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Theorem sylow2blem2 15210
Description: Lemma for sylow2b 15212. 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 2404 . . . . 5  |-  ( Gs  H )  =  ( Gs  H )
32subggrp 14902 . . . 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 7360 . . . . 5  |-  ( X  e.  Fin  <->  ~P X  e.  Fin )
75, 6sylib 189 . . . 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 14945 . . . . . 6  |-  ( K  e.  (SubGrp `  G
)  ->  .~  Er  X
)
128, 11syl 16 . . . . 5  |-  ( ph  ->  .~  Er  X )
1312qsss 6924 . . . 4  |-  ( ph  ->  ( X /.  .~  )  C_  ~P X )
147, 13ssexd 4310 . . 3  |-  ( ph  ->  ( X /.  .~  )  e.  _V )
154, 14jca 519 . 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 2919 . . . . . . . . 9  |-  y  e. 
_V
1817mptex 5925 . . . . . . . 8  |-  ( z  e.  y  |->  ( x 
.+  z ) )  e.  _V
1918rnex 5092 . . . . . . 7  |-  ran  (
z  e.  y  |->  ( x  .+  z ) )  e.  _V
2016, 19fnmpt2i 6379 . . . . . 6  |-  .x.  Fn  ( H  X.  ( X /.  .~  ) )
2120a1i 11 . . . . 5  |-  ( ph  ->  .x.  Fn  ( H  X.  ( X /.  .~  ) ) )
22 eqid 2404 . . . . . . . 8  |-  ( X /.  .~  )  =  ( X /.  .~  )
23 oveq2 6048 . . . . . . . . 9  |-  ( [ s ]  .~  =  v  ->  ( u  .x.  [ s ]  .~  )  =  ( u  .x.  v ) )
2423eleq1d 2470 . . . . . . . 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 15209 . . . . . . . . . 10  |-  ( (
ph  /\  u  e.  H  /\  s  e.  X
)  ->  ( u  .x.  [ s ]  .~  )  =  [ (
u  .+  s ) ]  .~  )
27 ovex 6065 . . . . . . . . . . . 12  |-  ( G ~QG  K )  e.  _V
2810, 27eqeltri 2474 . . . . . . . . . . 11  |-  .~  e.  _V
29 subgrcl 14904 . . . . . . . . . . . . . 14  |-  ( H  e.  (SubGrp `  G
)  ->  G  e.  Grp )
301, 29syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  G  e.  Grp )
31303ad2ant1 978 . . . . . . . . . . . 12  |-  ( (
ph  /\  u  e.  H  /\  s  e.  X
)  ->  G  e.  Grp )
329subgss 14900 . . . . . . . . . . . . . . 15  |-  ( H  e.  (SubGrp `  G
)  ->  H  C_  X
)
331, 32syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  H  C_  X )
3433sselda 3308 . . . . . . . . . . . . 13  |-  ( (
ph  /\  u  e.  H )  ->  u  e.  X )
35343adant3 977 . . . . . . . . . . . 12  |-  ( (
ph  /\  u  e.  H  /\  s  e.  X
)  ->  u  e.  X )
36 simp3 959 . . . . . . . . . . . 12  |-  ( (
ph  /\  u  e.  H  /\  s  e.  X
)  ->  s  e.  X )
379, 25grpcl 14773 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  u  e.  X  /\  s  e.  X )  ->  ( u  .+  s
)  e.  X )
3831, 35, 36, 37syl3anc 1184 . . . . . . . . . . 11  |-  ( (
ph  /\  u  e.  H  /\  s  e.  X
)  ->  ( u  .+  s )  e.  X
)
39 ecelqsg 6918 . . . . . . . . . . 11  |-  ( (  .~  e.  _V  /\  ( u  .+  s )  e.  X )  ->  [ ( u  .+  s ) ]  .~  e.  ( X /.  .~  ) )
4028, 38, 39sylancr 645 . . . . . . . . . 10  |-  ( (
ph  /\  u  e.  H  /\  s  e.  X
)  ->  [ (
u  .+  s ) ]  .~  e.  ( X /.  .~  ) )
4126, 40eqeltrd 2478 . . . . . . . . 9  |-  ( (
ph  /\  u  e.  H  /\  s  e.  X
)  ->  ( u  .x.  [ s ]  .~  )  e.  ( X /.  .~  ) )
42413expa 1153 . . . . . . . 8  |-  ( ( ( ph  /\  u  e.  H )  /\  s  e.  X )  ->  (
u  .x.  [ s ]  .~  )  e.  ( X /.  .~  )
)
