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Theorem sylow2blem2 16113
Description: Lemma for sylow2b 16115. 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 2441 . . . . 5  |-  ( Gs  H )  =  ( Gs  H )
32subggrp 15677 . . . 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 7602 . . . . 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 15724 . . . . . 6  |-  ( K  e.  (SubGrp `  G
)  ->  .~  Er  X
)
128, 11syl 16 . . . . 5  |-  ( ph  ->  .~  Er  X )
1312qsss 7157 . . . 4  |-  ( ph  ->  ( X /.  .~  )  C_  ~P X )
147, 13ssexd 4436 . . 3  |-  ( ph  ->  ( X /.  .~  )  e.  _V )
154, 14jca 529 . 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 2973 . . . . . . . . 9  |-  y  e. 
_V
1817mptex 5945 . . . . . . . 8  |-  ( z  e.  y  |->  ( x 
.+  z ) )  e.  _V
1918rnex 6511 . . . . . . 7  |-  ran  (
z  e.  y  |->  ( x  .+  z ) )  e.  _V
2016, 19fnmpt2i 6642 . . . . . 6  |-  .x.  Fn  ( H  X.  ( X /.  .~  ) )
2120a1i 11 . . . . 5  |-  ( ph  ->  .x.  Fn  ( H  X.  ( X /.  .~  ) ) )
22 eqid 2441 . . . . . . . 8  |-  ( X /.  .~  )  =  ( X /.  .~  )
23 oveq2 6098 . . . . . . . . 9  |-  ( [ s ]  .~  =  v  ->  ( u  .x.  [ s ]  .~  )  =  ( u  .x.  v ) )
2423eleq1d 2507 . . . . . . . 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 16112 . . . . . . . . . 10  |-  ( (
ph  /\  u  e.  H  /\  s  e.  X
)  ->  ( u  .x.  [ s ]  .~  )  =  [ (
u  .+  s ) ]  .~  )
27 ovex 6115 . . . . . . . . . . . 12  |-  ( G ~QG  K )  e.  _V
2810, 27eqeltri 2511 . . . . . . . . . . 11  |-  .~  e.  _V
29 subgrcl 15679 . . . . . . . . . . . . . 14  |-  ( H  e.  (SubGrp `  G
)  ->  G  e.  Grp )
301, 29syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  G  e.  Grp )
31303ad2ant1 1004 . . . . . . . . . . . 12  |-  ( (
ph  /\  u  e.  H  /\  s  e.  X
)  ->  G  e.  Grp )
329subgss 15675 . . . . . . . . . . . . . . 15  |-  ( H  e.  (SubGrp `  G
)  ->  H  C_  X
)
331, 32syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  H  C_  X )
3433sselda 3353 . . . . . . . . . . . . 13  |-  ( (
ph  /\  u  e.  H )  ->  u  e.  X )
35343adant3 1003 . . . . . . . . . . . 12  |-  ( (
ph  /\  u  e.  H  /\  s  e.  X
)  ->  u  e.  X )
36 simp3 985 . . . . . . . . . . . 12  |-  ( (
ph  /\  u  e.  H  /\  s  e.  X
)  ->  s  e.  X )
379, 25grpcl 15544 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  u  e.  X  /\  s  e.  X )  ->  ( u  .+  s
)  e.  X )
3831, 35, 36, 37syl3anc 1213 . . . . . . . . . . 11  |-  ( (
ph  /\  u  e.  H  /\  s  e.  X
)  ->  ( u  .+  s )  e.  X
)
39 ecelqsg 7151 . . . . . . . . . . 11  |-  ( (  .~  e.  _V  /\  ( u  .+  s )  e.  X )  ->  [ ( u  .+  s ) ]  .~  e.  ( X /.  .~  ) )
4028, 38, 39sylancr 658 . . . . . . . . . 10  |-  ( (
ph  /\  u  e.  H  /\  s  e.  X
)  ->  [ (
u  .+  s ) ]  .~  e.  ( X /.  .~  ) )
4126, 40eqeltrd 2515 . . . . . . . . 9  |-  ( (
ph  /\  u  e.  H  /\  s  e.  X
)  ->  ( u  .x.  [ s ]  .~  )  e.  ( X /.  .~  ) )
42413expa 1182 . . . . . . . 8  |-  ( ( ( ph  /\  u  e.  H )  /\  s  e.  X )  ->  (
u  .x.  [ s ]  .~  )  e.  ( X /.  .~  )
)
4322, 24, 42ectocld 7163 . . . . . . 7  |-  ( ( ( ph  /\  u  e.  H )  /\  v  e.  ( X /.  .~  ) )  ->  (
u  .x.  v )  e.  ( X /.  .~  ) )
4443ralrimiva 2797 . . . . . 6  |-  ( (
