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Theorem slwhash 16518
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 16504 . . . . 5  |-  ( H  e.  ( P pSyl  G
)  ->  H  e.  (SubGrp `  G ) )
42, 3syl 16 . . . 4  |-  ( ph  ->  H  e.  (SubGrp `  G ) )
5 subgrcl 16080 . . . 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 16503 . . . 4  |-  ( H  e.  ( P pSyl  G
)  ->  P  e.  Prime )
92, 8syl 16 . . 3  |-  ( ph  ->  P  e.  Prime )
101grpbn0 15953 . . . . . 6  |-  ( G  e.  Grp  ->  X  =/=  (/) )
116, 10syl 16 . . . . 5  |-  ( ph  ->  X  =/=  (/) )
12 hashnncl 12415 . . . . . 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 14251 . . 3  |-  ( ph  ->  ( P  pCnt  ( # `
 X ) )  e.  NN0 )
16 pcdvds 14264 . . . 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 16497 . 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 756 . . . 4  |-  ( (
ph  /\  ( k  e.  (SubGrp `  G )  /\  ( # `  k
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  ->  k  e.  (SubGrp `  G ) )
22 eqid 2443 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
23 eqid 2443 . . . . . . 7  |-  ( Gs  H )  =  ( Gs  H )
2423slwpgp 16507 . . . . . 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 757 . . . 4  |-  ( (
ph  /\  ( k  e.  (SubGrp `  G )  /\  ( # `  k
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  ->  ( # `  k
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
28 eqid 2443 . . . 4  |-  ( -g `  G )  =  (
-g `  G )
291, 19, 20, 21, 22, 26, 27, 28sylow2b 16517 . . 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 757 . . . . . 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 756 . . . . . . . . . . . 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 2443 . . . . . . . . . . . . 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 16172 . . . . . . . . . . . 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 2443 . . . . . . . . . . . 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 16079 . . . . . . . . . . 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 5860 . . . . . . . . 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 16173 . . . . . . . . . . . 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 16076 . . . . . . . . . . . . . 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 7742 . . . . . . . . . . . . 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 16076 . . . . . . . . . . . . . 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 7742 . . . . . . . . . . . . 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 12399 . . . . . . . . . . . 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 762 . . . . . . . . . 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 2486 . . . . . . . . 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 2486 . . . . . . . 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 6289 . . . . . . . . . 10  |-  ( n  =  ( P  pCnt  (
# `  X )
)  ->  ( P ^ n )  =  ( P ^ ( P  pCnt  ( # `  X
) ) ) )
6160eqeq2d 2457 . . . . . . . . 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 3196 . . . . . . . 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 16078 . . . . . . . . 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 2532 . . . . . . . 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 2443 . . . . . . . . 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 16499 . . . . . . . 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 922 . . . . . 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 16505 . . . . . . 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 921 . . . . 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 5860 . . . 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 2484 . . 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 2939 . 2  |-  ( (
ph  /\  ( k  e.  (SubGrp `  G )  /\  ( # `  k
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) ) )  ->  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
7718, 76rexlimddv 2939 1  |-  ( ph  ->  ( # `  H
)  =  ( P ^ ( P  pCnt  (
# `  X )
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1383    e. wcel 1804    =/= wne 2638   E.wrex 2794    C_ wss 3461   (/)c0 3770   class class class wbr 4437    |-> cmpt 4495   ran crn 4990   ` cfv 5578  (class class class)co 6281    ~~ cen 7515   Fincfn 7518   NNcn 10542   NN0cn0 10801   ^cexp 12145   #chash 12384    || cdvds 13863   Primecprime 14094    pCnt cpc 14237   Basecbs 14509   ↾s cress 14510   +g cplusg 14574   Grpcgrp 15927   -gcsg 15929  SubGrpcsubg 16069   pGrp cpgp 16425   pSyl cslw 16426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-disj 4408  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-omul 7137  df-er 7313  df-ec 7315  df-qs 7319  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-oi 7938  df-card 8323  df-acn 8326  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-n0 10802  df-z 10871  df-uz 11091  df-q 11192  df-rp 11230  df-fz 11682  df-fzo 11804  df-fl 11908  df-mod 11976  df-seq 12087  df-exp 12146  df-fac 12333  df-bc 12360  df-hash 12385  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-clim 13290  df-sum 13488  df-dvds 13864  df-gcd 14022  df-prm 14095  df-pc 14238  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-0g 14716  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15841  df-grp 15931  df-minusg 15932  df-sbg 15933  df-mulg 15934  df-subg 16072  df-eqg 16074  df-ghm 16139  df-ga 16202  df-od 16427  df-pgp 16429  df-slw 16430
This theorem is referenced by:  fislw  16519  sylow2  16520  sylow3lem4  16524
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