.  Fin )
67 eqid 2441 . . . . . . . . 9  |-  ( Base `  ( Gs  ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )  =  ( Base `  ( Gs  ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )
6867pgpfi 16102 . . . . . . . 8  |-  ( ( ( Gs  ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) )  e.  Grp  /\  ( Base `  ( Gs  ran  (
x  e.  k  |->  ( ( g ( +g  `  G ) x ) ( -g `  G
) g ) ) ) )  e.  Fin )  ->  ( P pGrp  ( Gs  ran  ( x  e.  k 
|->  ( ( g ( +g  `  G ) x ) ( -g `  G ) g ) ) )  <->  ( P  e.  Prime  /\  E. n  e.  NN0  ( # `  ( Base `  ( Gs  ran  (
x  e.  k  |->  ( ( g ( +g  `  G ) x ) ( -g `  G
) g ) ) ) ) )  =  ( P ^ n
) ) ) )
6965, 66, 68syl2anc 661 . . . . . . 7  |-  ( ( ( ph  /\  (
k  e.  (SubGrp `  G )  /\  ( # `
 k )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  /\  ( g  e.  X  /\  H  C_ 
ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )  ->  ( P pGrp  ( Gs  ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) )  <-> 
( P  e.  Prime  /\ 
E. n  e.  NN0  ( # `  ( Base `  ( Gs  ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) ) )  =  ( P ^ n ) ) ) )
7032, 63, 69mpbir2and 913 . . . . . 6  |-  ( ( ( ph  /\  (
k  e.  (SubGrp `  G )  /\  ( # `
 k )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  /\  ( g  e.  X  /\  H  C_ 
ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )  ->  P pGrp  ( Gs  ran  ( x  e.  k 
|->  ( ( g ( +g  `  G ) x ) ( -g `  G ) g ) ) ) )
7139slwispgp 16108 . . . . . . 7  |-  ( ( H  e.  ( P pSyl 
G )  /\  ran  ( x  e.  k  |->  ( ( g ( +g  `  G ) x ) ( -g `  G ) g ) )  e.  (SubGrp `  G ) )  -> 
( ( H  C_  ran  ( x  e.  k 
|->  ( ( g ( +g  `  G ) x ) ( -g `  G ) g ) )  /\  P pGrp  ( Gs  ran  ( x  e.  k 
|->  ( ( g ( +g  `  G ) x ) ( -g `  G ) g ) ) ) )  <->  H  =  ran  ( x  e.  k 
|->  ( ( g ( +g  `  G ) x ) ( -g `  G ) g ) ) ) )
7231, 38, 71syl2anc 661 . . . . . 6  |-  ( ( ( ph  /\  (
k  e.  (SubGrp `  G )  /\  ( # `
 k )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  /\  ( g  e.  X  /\  H  C_ 
ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )  ->  ( ( H  C_  ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) )  /\  P pGrp  ( Gs  ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )  <->  H  =  ran  ( x  e.  k  |->  ( ( g ( +g  `  G ) x ) ( -g `  G ) g ) ) ) )
7330, 70, 72mpbi2and 912 . . . . 5  |-  ( ( ( ph  /\  (
k  e.  (SubGrp `  G )  /\  ( # `
 k )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  /\  ( g  e.  X  /\  H  C_ 
ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )  ->  H  =  ran  ( x  e.  k 
|->  ( ( g ( +g  `  G ) x ) ( -g `  G ) g ) ) )
7473fveq2d 5693 . . . 4  |-  ( ( ( ph  /\  (
k  e.  (SubGrp `  G )  /\  ( # `
 k )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  /\  ( g  e.  X  /\  H  C_ 
ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )  ->  ( # `  H
)  =  ( # `  ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )
7574, 58eqtrd 2473 . . 3  |-  ( ( ( ph  /\  (
k  e.  (SubGrp `  G )  /\  ( # `
 k )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) ) )  /\  ( g  e.  X  /\  H  C_ 
ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) ) )  ->  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
7629, 75rexlimddv 2843 . 2  |-  ( (
ph  /\  ( k  e.  (SubGrp `  G )  /\  ( # `  k
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  ->  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
7718, 76rexlimddv 2843 1  |-  ( ph  ->  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2604   E.wrex 2714    C_ wss 3326   (/)c0 3635   class class class wbr 4290    e. cmpt 4348   ran crn 4839   ` cfv 5416  (class class class)co 6089    ~~ cen 7305   Fincfn 7308   NNcn 10320   NN0cn0 10577   ^cexp 11863   #chash 12101    || cdivides 13533   Primecprime 13761    pCnt cpc 13901   Basecbs 14172   ↾s cress 14173   +g cplusg 14236   Grpcgrp 15408   -gcsg 15411  SubGrpcsubg 15673   pGrp cpgp 16028   pSyl cslw 16029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-inf2 7845  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357  ax-pre-sup 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-int 4127  df-iun 4171  df-disj 4261  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-se 4678  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-isom 5425  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-2o 6919  df-oadd 6922  df-omul 6923  df-er 7099  df-ec 7101  df-qs 7105  df-map 7214  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-sup 7689  df-oi 7722  df-card 8107  df-acn 8110  df-cda 8335  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992  df-nn 10321  df-2 10378  df-3 10379  df-n0 10578  df-z 10645  df-uz 10860  df-q 10952  df-rp 10990  df-fz 11436  df-fzo 11547  df-fl 11640  df-mod 11707  df-seq 11805  df-exp 11864  df-fac 12050  df-bc 12077  df-hash 12102  df-cj 12586  df-re 12587  df-im 12588  df-sqr 12722  df-abs 12723  df-clim 12964  df-sum 13162  df-dvds 13534  df-gcd 13689  df-prm 13762  df-pc 13902  df-ndx 14175  df-slot 14176  df-base 14177  df-sets 14178  df-ress 14179  df-plusg 14249  df-0g 14378  df-mnd 15413  df-submnd 15463  df-grp 15543  df-minusg 15544  df-sbg 15545  df-mulg 15546  df-subg 15676  df-eqg 15678  df-ghm 15743  df-ga 15806  df-od 16030  df-pgp 16032  df-slw 16033
This theorem is referenced by:  fislw  16122  sylow2  16123  sylow3lem4  16127
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