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Theorem slwhash 15213
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 15199 . . . . 5  |-  ( H  e.  ( P pSyl  G
)  ->  H  e.  (SubGrp `  G ) )
42, 3syl 16 . . . 4  |-  ( ph  ->  H  e.  (SubGrp `  G ) )
5 subgrcl 14904 . . . 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 15198 . . . 4  |-  ( H  e.  ( P pSyl  G
)  ->  P  e.  Prime )
92, 8syl 16 . . 3  |-  ( ph  ->  P  e.  Prime )
101grpbn0 14789 . . . . . 6  |-  ( G  e.  Grp  ->  X  =/=  (/) )
116, 10syl 16 . . . . 5  |-  ( ph  ->  X  =/=  (/) )
12 hashnncl 11600 . . . . . 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 13179 . . 3  |-  ( ph  ->  ( P  pCnt  ( # `
 X ) )  e.  NN0 )
16 pcdvds 13192 . . . 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 15192 . 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 2404 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
23 eqid 2404 . . . . . . 7  |-  ( Gs  H )  =  ( Gs  H )
2423slwpgp 15202 . . . . . 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 2404 . . . 4  |-  ( -g `  G )  =  (
-g `  G )
291, 19, 20, 21, 22, 26, 27, 28sylow2b 15212 . . 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 2404 . . . . . . . . . . . . 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 14992 . . . . . . . . . . . 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 2404 . . . . . . . . . . . 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 14903 . . . . . . . . . . 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 5691 . . . . . . . . 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 14993 . . . . . . . . . . . 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 14900 . . . . . . . . . . . . . 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 7288 . . . . . . . . . . . . 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 14900 . . . . . . . . . . . . . 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 7288 . . . . . . . . . . . . 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 11586 . . . . . . . . . . . 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 2438 . . . . . . . . 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 2438 . . . . . . . 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 6048 . . . . . . . . . 10  |-  ( n  =  ( P  pCnt  (
# `  X )
)  ->  ( P ^ n )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) )
6160eqeq2d 2415 . . . . . . . . 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 3012 . . . . . . . 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 14902 . . . . . . . . 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 2479 . . . . . . . 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 2404 . . . . . . . . 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 15194 . . . . . . . 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 15200 . . . . . . 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 5691 . . . 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 2436 . . 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 2794 . 2  |-  ( (
ph  /\  ( k  e.  (SubGrp `  G )  /\  ( # `  k
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  ->  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
7718, 76rexlimddv 2794 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 2567   E.wrex 2667    C_ wss 3280   (/)c0 3588   class class class wbr 4172    e. cmpt 4226   ran crn 4838   ` cfv 5413  (class class class)co 6040    ~~ cen 7065   Fincfn 7068   NNcn 9956   NN0cn0 10177   ^cexp 11337   #chash 11573    || cdivides 12807   Primecprime 13034    pCnt cpc 13165   Basecbs 13424   ↾s cress 13425   +g cplusg 13484   Grpcgrp 14640   -gcsg 14643  SubGrpcsubg 14893   pGrp cpgp 15120   pSyl cslw 15121
This theorem is referenced by:  fislw  15214  sylow2  15215  sylow3lem4  15219
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 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-disj 4143  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-omul 6688  df-er 6864  df-ec 6866  df-qs 6870  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-acn 7785  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-q 10531  df-rp 10569  df-fz 11000  df-fzo 11091  df-fl 11157  df-mod 11206  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435  df-dvds 12808  df-gcd 12962  df-prm 13035  df-pc 13166  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-0g 13682  df-mnd 14645  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-eqg 14898  df-ghm 14959  df-ga 15022  df-od 15122  df-pgp 15124  df-slw 15125
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