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Theorem sylow2blem2 16514
Description: Lemma for sylow2b 16516. Left multiplication in a subgroup  H is a group action on the set of all left cosets of  K. (Contributed by Mario Carneiro, 17-Jan-2015.)
Hypotheses
Ref Expression
sylow2b.x  |-  X  =  ( Base `  G
)
sylow2b.xf  |-  ( ph  ->  X  e.  Fin )
sylow2b.h  |-  ( ph  ->  H  e.  (SubGrp `  G ) )
sylow2b.k  |-  ( ph  ->  K  e.  (SubGrp `  G ) )
sylow2b.a  |-  .+  =  ( +g  `  G )
sylow2b.r  |-  .~  =  ( G ~QG  K )
sylow2b.m  |-  .x.  =  ( x  e.  H ,  y  e.  ( X /.  .~  )  |->  ran  ( z  e.  y 
|->  ( x  .+  z
) ) )
Assertion
Ref Expression
sylow2blem2  |-  ( ph  ->  .x.  e.  ( ( Gs  H )  GrpAct  ( X /.  .~  ) ) )
Distinct variable groups:    x, y,
z, G    x, K, y, z    x,  .x. , y,
z    x,  .+ , y, z   
x,  .~ , y, z    ph, z    x, H, y, z    x, X, y, z
Allowed substitution hints:    ph( x, y)

Proof of Theorem sylow2blem2
Dummy variables  a 
b  s  u  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sylow2b.h . . . 4  |-  ( ph  ->  H  e.  (SubGrp `  G ) )
2 eqid 2467 . . . . 5  |-  ( Gs  H )  =  ( Gs  H )
32subggrp 16076 . . . 4  |-  ( H  e.  (SubGrp `  G
)  ->  ( Gs  H
)  e.  Grp )
41, 3syl 16 . . 3  |-  ( ph  ->  ( Gs  H )  e.  Grp )
5 sylow2b.xf . . . . 5  |-  ( ph  ->  X  e.  Fin )
6 pwfi 7827 . . . . 5  |-  ( X  e.  Fin  <->  ~P X  e.  Fin )
75, 6sylib 196 . . . 4  |-  ( ph  ->  ~P X  e.  Fin )
8 sylow2b.k . . . . . 6  |-  ( ph  ->  K  e.  (SubGrp `  G ) )
9 sylow2b.x . . . . . . 7  |-  X  =  ( Base `  G
)
10 sylow2b.r . . . . . . 7  |-  .~  =  ( G ~QG  K )
119, 10eqger 16123 . . . . . 6  |-  ( K  e.  (SubGrp `  G
)  ->  .~  Er  X
)
128, 11syl 16 . . . . 5  |-  ( ph  ->  .~  Er  X )
1312qsss 7384 . . . 4  |-  ( ph  ->  ( X /.  .~  )  C_  ~P X )
147, 13ssexd 4600 . . 3  |-  ( ph  ->  ( X /.  .~  )  e.  _V )
154, 14jca 532 . 2  |-  ( ph  ->  ( ( Gs  H )  e.  Grp  /\  ( X /.  .~  )  e. 
_V ) )
16 sylow2b.m . . . . . . 7  |-  .x.  =  ( x  e.  H ,  y  e.  ( X /.  .~  )  |->  ran  ( z  e.  y 
|->  ( x  .+  z
) ) )
17 vex 3121 . . . . . . . . 9  |-  y  e. 