ph  /\  u  e.  H )  ->  A. v  e.  ( X /.  .~  ) ( u  .x.  v )  e.  ( X /.  .~  )
)
4544ralrimiva 2797 . . . . 5  |-  ( ph  ->  A. u  e.  H  A. v  e.  ( X /.  .~  ) ( u  .x.  v )  e.  ( X /.  .~  ) )
46 ffnov 6193 . . . . 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 659 . . . 4  |-  ( ph  ->  .x.  : ( H  X.  ( X /.  .~  ) ) --> ( X /.  .~  ) )
482subgbas 15678 . . . . . . 7  |-  ( H  e.  (SubGrp `  G
)  ->  H  =  ( Base `  ( Gs  H
) ) )
491, 48syl 16 . . . . . 6  |-  ( ph  ->  H  =  ( Base `  ( Gs  H ) ) )
5049xpeq1d 4859 . . . . 5  |-  ( ph  ->  ( H  X.  ( X /.  .~  ) )  =  ( ( Base `  ( Gs  H ) )  X.  ( X /.  .~  ) ) )
5150feq2d 5544 . . . 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 6098 . . . . . . 7  |-  ( [ s ]  .~  =  u  ->  ( ( 0g
`  ( Gs  H ) )  .x.  [ s ]  .~  )  =  ( ( 0g `  ( Gs  H ) )  .x.  u ) )
54 id 22 . . . . . . 7  |-  ( [ s ]  .~  =  u  ->  [ s ]  .~  =  u )
5553, 54eqeq12d 2455 . . . . . 6  |-  ( [ s ]  .~  =  u  ->  ( ( ( 0g `  ( Gs  H ) )  .x.  [ s ]  .~  )  =  [ s ]  .~  <->  ( ( 0g `  ( Gs  H ) )  .x.  u )  =  u ) )
56 oveq2 6098 . . . . . . . 8  |-  ( [ s ]  .~  =  u  ->  ( ( a ( +g  `  ( Gs  H ) ) b )  .x.  [ s ]  .~  )  =  ( ( a ( +g  `  ( Gs  H ) ) b ) 
.x.  u ) )
57 oveq2 6098 . . . . . . . . 9  |-  ( [ s ]  .~  =  u  ->  ( b  .x.  [ s ]  .~  )  =  ( b  .x.  u ) )
5857oveq2d 6106 . . . . . . . 8  |-  ( [ s ]  .~  =  u  ->  ( a  .x.  ( b  .x.  [ s ]  .~  ) )  =  ( a  .x.  ( b  .x.  u
) ) )
5956, 58eqeq12d 2455 . . . . . . 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 2755 . . . . . 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 705 . . . . 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 454 . . . . . . . 8  |-  ( (
ph  /\  s  e.  X )  ->  ph )
631adantr 462 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  X )  ->  H  e.  (SubGrp `  G )
)
64 eqid 2441 . . . . . . . . . 10  |-  ( 0g
`  G )  =  ( 0g `  G
)
6564subg0cl 15682 . . . . . . . . 9  |-  ( H  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  H
)
6663, 65syl 16 . . . . . . . 8  |-  ( (
ph  /\  s  e.  X )  ->  ( 0g `  G )  e.  H )
67 simpr 458 . . . . . . . 8  |-  ( (
ph  /\  s  e.  X )  ->  s  e.  X )
689, 5, 1, 8, 25, 10, 16sylow2blem1 16112 . . . . . . . 8  |-  ( (
ph  /\  ( 0g `  G )  e.  H  /\  s  e.  X
)  ->  ( ( 0g `  G )  .x.  [ s ]  .~  )  =  [ ( ( 0g
`  G )  .+  s ) ]  .~  )
6962, 66, 67, 68syl3anc 1213 . . . . . . 7  |-  ( (
ph  /\  s  e.  X )  ->  (
( 0g `  G
)  .x.  [ s ]  .~  )  =  [
( ( 0g `  G )  .+  s
) ]  .~  )
702, 64subg0 15680 . . . . . . . . 9  |-  ( H  e.  (SubGrp `  G
)  ->  ( 0g `  G )  =  ( 0g `  ( Gs  H ) ) )
7163, 70syl 16 . . . . . . . 8  |-  ( (
ph  /\  s  e.  X )  ->  ( 0g `  G )  =  ( 0g `  ( Gs  H ) ) )
7271oveq1d 6105 . . . . . . 7  |-  ( (
ph  /\  s  e.  X )  ->  (
( 0g `  G
)  .x.  [ s ]  .~  )  =  ( ( 0g `  ( Gs  H ) )  .x.  [ s ]  .~  )
)
739, 25, 64grplid 15561 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  s  e.  X )  ->  ( ( 0g `  G )  .+  s
)  =  s )
7430, 73sylan 468 . . . . . . . 8  |-  ( (
ph  /\  s  e.  X )  ->  (
( 0g `  G
)  .+  s )  =  s )
75 eceq1 7133 . . . . . . . 8  |-  ( ( ( 0g `  G