4322, 24, 42ectocld 6930 . . . . . . 7  |-  ( ( ( ph  /\  u  e.  H )  /\  v  e.  ( X /.  .~  ) )  ->  (
u  .x.  v )  e.  ( X /.  .~  ) )
4443ralrimiva 2749 . . . . . 6  |-  ( (
ph  /\  u  e.  H )  ->  A. v  e.  ( X /.  .~  ) ( u  .x.  v )  e.  ( X /.  .~  )
)
4544ralrimiva 2749 . . . . 5  |-  ( ph  ->  A. u  e.  H  A. v  e.  ( X /.  .~  ) ( u  .x.  v )  e.  ( X /.  .~  ) )
46 ffnov 6133 . . . . 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 646 . . . 4  |-  ( ph  ->  .x.  : ( H  X.  ( X /.  .~  ) ) --> ( X /.  .~  ) )
482subgbas 14903 . . . . . . 7  |-  ( H  e.  (SubGrp `  G
)  ->  H  =  ( Base `  ( Gs  H
) ) )
491, 48syl 16 . . . . . 6  |-  ( ph  ->  H  =  ( Base `  ( Gs  H ) ) )
5049xpeq1d 4860 . . . . 5  |-  ( ph  ->  ( H  X.  ( X /.  .~  ) )  =  ( ( Base `  ( Gs  H ) )  X.  ( X /.  .~  ) ) )
5150feq2d 5540 . . . 4  |-  ( ph  ->  (  .x.  : ( H  X.  ( X /.  .~  ) ) --> ( X /.  .~  ) 
<-> 
.x.  : ( ( Base `  ( Gs  H ) )  X.  ( X /.  .~  ) ) --> ( X /.  .~  ) ) )
5247, 51mpbid 202 . . 3  |-  ( ph  ->  .x.  : ( (
Base `  ( Gs  H
) )  X.  ( X /.  .~  ) ) --> ( X /.  .~  ) )
53 oveq2 6048 . . . . . . 7  |-  ( [ s ]  .~  =  u  ->  ( ( 0g
`  ( Gs  H ) )  .x.  [ s ]  .~  )  =  ( ( 0g `  ( Gs  H ) )  .x.  u ) )
54 id 20 . . . . . . 7  |-  ( [ s ]  .~  =  u  ->  [ s ]  .~  =  u )
5553, 54eqeq12d 2418 . . . . . 6  |-  ( [ s ]  .~  =  u  ->  ( ( ( 0g `  ( Gs  H ) )  .x.  [ s ]  .~  )  =  [ s ]  .~  <->  ( ( 0g `  ( Gs  H ) )  .x.  u )  =  u ) )
56 oveq2 6048 . . . . . . . 8  |-  ( [ s ]  .~  =  u  ->  ( ( a ( +g  `  ( Gs  H ) ) b )  .x.  [ s ]  .~  )  =  ( ( a ( +g  `  ( Gs  H ) ) b ) 
.x.  u ) )
57 oveq2 6048 . . . . . . . . 9  |-  ( [ s ]  .~  =  u  ->  ( b  .x.  [ s ]  .~  )  =  ( b  .x.  u ) )
5857oveq2d 6056 . . . . . . . 8  |-  ( [ s ]  .~  =  u  ->  ( a  .x.  ( b  .x.  [ s ]  .~  ) )  =  ( a  .x.  ( b  .x.  u
) ) )
5956, 58eqeq12d 2418 . . . . . . 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 2708 . . . . . 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 692 . . . . 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 444 . . . . . . . 8  |-  ( (
ph  /\  s  e.  X )  ->  ph )
631adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  s  e.  X )  ->  H  e.  (SubGrp `  G )
)
64 eqid 2404 . . . . . . . . . 10  |-  ( 0g
`  G )  =  ( 0g `  G
)
6564subg0cl 14907 . . . . . . . . 9  |-  ( H  e.  (SubGrp `  G
)  ->  ( 0g `  G )  e.  H
)
6663, 65syl 16 . . . . . . . 8  |-  ( (
ph  /\  s  e.  X )  ->  ( 0g `  G )  e.  H )
67 simpr 448 . . . . . . . 8  |-  ( (
ph  /\  s  e.  X )  ->  s  e.  X )
689, 5, 1, 8, 25, 10, 16sylow2blem1 15209 . . . . . . . 8  |-  ( (
ph  /\  ( 0g `  G )  e.  H  /\  s  e.  X
)  ->  ( ( 0g `  G )  .x.  [ s ]  .~  )  =  [ ( ( 0g
`  G )  .+  s ) ]  .~  )
6962, 66, 67, 68syl3anc 1184 . . . . . . 7  |-  ( (
ph  /\  s  e.  X )  ->  (
( 0g `  G
)  .x.  [ s ]  .~  )  =  [
( ( 0g `  G )  .+  s
) ]  .~  )
702, 64subg0 14905 . . . . . . . . 9  |-  ( H  e.  (SubGrp `  G
)  ->  ( 0g `  G )  =  ( 0g `  ( Gs  H ) ) )
7163, 70syl 16 . . . . . . . 8  |-  ( (
ph  /\  s  e.  X )  ->  ( 0g `  G )  =  ( 0g `  ( Gs  H ) ) )