_V
1817mptex 6142 . . . . . . . 8  |-  ( z  e.  y  |->  ( x 
.+  z ) )  e.  _V
1918rnex 6729 . . . . . . 7  |-  ran  (
z  e.  y  |->  ( x  .+  z ) )  e.  _V
2016, 19fnmpt2i 6864 . . . . . 6  |-  .x.  Fn  ( H  X.  ( X /.  .~  ) )
2120a1i 11 . . . . 5  |-  ( ph  ->  .x.  Fn  ( H  X.  ( X /.  .~  ) ) )
22 eqid 2467 . . . . . . . 8  |-  ( X /.  .~  )  =  ( X /.  .~  )
23 oveq2 6303 . . . . . . . . 9  |-  ( [ s ]  .~  =  v  ->  ( u  .x.  [ s ]  .~  )  =  ( u  .x.  v ) )
2423eleq1d 2536 . . . . . . . 8  |-  ( [ s ]  .~  =  v  ->  ( ( u 
.x.  [ s ]  .~  )  e.  ( X /.  .~  )  <->  ( u  .x.  v )  e.  ( X /.  .~  )
) )
25 sylow2b.a . . . . . . . . . . 11  |-  .+  =  ( +g  `  G )
269, 5, 1, 8, 25, 10, 16sylow2blem1 16513 . . . . . . . . . 10  |-  ( (
ph  /\  u  e.  H  /\  s  e.  X
)  ->  ( u  .x.  [ s ]  .~  )  =  [ (
u  .+  s ) ]  .~  )
27 ovex 6320 . . . . . . . . . . . 12  |-  ( G ~QG  K )  e.  _V
2810, 27eqeltri 2551 . . . . . . . . . . 11  |-  .~  e.  _V
29 subgrcl 16078 . . . . . . . . . . . . . 14  |-  ( H  e.  (SubGrp `  G
)  ->  G  e.  Grp )
301, 29syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  G  e.  Grp )
31303ad2ant1 1017 . . . . . . . . . . . 12  |-  ( (
ph  /\  u  e.  H  /\  s  e.  X
)  ->  G  e.  Grp )
329subgss 16074 . . . . . . . . . . . . . . 15  |-  ( H  e.  (SubGrp `  G
)  ->  H  C_  X
)
331, 32syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  H  C_  X )
3433sselda 3509 . . . . . . . . . . . . 13  |-  ( (
ph  /\  u  e.  H )  ->  u  e.  X )
35343adant3 1016 . . . . . . . . . . . 12  |-  ( (
ph  /\  u  e.  H  /\  s  e.  X
)  ->  u  e.  X )
36 simp3 998 . . . . . . . . . . . 12  |-  ( (
ph  /\  u  e.  H  /\  s  e.  X
)  ->  s  e.  X )
379, 25grpcl 15935 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  u  e.  X  /\  s  e.  X )  ->  ( u  .+  s
)  e.  X )
3831, 35, 36, 37syl3anc 1228 . . . . . . . . . . 11  |-  ( (
ph  /\  u  e.  H  /\  s  e.  X
)  ->  ( u  .+  s )  e.  X
)
39 ecelqsg 7378 . . . . . . . . . . 11  |-  ( (  .~  e.  _V  /\  ( u  .+  s )  e.  X )  ->  [ ( u  .+  s ) ]  .~  e.  ( X /.  .~  ) )
4028, 38, 39sylancr 663 . . . . . . . . . 10  |-  ( (
ph  /\  u  e.  H  /\  s  e.  X
)  ->  [ (
u  .+  s ) ]  .~  e.  ( X /.  .~  ) )
4126, 40eqeltrd 2555 . . . . . . . . 9  |-  ( (
ph  /\  u  e.  H  /\  s  e.  X
)  ->  ( u  .x.  [ s ]  .~  )  e.  ( X /.  .~  ) )
42413expa 1196 . . . . . . . 8  |-  ( ( ( ph  /\  u  e.  H )  /\  s  e.  X )  ->  (
u  .x.  [ s ]  .~  )  e.  ( X /.  .~  )
)
4322, 24, 42ectocld 7390 . . . . . . 7  |-  ( ( ( ph  /\  u  e.  H )  /\  v  e.  ( X /.  .~  ) )  ->  (
u  .x.  v )  e.  ( X /.  .~  ) )
4443ralrimiva 2881 . . . . . 6  |-  ( (
ph  /\  u  e.  H )  ->  A. v  e.  ( X /.  .~  ) ( u  .x.  v )  e.  ( X /.  .~  )
)
4544ralrimiva 2881 . . . . 5  |-  ( ph  ->  A. u  e.  H  A. v  e.  ( X /.  .~  ) ( u  .x.  v )  e.  ( X /.  .~  ) )
46 ffnov 6401 . . . . 5  |-  (  .x.  : ( H  X.  ( X /.  .~  ) ) --> ( X /.  .~  ) 