)  .+  s )  =  s  ->  [ ( ( 0g `  G
)  .+  s ) ]  .~  =  [ s ]  .~  )
7674, 75syl 16 . . . . . . 7  |-  ( (
ph  /\  s  e.  X )  ->  [ ( ( 0g `  G
)  .+  s ) ]  .~  =  [ s ]  .~  )
7769, 72, 763eqtr3d 2481 . . . . . 6  |-  ( (
ph  /\  s  e.  X )  ->  (
( 0g `  ( Gs  H ) )  .x.  [ s ]  .~  )  =  [ s ]  .~  )
7863adantr 462 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  H  e.  (SubGrp `  G ) )
7978, 29syl 16 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  G  e.  Grp )
8078, 32syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  H  C_  X
)
81 simprl 750 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  a  e.  H )
8280, 81sseldd 3354 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  a  e.  X )
83 simprr 751 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  b  e.  H )
8480, 83sseldd 3354 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  b  e.  X )
8567adantr 462 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  s  e.  X )
869, 25grpass 15545 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  ( a  e.  X  /\  b  e.  X  /\  s  e.  X
) )  ->  (
( a  .+  b
)  .+  s )  =  ( a  .+  ( b  .+  s
) ) )
8779, 82, 84, 85, 86syl13anc 1215 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  ( (
a  .+  b )  .+  s )  =  ( a  .+  ( b 
.+  s ) ) )
88 eceq1 7133 . . . . . . . . . . 11  |-  ( ( ( a  .+  b
)  .+  s )  =  ( a  .+  ( b  .+  s
) )  ->  [ ( ( a  .+  b
)  .+  s ) ]  .~  =  [ ( a  .+  ( b 
.+  s ) ) ]  .~  )
8987, 88syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  [ (
( a  .+  b
)  .+  s ) ]  .~  =  [ ( a  .+  ( b 
.+  s ) ) ]  .~  )
9062adantr 462 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  ph )
919, 25grpcl 15544 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  b  e.  X  /\  s  e.  X )  ->  ( b  .+  s
)  e.  X )
9279, 84, 85, 91syl3anc 1213 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  ( b  .+  s )  e.  X
)
939, 5, 1, 8, 25, 10, 16sylow2blem1 16112 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  H  /\  ( b  .+  s )  e.  X
)  ->  ( a  .x.  [ ( b  .+  s ) ]  .~  )  =  [ (
a  .+  ( b  .+  s ) ) ]  .~  )
9490, 81, 92, 93syl3anc 1213 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  ( a  .x.  [ ( b  .+  s ) ]  .~  )  =  [ (
a  .+  ( b  .+  s ) ) ]  .~  )
9589, 94eqtr4d 2476 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  [ (
( a  .+  b
)  .+  s ) ]  .~  =  ( a 
.x.  [ ( b  .+  s ) ]  .~  ) )
9625subgcl 15684 . . . . . . . . . . 11  |-  ( ( H  e.  (SubGrp `  G )  /\  a  e.  H  /\  b  e.  H )  ->  (
a  .+  b )  e.  H )
9778, 81, 83, 96syl3anc 1213 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  ( a  .+  b )  e.  H
)
989, 5, 1, 8, 25, 10, 16sylow2blem1 16112 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  .+  b )  e.  H  /\  s  e.  X
)  ->  ( (
a  .+  b )  .x.  [ s ]  .~  )  =  [ (
( a  .+  b
)  .+  s ) ]  .~  )
9990, 97, 85, 98syl3anc 1213 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  ( (
a  .+  b )  .x.  [ s ]  .~  )  =  [ (
( a  .+  b
)  .+  s ) ]  .~  )
1009, 5, 1, 8, 25, 10, 16sylow2blem1 16112 . . . . . . . . . . 11  |-  ( (
ph  /\  b  e.  H  /\  s  e.  X
)  ->  ( b  .x.  [ s ]  .~  )  =  [ (
b  .+  s ) ]  .~  )
