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Theorem sylow2blem2 15218
Description: Lemma for sylow2b 15220. 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 2412 . . . . 5  |-  ( Gs  H )  =  ( Gs  H )
32subggrp 14910 . . . 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 7368 . . . . 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 14953 . . . . . 6  |-  ( K  e.  (SubGrp `  G
)  ->  .~  Er  X
)
128, 11syl 16 . . . . 5  |-  ( ph  ->  .~  Er  X )
1312qsss 6932 . . . 4  |-  ( ph  ->  ( X /.  .~  )  C_  ~P X )
147, 13ssexd 4318 . . 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 2927 . . . . . . . . 9  |-  y  e. 
_V
1817mptex 5933 . . . . . . . 8  |-  ( z  e.  y  |->  ( x 
.+  z ) )  e.  _V
1918rnex 5100 . . . . . . 7  |-  ran  (
z  e.  y  |->  ( x  .+  z ) )  e.  _V
2016, 19fnmpt2i 6387 . . . . . 6  |-  .x.  Fn  ( H  X.  ( X /.  .~  ) )
2120a1i 11 . . . . 5  |-  ( ph  ->  .x.  Fn  ( H  X.  ( X /.  .~  ) ) )
22 eqid 2412 . . . . . . . 8  |-  ( X /.  .~  )  =  ( X /.  .~  )
23 oveq2 6056 . . . . . . . . 9  |-  ( [ s ]  .~  =  v  ->  ( u  .x.  [ s ]  .~  )  =  ( u  .x.  v ) )
2423eleq1d 2478 . . . . . . . 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 15217 . . . . . . . . . 10  |-  ( (
ph  /\  u  e.  H  /\  s  e.  X
)  ->  ( u  .x.  [ s ]  .~  )  =  [ (
u  .+  s ) ]  .~  )
27 ovex 6073 . . . . . . . . . . . 12  |-  ( G ~QG  K )  e.  _V
2810, 27eqeltri 2482 . . . . . . . . . . 11  |-  .~  e.  _V
29 subgrcl 14912 . . . . . . . . . . . . . 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 14908 . . . . . . . . . . . . . . 15  |-  ( H  e.  (SubGrp `  G
)  ->  H  C_  X
)
331, 32syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  H  C_  X )
3433sselda 3316 . . . . . . . . . . . . 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 14781 . . . . . . . . . . . 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 6926 . . . . . . . . . . 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 2486 . . . . . . . . 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 6938 . . . . . . 7  |-  ( ( ( ph  /\  u  e.  H )  /\  v  e.  ( X /.  .~  ) )  ->  (
u  .x.  v )  e.  ( X /.  .~  ) )
4443ralrimiva 2757 . . . . . 6  |-  ( (
ph  /\  u  e.  H )  ->  A. v  e.  ( X /.  .~  ) ( u  .x.  v )  e.  ( X /.  .~  )
)
4544ralrimiva 2757 . . . . 5  |-  ( ph  ->  A. u  e.  H  A. v  e.  ( X /.  .~  ) ( u  .x.  v )  e.  ( X /.  .~  ) )
46 ffnov 6141 . . . . 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 14911 . . . . . . 7  |-  ( H  e.  (SubGrp `  G
)  ->  H  =  ( Base `  ( Gs  H
) ) )
491, 48syl 16 . . . . . 6  |-  ( ph  ->  H  =  ( Base `  ( Gs  H ) ) )
5049xpeq1d 4868 . . . . 5  |-  ( ph  ->  ( H  X.  ( X /.  .~  ) )  =  ( ( Base `  ( Gs  H ) )  X.  ( X /.  .~  ) ) )
5150feq2d 5548 . . . 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 6056 . . . . . . 7  |-  ( [ s ]  .~  =  u  ->  ( ( 0g
`  ( Gs  H ) )  .x.  [ s ]  .~  )  =  ( ( 0g `  ( Gs  H ) )  .x.  u ) )
54 id 20 . . . . . . 7  |-  ( [ s ]  .~  =  u  ->  [ s ]  .~  =  u )
5553, 54eqeq12d 2426 . . . . . 6  |-  ( [ s ]  .~  =  u  ->  ( ( ( 0g `  ( Gs  H ) )  .x.  [ s ]  .~  )  =  [ s ]  .~  <->  ( ( 0g `  ( Gs  H ) )  .x.  u )  =  u ) )
56 oveq2 6056 . . . . . . . 8  |-  ( [ s ]  .~  =  u  ->  ( ( a ( +g  `  ( Gs  H ) ) b )  .x.  [ s ]  .~  )  =  ( ( a ( +g  `  ( Gs  H ) ) b ) 
.x.  u ) )
57 oveq2 6056 . . . . . . . . 9  |-  ( [ s ]  .~  =  u  ->  ( b  .x.  [ s ]  .~  )  =  ( b  .x.  u ) )
5857oveq2d 6064 . . . . . . . 8  |-  ( [ s ]  .~  =  u  ->  ( a  .x.  ( b  .x.  [ s ]  .~  ) )  =  ( a  .x.  ( b  .x.  u
) ) )
5956, 58eqeq12d 2426 . . . . . . 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 2716 . . . . . 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 2412 . . . . . . . . . 10  |-  ( 0g
`  G )  =  ( 0g `  G
)
6564subg0cl 14915 . . . . . . . . 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 15217 . . . . . . . 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 14913 . . . . . . . . 9  |-  ( H  e.  (SubGrp `  G