<->  (  .x.  Fn  ( H  X.  ( X /.  .~  ) )  /\  A. u  e.  H  A. v  e.  ( X /.  .~  ) ( u 
.x.  v )  e.  ( X /.  .~  ) ) )
4721, 45, 46sylanbrc 664 . . . 4  |-  ( ph  ->  .x.  : ( H  X.  ( X /.  .~  ) ) --> ( X /.  .~  ) )
482subgbas 16077 . . . . . . 7  |-  ( H  e.  (SubGrp `  G
)  ->  H  =  ( Base `  ( Gs  H
) ) )
491, 48syl 16 . . . . . 6  |-  ( ph  ->  H  =  ( Base `  ( Gs  H ) ) )
5049xpeq1d 5028 . . . . 5  |-  ( ph  ->  ( H  X.  ( X /.  .~  ) )  =  ( ( Base `  ( Gs  H ) )  X.  ( X /.  .~  ) ) )
5150feq2d 5724 . . . 4  |-  ( ph  ->  (  .x.  : ( H  X.  ( X /.  .~  ) ) --> ( X /.  .~  ) 
<-> 
.x.  : ( ( Base `  ( Gs  H ) )  X.  ( X /.  .~  ) ) --> ( X /.  .~  ) ) )
5247, 51mpbid 210 . . 3  |-  ( ph  ->  .x.  : ( (
Base `  ( Gs  H
) )  X.  ( X /.  .~  ) ) --> ( X /.  .~  ) )
53 oveq2 6303 . . . . . . 7  |-  ( [ s ]  .~  =  u  ->  ( ( 0g
`  ( Gs  H ) )  .x.  [ s ]  .~  )  =  ( ( 0g `  ( Gs  H ) )  .x.  u ) )
54 id 22 . . . . . . 7  |-  ( [ s ]  .~  =  u  ->  [ s ]  .~  =  u )
5553, 54eqeq12d 2489 . . . . . 6  |-  ( [ s ]  .~  =  u  ->  ( ( ( 0g `  ( Gs  H ) )  .x.  [ s ]  .~  )  =  [ s ]  .~  <->  ( ( 0g `  ( Gs  H ) )  .x.  u )  =  u ) )
56 oveq2 6303 . . . . . . . 8  |-  ( [ s ]  .~  =  u  ->  ( ( a ( +g  `  ( Gs  H ) ) b )  .x.  [ s ]  .~  )  =  ( ( a ( +g  `  ( Gs  H ) ) b ) 
.x.  u ) )
57 oveq2 6303 . . . . . . . . 9  |-  ( [ s ]  .~  =  u  ->  ( b  .x.  [ s ]  .~  )  =  ( b  .x.  u ) )
5857oveq2d 6311 . . . . . . . 8  |-  ( [ s ]  .~  =  u  ->  ( a  .x.  ( b  .x.  [ s ]  .~  ) )  =  ( a  .x.  ( b  .x.  u
) ) )
5956, 58eqeq12d 2489 . . . . . . 7  |-  ( [ s ]  .~  =  u  ->  ( ( ( a ( +g  `  ( Gs  H ) ) b )  .x.  [ s ]  .~  )  =  ( a  .x.  (
b  .x.  [ s ]  .~  ) )  <->  ( (
a ( +g  `  ( Gs  H ) ) b )  .x.  u )  =  ( a  .x.  ( b  .x.  u
) ) ) )
60592ralbidv 2911 . . . . . 6  |-  ( [ s ]  .~  =  u  ->  ( A. a  e.  ( Base `  ( Gs  H ) ) A. b  e.  ( Base `  ( Gs  H ) ) ( ( a ( +g  `  ( Gs  H ) ) b )  .x.  [ s ]  .~  )  =  ( a  .x.  (
b  .x.  [ s ]  .~  ) )  <->  A. a  e.  ( Base `  ( Gs  H ) ) A. b  e.  ( Base `  ( Gs  H ) ) ( ( a ( +g  `  ( Gs  H ) ) b )  .x.  u )  =  ( a  .x.  ( b  .x.  u
) ) ) )
6155, 60anbi12d 710 . . . . 5  |-  ( [ s ]  .~  =  u  ->  ( ( ( ( 0g `  ( Gs  H ) )  .x.  [ s ]  .~  )  =  [ s ]  .~  /\ 
A. a  e.  (
Base `  ( Gs  H
) ) A. b  e.  ( Base `  ( Gs  H ) ) ( ( a ( +g  `  ( Gs  H ) ) b )  .x.  [ s ]  .~  )  =  ( a  .x.  (
b  .x.  [ s ]  .~  ) ) )  <-> 
( ( ( 0g
`  ( Gs  H ) )  .x.  u )  =  u  /\  A. a  e.  ( Base `  ( Gs  H ) ) A. b  e.  ( Base `  ( Gs  H ) ) ( ( a ( +g  `  ( Gs  H ) ) b )  .x.  u )  =  ( a  .x.  ( b  .x.  u
) ) ) ) )
62 simpl 457 . . . . . . . 8  |-  ( (
ph  /\  s  e.  X )  ->  ph )
631adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  X )  ->  H  e.  (SubGrp `  G )
)
64 eqid 2467 . . . . . . . . . 10  |-  ( 0g
`  G )  =  ( 0g `  G
)
6564subg0cl 16081 . . . . . . . . 9  |-  ( H  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  H
)
6663, 65syl 16 . . . . . . . 8  |-  ( (
