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Theorem slwhash 15258
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 15244 . . . . 5  |-  ( H  e.  ( P pSyl  G
)  ->  H  e.  (SubGrp `  G ) )
42, 3syl 16 . . . 4  |-  ( ph  ->  H  e.  (SubGrp `  G ) )
5 subgrcl 14949 . . . 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 15243 . . . 4  |-  ( H  e.  ( P pSyl  G
)  ->  P  e.  Prime )
92, 8syl 16 . . 3  |-  ( ph  ->  P  e.  Prime )
101grpbn0 14834 . . . . . 6  |-  ( G  e.  Grp  ->  X  =/=  (/) )
116, 10syl 16 . . . . 5  |-  ( ph  ->  X  =/=  (/) )
12 hashnncl 11645 . . . . . 6  |-  ( X  e.  Fin  ->  (
( # `  X )  e.  NN  <->  X  =/=  (/) ) )
137, 12syl 16 . . . . 5  |-  ( ph  ->  ( ( # `  X
)  e.  NN  <->  X  =/=  (/) ) )
1411, 13mpbird 224 . . . 4  |-  ( ph  ->  ( # `  X
)  e.  NN )
159, 14pccld 13224 . . 3  |-  ( ph  ->  ( P  pCnt  ( # `
 X ) )  e.  NN0 )
16 pcdvds 13237 . . . 4  |-  ( ( P  e.  Prime  /\  ( # `
 X )  e.  NN )  ->  ( P ^ ( P  pCnt  (
# `  X )
) )  ||  ( # `
 X ) )
179, 14, 16syl2anc 643 . . 3  |-  ( ph  ->  ( P ^ ( P  pCnt  ( # `  X
) ) )  ||  ( # `  X ) )
181, 6, 7, 9, 15, 17sylow1 15237 . 2  |-  ( ph  ->  E. k  e.  (SubGrp `  G ) ( # `  k )  =  ( P ^ ( P 
pCnt  ( # `  X
) ) ) )
197adantr 452 . . . 4  |-  ( (
ph  /\  ( k  e.  (SubGrp `  G )  /\  ( # `  k
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  ->  X  e.  Fin )
204adantr 452 . . . 4  |-  ( (
ph  /\  ( k  e.  (SubGrp `  G )  /\  ( # `  k
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  ->  H  e.  (SubGrp `  G ) )
21 simprl 733 . . . 4  |-  ( (
ph  /\  ( k  e.  (SubGrp `  G )  /\  ( # `  k
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  ->  k  e.  (SubGrp `  G ) )
22 eqid 2436 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
23 eqid 2436 . . . . . . 7  |-  ( Gs  H )  =  ( Gs  H )
2423slwpgp 15247 . . . . . 6  |-  ( H  e.  ( P pSyl  G
)  ->  P pGrp  ( Gs  H ) )
252, 24syl 16 . . . . 5  |-  ( ph  ->  P pGrp  ( Gs  H ) )
2625adantr 452 . . . 4  |-  ( (
ph  /\  ( k  e.  (SubGrp `  G )  /\  ( # `  k
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  ->  P pGrp  ( Gs  H
) )
27 simprr 734 . . . 4  |-  ( (
ph  /\  ( k  e.  (SubGrp `  G )  /\  ( # `  k
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  ->  ( # `  k
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
28 eqid 2436 . . . 4  |-  ( -g `  G )  =  (
-g `  G )
291, 19, 20, 21, 22, 26, 27, 28sylow2b 15257 . . 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 734 . . . . . 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 707 . . . . . . . 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 707 . . . . . . . 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 452 . . . . . . . . . . . 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 733 . . . . . . . . . . . 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 2436 . . . . . . . . . . . . 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 15037 . . . . . . . . . . . 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 643 . . . . . . . . . . 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 2436 . . . . . . . . . . . 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 14948 . . . . . . . . . . 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 5732 . . . . . . . . 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 15038 . . . . . . . . . . . 12  |-  ( ( k  e.  (SubGrp `  G )  /\  g  e.  X )  ->  k  ~~  ran  ( x  e.  k  |->  ( ( g ( +g  `  G
) x ) (
-g `  G )
g ) ) )
4434, 35, 43syl2anc 643 . . . . . . . . . . 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 707 . . . . . . . . . . . . 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 14945 . . . . . . . . . . . . . 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 7329 . . . . . . . . . . . . 13  |-  ( ( X  e.  Fin  /\  k  C_  X )  -> 
k  e.  Fin )
4945, 47, 48syl2anc 643 . . . . . . . . . . . 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 14945 . . . . . . . . . . . . . 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 7329 . . . . . . . . . . . . 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 643 . . . . . . . . . . . 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 11631 . . . . . . . . . . . 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 643 . . . . . . . . . . 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 224 . . . . . . . . . 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 738 . . . . . . . . . 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 2470 . . . . . . . . 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 2470 . . . . . . . 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 6089 . . . . . . . . . 10  |-  ( n  =  ( P  pCnt  (
# `  X )
)  ->  ( P ^ n )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) )
6160eqeq2d 2447 . . . . . . . . 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 3052 . . . . . . . 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 643 . . . . . . 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 14947 . . . . . . . . 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 2511 . . . . . . . 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 2436 . . . . . . . . 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 15239 . . . . . . . 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 643 . . . . . . 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 889 . . . . . 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 15245 . . . . . . 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 643 . . . . . 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 888 . . . . 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 5732 . . . 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 2468 . . 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 2834 . 2  |-  ( (
ph  /\  ( k  e.  (SubGrp `  G )  /\  ( # `  k
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  ->  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
7718, 76rexlimddv 2834 1  |-  ( ph  ->  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2706    C_ wss 3320   (/)c0 3628   class class class wbr 4212    e. cmpt 4266   ran crn 4879   ` cfv 5454  (class class class)co 6081    ~~ cen 7106   Fincfn 7109   NNcn 10000   NN0cn0 10221   ^cexp 11382   #chash 11618    || cdivides 12852   Primecprime 13079    pCnt cpc 13210   Basecbs 13469   ↾s cress 13470   +g cplusg 13529   Grpcgrp 14685   -gcsg 14688  SubGrpcsubg 14938   pGrp cpgp 15165   pSyl cslw 15166
This theorem is referenced by:  fislw  15259  sylow2  15260  sylow3lem4  15264
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-disj 4183  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-omul 6729  df-er 6905  df-ec 6907  df-qs 6911  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-acn 7829  df-cda 8048  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575  df-rp 10613  df-fz 11044  df-fzo 11136  df-fl 11202  df-mod 11251  df-seq 11324  df-exp 11383  df-fac 11567  df-bc 11594  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-sum 12480  df-dvds 12853  df-gcd 13007  df-prm 13080  df-pc 13211  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-0g 13727  df-mnd 14690  df-submnd 14739  df-grp 14812  df-minusg 14813  df-sbg 14814  df-mulg 14815  df-subg 14941  df-eqg 14943  df-ghm 15004  df-ga 15067  df-od 15167  df-pgp 15169  df-slw 15170
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