10190, 83, 85, 100syl3anc 1213 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  ( b  .x.  [ s ]  .~  )  =  [ (
b  .+  s ) ]  .~  )
102101oveq2d 6106 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  ( a  .x.  ( b  .x.  [ s ]  .~  ) )  =  ( a  .x.  [ ( b  .+  s
) ]  .~  )
)
10395, 99, 1023eqtr4d 2483 . . . . . . . 8  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  ( (
a  .+  b )  .x.  [ s ]  .~  )  =  ( a  .x.  ( b  .x.  [ s ]  .~  ) ) )
104103ralrimivva 2806 . . . . . . 7  |-  ( (
ph  /\  s  e.  X )  ->  A. a  e.  H  A. b  e.  H  ( (
a  .+  b )  .x.  [ s ]  .~  )  =  ( a  .x.  ( b  .x.  [ s ]  .~  ) ) )
10563, 48syl 16 . . . . . . . 8  |-  ( (
ph  /\  s  e.  X )  ->  H  =  ( Base `  ( Gs  H ) ) )
1062, 25ressplusg 14276 . . . . . . . . . . . . 13  |-  ( H  e.  (SubGrp `  G
)  ->  .+  =  ( +g  `  ( Gs  H ) ) )
1071, 106syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  .+  =  ( +g  `  ( Gs  H ) ) )
108107proplem3 14625 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  X )  ->  (
a  .+  b )  =  ( a ( +g  `  ( Gs  H ) ) b ) )
109108oveq1d 6105 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  X )  ->  (
( a  .+  b
)  .x.  [ s ]  .~  )  =  ( ( a ( +g  `  ( Gs  H ) ) b )  .x.  [ s ]  .~  ) )
110109eqeq1d 2449 . . . . . . . . 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 ]  .~  ) ) ) )
111105, 110raleqbidv 2929 . . . . . . . 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 ]  .~  ) ) ) )
112105, 111raleqbidv 2929 . . . . . . 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 ]  .~  ) ) ) )
113104, 112mpbid 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 ]  .~  ) ) )
11477, 113jca 529 . . . . 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 ]  .~  ) ) ) )
11522, 61, 114ectocld 7163 . . . 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
) ) ) )
116115ralrimiva 2797 . . 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
) ) ) )
11752, 116jca 529 . 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
) ) ) ) )
118 eqid 2441 . . 3  |-  ( Base `  ( Gs  H ) )  =  ( Base `  ( Gs  H ) )
119 eqid 2441 . . 3  |-  ( +g  `  ( Gs  H ) )  =  ( +g  `  ( Gs  H ) )
120 eqid 2441 . . 3  |-  ( 0g
`  ( Gs  H ) )  =  ( 0g
`  ( Gs  H ) )
121118, 119, 120isga 15802 . 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
) ) ) ) ) )
12215, 117, 121sylanbrc 659 1  |-  ( ph  ->  .x.  e.  ( ( Gs  H )  GrpAct  ( X /.  .~  ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   A.wral 2713   _Vcvv 2970    C_ wss 3325   ~Pcpw 3857    e. cmpt 4347    X. cxp 4834   ran crn 4837    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090    e. cmpt2 6092    Er wer 7094   [cec 7095   /.cqs 7096   Fincfn 7306   Basecbs 14170   ↾s cress 14171   +g cplusg 14234   0gc0g 14374   Grpcgrp 15406  SubGrpcsubg 15668   ~QG cqg 15670    GrpAct cga 15800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-ec 7099  df-qs 7103  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-0g 14376  df-mnd 15411  df-grp 15538  df-minusg 15539  df-sbg 15540  df-subg 15671  df-eqg 15673  df-ga 15801
This theorem is referenced by:  sylow2blem3  16114
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