ph  /\  s  e.  X )  ->  ( 0g `  G )  e.  H )
67 simpr 461 . . . . . . . 8  |-  ( (
ph  /\  s  e.  X )  ->  s  e.  X )
689, 5, 1, 8, 25, 10, 16sylow2blem1 16513 . . . . . . . 8  |-  ( (
ph  /\  ( 0g `  G )  e.  H  /\  s  e.  X
)  ->  ( ( 0g `  G )  .x.  [ s ]  .~  )  =  [ ( ( 0g
`  G )  .+  s ) ]  .~  )
6962, 66, 67, 68syl3anc 1228 . . . . . . 7  |-  ( (
ph  /\  s  e.  X )  ->  (
( 0g `  G
)  .x.  [ s ]  .~  )  =  [
( ( 0g `  G )  .+  s
) ]  .~  )
702, 64subg0 16079 . . . . . . . . 9  |-  ( H  e.  (SubGrp `  G
)  ->  ( 0g `  G )  =  ( 0g `  ( Gs  H ) ) )
7163, 70syl 16 . . . . . . . 8  |-  ( (
ph  /\  s  e.  X )  ->  ( 0g `  G )  =  ( 0g `  ( Gs  H ) ) )
7271oveq1d 6310 . . . . . . 7  |-  ( (
ph  /\  s  e.  X )  ->  (
( 0g `  G
)  .x.  [ s ]  .~  )  =  ( ( 0g `  ( Gs  H ) )  .x.  [ s ]  .~  )
)
739, 25, 64grplid 15952 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  s  e.  X )  ->  ( ( 0g `  G )  .+  s
)  =  s )
7430, 73sylan 471 . . . . . . . 8  |-  ( (
ph  /\  s  e.  X )  ->  (
( 0g `  G
)  .+  s )  =  s )
7574eceq1d 7360 . . . . . . 7  |-  ( (
ph  /\  s  e.  X )  ->  [ ( ( 0g `  G
)  .+  s ) ]  .~  =  [ s ]  .~  )
7669, 72, 753eqtr3d 2516 . . . . . 6  |-  ( (
ph  /\  s  e.  X )  ->  (
( 0g `  ( Gs  H ) )  .x.  [ s ]  .~  )  =  [ s ]  .~  )
7763adantr 465 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  H  e.  (SubGrp `  G ) )
7877, 29syl 16 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  G  e.  Grp )
7977, 32syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  H  C_  X
)
80 simprl 755 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  a  e.  H )
8179, 80sseldd 3510 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  a  e.  X )
82 simprr 756 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  b  e.  H )
8379, 82sseldd 3510 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  b  e.  X )
8467adantr 465 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  s  e.  X )
859, 25grpass 15936 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  ( a  e.  X  /\  b  e.  X  /\  s  e.  X
) )  ->  (
( a  .+  b
)  .+  s )  =  ( a  .+  ( b  .+  s
) ) )
8678, 81, 83, 84, 85syl13anc 1230 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  ( (
a  .+  b )  .+  s )  =  ( a  .+  ( b 
.+  s ) ) )
8786eceq1d 7360 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  [ (
( a  .+  b
)  .+  s ) ]  .~  =  [ ( a  .+  ( b 
.+  s ) ) ]  .~  )
8862adantr 465 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  ph )
899, 25grpcl 15935 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  b  e.  X  /\  s  e.  X )  ->  ( b  .+  s
)  e.  X )
9078, 83, 84, 89syl3anc 1228 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  ( b  .+  s )  e.  X
)
919, 5, 1, 8, 25, 10, 16sylow2blem1 16513 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  H  /\  ( b  .+  s )  e.  X
)  ->  ( a  .x.  [ ( b  .+  s ) ]  .~  )  =  [ (
a  .+  ( b  .+  s ) ) ]  .~  )