7271oveq1d 6055 . . . . . . 7  |-  ( (
ph  /\  s  e.  X )  ->  (
( 0g `  G
)  .x.  [ s ]  .~  )  =  ( ( 0g `  ( Gs  H ) )  .x.  [ s ]  .~  )
)
739, 25, 64grplid 14790 . . . . . . . . 9  |-  ( ( G  e.  Grp  /\  s  e.  X )  ->  ( ( 0g `  G )  .+  s
)  =  s )
7430, 73sylan 458 . . . . . . . 8  |-  ( (
ph  /\  s  e.  X )  ->  (
( 0g `  G
)  .+  s )  =  s )
75 eceq1 6900 . . . . . . . 8  |-  ( ( ( 0g `  G
)  .+  s )  =  s  ->  [ ( ( 0g `  G
)  .+  s ) ]  .~  =  [ s ]  .~  )
7674, 75syl 16 . . . . . . 7  |-  ( (
ph  /\  s  e.  X )  ->  [ ( ( 0g `  G
)  .+  s ) ]  .~  =  [ s ]  .~  )
7769, 72, 763eqtr3d 2444 . . . . . 6  |-  ( (
ph  /\  s  e.  X )  ->  (
( 0g `  ( Gs  H ) )  .x.  [ s ]  .~  )  =  [ s ]  .~  )
7863adantr 452 . . . . . . . . . . . . 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 733 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  a  e.  H )
8280, 81sseldd 3309 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  a  e.  X )
83 simprr 734 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  b  e.  H )
8480, 83sseldd 3309 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  b  e.  X )
8567adantr 452 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  s  e.  X )
869, 25grpass 14774 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  ( a  e.  X  /\  b  e.  X  /\  s  e.  X
) )  ->  (
( a  .+  b
)  .+  s )  =  ( a  .+  ( b  .+  s
) ) )
8779, 82, 84, 85, 86syl13anc 1186 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  ( (
a  .+  b )  .+  s )  =  ( a  .+  ( b 
.+  s ) ) )
88 eceq1 6900 . . . . . . . . . . 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 452 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  ph )
919, 25grpcl 14773 . . . . . . . . . . . 12  |-  ( ( G  e.  Grp  /\  b  e.  X  /\  s  e.  X )  ->  ( b  .+  s
)  e.  X )
9279, 84, 85, 91syl3anc 1184 . . . . . . . . . . 11  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  ( b  .+  s )  e.  X
)
939, 5, 1, 8, 25, 10, 16sylow2blem1 15209 . . . . . . . . . . 11  |-  ( (
ph  /\  a  e.  H  /\  ( b  .+  s )  e.  X
)  ->  ( a  .x.  [ ( b  .+  s ) ]  .~  )  =  [ (
a  .+  ( b  .+  s ) ) ]  .~  )
9490, 81, 92, 93syl3anc 1184 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  ( a  .x.  [ ( b  .+  s ) ]  .~  )  =  [ (
a  .+  ( b  .+  s ) ) ]  .~  )
9589, 94eqtr4d 2439 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  [ (
( a  .+  b
)  .+  s ) ]  .~  =  ( a 
.x.  [ ( b  .+  s ) ]  .~  ) )
9625subgcl 14909 . . . . . . . . . . 11  |-  ( ( H  e.  (SubGrp `  G )  /\  a  e.  H  /\  b  e.  H )  ->  (
a  .+  b )  e.  H )
9778, 81, 83, 96syl3anc 1184 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  ( a  .+  b )  e.  H
)
989, 5, 1, 8, 25, 10, 16sylow2blem1 15209 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  .+  b )  e.  H  /\  s  e.  X
)  ->  ( (
a  .+  b )  .x.  [ s ]  .~  )  =  [ (
( a  .+  b
)  .+  s ) ]  .~  )
9990, 97, 85, 98syl3anc 1184 . . . . . . . . 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 15209 . . . . . . . . . . 11  |-  ( (
ph  /\  b  e.  H  /\  s  e.  X