)  ->  ( 0g `  G )  =  ( 0g `  ( Gs  H ) ) )
7163, 70syl 16 . . . . . . . 8  |-  ( (
ph  /\  s  e.  X )  ->  ( 0g `  G )  =  ( 0g `  ( Gs  H ) ) )
7271oveq1d 6063 . . . . . . 7  |-  ( (
ph  /\  s  e.  X )  ->  (
( 0g `  G
)  .x.  [ s ]  .~  )  =  ( ( 0g `  ( Gs  H ) )  .x.  [ s ]  .~  )
)
739, 25, 64grplid 14798 . . . . . . . . 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 6908 . . . . . . . 8  |-  ( ( ( 0g `  G
)  .+  s )  =  s  ->  [ ( ( 0g `  G
)  .+  s ) ]  .~  =  [ s ]  .~  )
7674, 75syl 16 . . . . . . 7  |-  ( (
ph  /\  s  e.  X )  ->  [ ( ( 0g `  G
)  .+  s ) ]  .~  =  [ s ]  .~  )
7769, 72, 763eqtr3d 2452 . . . . . 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 3317 . . . . . . . . . . . 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 3317 . . . . . . . . . . . 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 14782 . . . . . . . . . . . 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 6908 . . . . . . . . . . 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 14781 . . . . . . . . . . . 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 15217 . . . . . . . . . . 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 2447 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  [ (
( a  .+  b
)  .+  s ) ]  .~  =  ( a 
.x.  [ ( b  .+  s ) ]  .~  ) )
9625subgcl 14917 . . . . . . . . . . 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 15217 . . . . . . . . . 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 15217 . . . . . . . . . . 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 6064 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  ( a  .x.  ( b  .x.  [ s ]  .~  ) )  =  ( a  .x.  [ ( b  .+  s
) ]  .~  )
)
10395, 99, 1023eqtr4d 2454 . . . . . . . 8  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  ( (
a  .+  b )  .x.  [ s ]  .~  )  =  ( a  .x.  ( b  .x.  [ s ]  .~  ) ) )
104103ralrimivva 2766 . . . . . . 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 13534 . . . . . . . . . . . . 13  |-  ( H  e.  (SubGrp `  G
)  ->  .+  =  ( +g  `  ( Gs  H ) ) )
1071, 106syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  .+  =  ( +g  `  ( Gs  H ) ) )
108107proplem3 13879 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  X )  ->  (
a  .+  b )  =  ( a ( +g  `  ( Gs  H ) ) b ) )
109108oveq1d 6063 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  X )  ->  (
( a  .+  b
)  .x.  [ s ]  .~  )  =  ( ( a ( +g  `  ( Gs  H ) ) b )  .x.  [ s ]  .~  ) )
110109eqeq1d 2420 . . . . . . . . 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 2884 . . . . . . . 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 2884 . . . . . . 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 6938 . . . 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 2757 . . 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 2412 . . 3  |-  ( Base `  ( Gs  H ) )  =  ( Base `  ( Gs  H ) )
119 eqid 2412 . . 3  |-  ( +g  `  ( Gs  H ) )  =  ( +g  `  ( Gs  H ) )
120 eqid 2412 . . 3  |-  ( 0g
`  ( Gs  H ) )  =  ( 0g
`  ( Gs  H ) )
121118, 119, 120isga 15031 . 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 2674   _Vcvv 2924    C_ wss 3288   ~Pcpw 3767    e. cmpt 4234    X. cxp 4843   ran crn 4846    Fn wfn 5416   -->wf 5417   ` cfv 5421  (class class class)co 6048    e. cmpt2 6050    Er wer 6869   [cec 6870   /.cqs 6871   Fincfn 7076   Basecbs 13432   ↾s cress 13433   +g cplusg 13492   0gc0g 13686   Grpcgrp 14648  SubGrpcsubg 14901   ~QG cqg 14903    GrpAct cga 15029
This theorem is referenced by:  sylow2blem3  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 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-2o 6692  df-oadd 6695  df-er 6872  df-ec 6874  df-qs 6878  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-0g 13690  df-mnd 14653  df-grp 14775  df-minusg 14776  df-sbg 14777  df-subg 14904  df-eqg 14906  df-ga 15030
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