9288, 80, 90, 91syl3anc 1228 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  ( a  .x.  [ ( b  .+  s ) ]  .~  )  =  [ (
a  .+  ( b  .+  s ) ) ]  .~  )
9387, 92eqtr4d 2511 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  [ (
( a  .+  b
)  .+  s ) ]  .~  =  ( a 
.x.  [ ( b  .+  s ) ]  .~  ) )
9425subgcl 16083 . . . . . . . . . . 11  |-  ( ( H  e.  (SubGrp `  G )  /\  a  e.  H  /\  b  e.  H )  ->  (
a  .+  b )  e.  H )
9577, 80, 82, 94syl3anc 1228 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  ( a  .+  b )  e.  H
)
969, 5, 1, 8, 25, 10, 16sylow2blem1 16513 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  .+  b )  e.  H  /\  s  e.  X
)  ->  ( (
a  .+  b )  .x.  [ s ]  .~  )  =  [ (
( a  .+  b
)  .+  s ) ]  .~  )
9788, 95, 84, 96syl3anc 1228 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  ( (
a  .+  b )  .x.  [ s ]  .~  )  =  [ (
( a  .+  b
)  .+  s ) ]  .~  )
989, 5, 1, 8, 25, 10, 16sylow2blem1 16513 . . . . . . . . . . 11  |-  ( (
ph  /\  b  e.  H  /\  s  e.  X
)  ->  ( b  .x.  [ s ]  .~  )  =  [ (
b  .+  s ) ]  .~  )
9988, 82, 84, 98syl3anc 1228 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  ( b  .x.  [ s ]  .~  )  =  [ (
b  .+  s ) ]  .~  )
10099oveq2d 6311 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  ( a  .x.  ( b  .x.  [ s ]  .~  ) )  =  ( a  .x.  [ ( b  .+  s
) ]  .~  )
)
10193, 97, 1003eqtr4d 2518 . . . . . . . 8  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  ( (
a  .+  b )  .x.  [ s ]  .~  )  =  ( a  .x.  ( b  .x.  [ s ]  .~  ) ) )
102101ralrimivva 2888 . . . . . . 7  |-  ( (
ph  /\  s  e.  X )  ->  A. a  e.  H  A. b  e.  H  ( (
a  .+  b )  .x.  [ s ]  .~  )  =  ( a  .x.  ( b  .x.  [ s ]  .~  ) ) )
10363, 48syl 16 . . . . . . . 8  |-  ( (
ph  /\  s  e.  X )  ->  H  =  ( Base `  ( Gs  H ) ) )
1042, 25ressplusg 14614 . . . . . . . . . . . . 13  |-  ( H  e.  (SubGrp `  G
)  ->  .+  =  ( +g  `  ( Gs  H ) ) )
1051, 104syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  .+  =  ( +g  `  ( Gs  H ) ) )
106105proplem3 14963 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  X )  ->  (
a  .+  b )  =  ( a ( +g  `  ( Gs  H ) ) b ) )
107106oveq1d 6310 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  X )  ->  (
( a  .+  b
)  .x.  [ s ]  .~  )  =  ( ( a ( +g  `  ( Gs  H ) ) b )  .x.  [ s ]  .~  ) )
108107eqeq1d 2469 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  X )  ->  (
( ( a  .+  b )  .x.  [ s ]  .~  )  =  ( a  .x.  (
b  .x.  [ s ]  .~  ) )  <->  ( (
a ( +g  `  ( Gs  H ) ) b )  .x.  [ s ]  .~  )  =  ( a  .x.  (
b  .x.  [ s ]  .~  ) ) ) )
109103, 108raleqbidv 3077 . . . . . . . 8  |-  ( (
ph  /\  s  e.  X )  ->  ( A. b  e.  H  ( ( a  .+  b )  .x.  [ s ]  .~  )  =  ( a  .x.  (
b  .x.  [ s ]  .~  ) )  <->  A. b  e.  ( Base `  ( Gs  H ) ) ( ( a ( +g  `  ( Gs  H ) ) b )  .x.  [ s ]  .~  )  =  ( a  .x.  (
b  .x.  [ s ]  .~  ) ) ) )
110103, 109raleqbidv 3077 . . . . . . 7  |-  ( (
ph  /\  s  e.  X )  ->  ( A. a  e.  H  A. b  e.  H  ( ( a  .+  b )  .x.  [ s ]  .~  )  =  ( a  .x.  (
