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Theorem slwhash 15217
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 15203 . . . . 5  |-  ( H  e.  ( P pSyl  G
)  ->  H  e.  (SubGrp `  G ) )
42, 3syl 16 . . . 4  |-  ( ph  ->  H  e.  (SubGrp `  G ) )
5 subgrcl 14908 . . . 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 15202 . . . 4  |-  ( H  e.  ( P pSyl  G
)  ->  P  e.  Prime )
92, 8syl 16 . . 3  |-  ( ph  ->  P  e.  Prime )
101grpbn0 14793 . . . . . 6  |-  ( G  e.  Grp  ->  X  =/=  (/) )
116, 10syl 16 . . . . 5  |-  ( ph  ->  X  =/=  (/) )
12 hashnncl 11604 . . . . . 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 13183 . . 3  |-  ( ph  ->  ( P  pCnt  ( # `
 X ) )  e.  NN0 )
16 pcdvds 13196 . . . 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 15196 . 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 2408 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
23 eqid 2408 . . . . . . 7  |-  ( Gs  H )  =  ( Gs  H )
2423slwpgp 15206 . . . . . 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 2408 . . . 4  |-  ( -g `  G )  =  (
-g `  G )
291, 19, 20, 21, 22, 26, 27, 28sylow2b 15216 . . 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 2408 . . . . . . . . . . . . 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 14996 . . . . . . . . . . . 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 2408 . . . . . . . . . . . 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 14907 . . . . . . . . . . 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 5695 . . . . . . . . 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 14997 . . . . . . . . . . . 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 14904 . . . . . . . . . . . . . 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 7292 . . . . . . . . . . . . 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 14904 . . . . . . . . . . . . . 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 7292 . . . . . . . . . . . . 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 11590 . . . . . . . . . . . 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 2442 . . . . . . . . 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 2442 . . . . . . . 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 6052 . . . . . . . . . 10  |-  ( n  =  ( P  pCnt  (
# `  X )
)  ->  ( P ^ n )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) )
6160eqeq2d 2419 . . . . . . . . 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 3016 . . . . . . . 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 14906 . . . . . . . . 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 2483 . . . . . . . 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 2408 . . . . . . . . 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 15198 . . . . . . . 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 15204 . . . . . . 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 5695 . . . 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 2440 . . 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 2798 . 2  |-  ( (
ph  /\  ( k  e.  (SubGrp `  G )  /\  ( # `  k
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  ->  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
7718, 76rexlimddv 2798 1  |-  ( ph  ->  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2571   E.wrex 2671    C_ wss 3284   (/)c0 3592   class class class wbr 4176    e. cmpt 4230   ran crn 4842   ` cfv 5417  (class class class)co 6044    ~~ cen 7069   Fincfn 7072   NNcn 9960   NN0cn0 10181   ^cexp 11341   #chash 11577    || cdivides 12811   Primecprime 13038    pCnt cpc 13169   Basecbs 13428   ↾s cress 13429   +g cplusg 13488   Grpcgrp 14644   -gcsg 14647  SubGrpcsubg 14897   pGrp cpgp 15124   pSyl cslw 15125
This theorem is referenced by:  fislw  15218  sylow2  15219  sylow3lem4  15223
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-disj 4147  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-2o 6688  df-oadd 6691  df-omul 6692  df-er 6868  df-ec 6870  df-qs 6874  df-map 6983  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-sup 7408  df-oi 7439  df-card 7786  df-acn 7789  df-cda 8008  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-3 10019  df-n0 10182  df-z 10243  df-uz 10449  df-q 10535  df-rp 10573  df-fz 11004  df-fzo 11095  df-fl 11161  df-mod 11210  df-seq 11283  df-exp 11342  df-fac 11526  df-bc 11553  df-hash 11578  df-cj 11863  df-re 11864  df-im 11865  df-sqr 11999  df-abs 12000  df-clim 12241  df-sum 12439  df-dvds 12812  df-gcd 12966  df-prm 13039  df-pc 13170  df-ndx 13431  df-slot 13432  df-base 13433  df-sets 13434  df-ress 13435  df-plusg 13501  df-0g 13686  df-mnd 14649  df-submnd 14698  df-grp 14771  df-minusg 14772  df-sbg 14773  df-mulg 14774  df-subg 14900  df-eqg 14902  df-ghm 14963  df-ga 15026  df-od 15126  df-pgp 15128  df-slw 15129
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