)  ->  ( b  .x.  [ s ]  .~  )  =  [ (
b  .+  s ) ]  .~  )
10190, 83, 85, 100syl3anc 1184 . . . . . . . . . 10  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  ( b  .x.  [ s ]  .~  )  =  [ (
b  .+  s ) ]  .~  )
102101oveq2d 6056 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  ( a  .x.  ( b  .x.  [ s ]  .~  ) )  =  ( a  .x.  [ ( b  .+  s
) ]  .~  )
)
10395, 99, 1023eqtr4d 2446 . . . . . . . 8  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  ( (
a  .+  b )  .x.  [ s ]  .~  )  =  ( a  .x.  ( b  .x.  [ s ]  .~  ) ) )
104103ralrimivva 2758 . . . . . . 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 13526 . . . . . . . . . . . . 13  |-  ( H  e.  (SubGrp `  G
)  ->  .+  =  ( +g  `  ( Gs  H ) ) )
1071, 106syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  .+  =  ( +g  `  ( Gs  H ) ) )
108107proplem3 13871 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  X )  ->  (
a  .+  b )  =  ( a ( +g  `  ( Gs  H ) ) b ) )
109108oveq1d 6055 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  X )  ->  (
( a  .+  b
)  .x.  [ s ]  .~  )  =  ( ( a ( +g  `  ( Gs  H ) ) b )  .x.  [ s ]  .~  ) )
110109eqeq1d 2412 . . . . . . . . 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 2876 . . . . . . . 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 2876 . . . . . . 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 202 . . . . . 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 519 . . . . 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 6930 . . . 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 2749 . . 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 519 . 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 2404 . . 3  |-  ( Base `  ( Gs  H ) )  =  ( Base `  ( Gs  H ) )
119 eqid 2404 . . 3  |-  ( +g  `  ( Gs  H ) )  =  ( +g  `  ( Gs  H ) )
120 eqid 2404 . . 3  |-  ( 0g
`  ( Gs  H ) )  =  ( 0g
`  ( Gs  H ) )
121118, 119, 120isga 15023 . 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 646 1  |-  ( ph  ->  .x.  e.  ( ( Gs  H )  GrpAct  ( X /.  .~  ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916    C_ wss 3280   ~Pcpw 3759    e. cmpt 4226    X. cxp 4835   ran crn 4838    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042    Er wer 6861   [cec 6862   /.cqs 6863   Fincfn 7068   Basecbs 13424   ↾s cress 13425   +g cplusg 13484   0gc0g 13678   Grpcgrp 14640  SubGrpcsubg 14893   ~QG cqg 14895    GrpAct cga 15021
This theorem is referenced by:  sylow2blem3  15211
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-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
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-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-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-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-er 6864  df-ec 6866  df-qs 6870  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  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-grp 14767  df-minusg 14768  df-sbg 14769  df-subg 14896  df-eqg 14898  df-ga 15022
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