b  .x.  [ s ]  .~  ) )  <->  A. a  e.  ( Base `  ( Gs  H ) ) A. b  e.  ( Base `  ( Gs  H ) ) ( ( a ( +g  `  ( Gs  H ) ) b )  .x.  [ s ]  .~  )  =  ( a  .x.  (
b  .x.  [ s ]  .~  ) ) ) )
111102, 110mpbid 210 . . . . . 6  |-  ( (
ph  /\  s  e.  X )  ->  A. a  e.  ( Base `  ( Gs  H ) ) A. b  e.  ( Base `  ( Gs  H ) ) ( ( a ( +g  `  ( Gs  H ) ) b )  .x.  [ s ]  .~  )  =  ( a  .x.  (
b  .x.  [ s ]  .~  ) ) )
11276, 111jca 532 . . . . 5  |-  ( (
ph  /\  s  e.  X )  ->  (
( ( 0g `  ( Gs  H ) )  .x.  [ s ]  .~  )  =  [ s ]  .~  /\ 
A. a  e.  (
Base `  ( Gs  H
) ) A. b  e.  ( Base `  ( Gs  H ) ) ( ( a ( +g  `  ( Gs  H ) ) b )  .x.  [ s ]  .~  )  =  ( a  .x.  (
b  .x.  [ s ]  .~  ) ) ) )
11322, 61, 112ectocld 7390 . . . 4  |-  ( (
ph  /\  u  e.  ( X /.  .~  )
)  ->  ( (
( 0g `  ( Gs  H ) )  .x.  u )  =  u  /\  A. a  e.  ( Base `  ( Gs  H ) ) A. b  e.  ( Base `  ( Gs  H ) ) ( ( a ( +g  `  ( Gs  H ) ) b )  .x.  u )  =  ( a  .x.  ( b  .x.  u
) ) ) )
114113ralrimiva 2881 . . 3  |-  ( ph  ->  A. u  e.  ( X /.  .~  )
( ( ( 0g
`  ( Gs  H ) )  .x.  u )  =  u  /\  A. a  e.  ( Base `  ( Gs  H ) ) A. b  e.  ( Base `  ( Gs  H ) ) ( ( a ( +g  `  ( Gs  H ) ) b )  .x.  u )  =  ( a  .x.  ( b  .x.  u
) ) ) )
11552, 114jca 532 . 2  |-  ( ph  ->  (  .x.  : ( ( Base `  ( Gs  H ) )  X.  ( X /.  .~  ) ) --> ( X /.  .~  )  /\  A. u  e.  ( X /.  .~  ) ( ( ( 0g `  ( Gs  H ) )  .x.  u )  =  u  /\  A. a  e.  ( Base `  ( Gs  H ) ) A. b  e.  ( Base `  ( Gs  H ) ) ( ( a ( +g  `  ( Gs  H ) ) b )  .x.  u )  =  ( a  .x.  ( b  .x.  u
) ) ) ) )
116 eqid 2467 . . 3  |-  ( Base `  ( Gs  H ) )  =  ( Base `  ( Gs  H ) )
117 eqid 2467 . . 3  |-  ( +g  `  ( Gs  H ) )  =  ( +g  `  ( Gs  H ) )
118 eqid 2467 . . 3  |-  ( 0g
`  ( Gs  H ) )  =  ( 0g
`  ( Gs  H ) )
119116, 117, 118isga 16201 . 2  |-  (  .x.  e.  ( ( Gs  H ) 
GrpAct  ( X /.  .~  ) )  <->  ( (
( Gs  H )  e.  Grp  /\  ( X /.  .~  )  e.  _V )  /\  (  .x.  : ( ( Base `  ( Gs  H ) )  X.  ( X /.  .~  ) ) --> ( X /.  .~  )  /\  A. u  e.  ( X /.  .~  ) ( ( ( 0g `  ( Gs  H ) )  .x.  u )  =  u  /\  A. a  e.  ( Base `  ( Gs  H ) ) A. b  e.  ( Base `  ( Gs  H ) ) ( ( a ( +g  `  ( Gs  H ) ) b )  .x.  u )  =  ( a  .x.  ( b  .x.  u
) ) ) ) ) )
12015, 115, 119sylanbrc 664 1  |-  ( ph  ->  .x.  e.  ( ( Gs  H )  GrpAct  ( X /.  .~  ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   _Vcvv 3118    C_ wss 3481   ~Pcpw 4016    |-> cmpt 4511    X. cxp 5003   ran crn 5006    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297    Er wer 7320   [cec 7321   /.cqs 7322   Fincfn 7528   Basecbs 14507   ↾s cress 14508   +g cplusg 14572   0gc0g 14712   Grpcgrp 15925  SubGrpcsubg 16067   ~QG cqg 16069    GrpAct cga 16199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-ec 7325  df-qs 7329  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-sbg 15931  df-subg 16070  df-eqg 16072  df-ga 16200
This theorem is referenced by:  sylow2blem3  16515
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