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Theorem sylow2blem2 15182
Description: Lemma for sylow2b 15184. 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 2387 . . . . 5  |-  ( Gs  H )  =  ( Gs  H )
32subggrp 14874 . . . 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 7337 . . . . 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 14917 . . . . . 6  |-  ( K  e.  (SubGrp `  G
)  ->  .~  Er  X
)
128, 11syl 16 . . . . 5  |-  ( ph  ->  .~  Er  X )
1312qsss 6901 . . . 4  |-  ( ph  ->  ( X /.  .~  )  C_  ~P X )
147, 13ssexd 4291 . . 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 2902 . . . . . . . . 9  |-  y  e. 
_V
1817mptex 5905 . . . . . . . 8  |-  ( z  e.  y  |->  ( x 
.+  z ) )  e.  _V
1918rnex 5073 . . . . . . 7  |-  ran  (
z  e.  y  |->  ( x  .+  z ) )  e.  _V
2016, 19fnmpt2i 6359 . . . . . 6  |-  .x.  Fn  ( H  X.  ( X /.  .~  ) )
2120a1i 11 . . . . 5  |-  ( ph  ->  .x.  Fn  ( H  X.  ( X /.  .~  ) ) )
22 eqid 2387 . . . . . . . 8  |-  ( X /.  .~  )  =  ( X /.  .~  )
23 oveq2 6028 . . . . . . . . 9  |-  ( [ s ]  .~  =  v  ->  ( u  .x.  [ s ]  .~  )  =  ( u  .x.  v ) )
2423eleq1d 2453 . . . . . . . 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 15181 . . . . . . . . . 10  |-  ( (
ph  /\  u  e.  H  /\  s  e.  X
)  ->  ( u  .x.  [ s ]  .~  )  =  [ (
u  .+  s ) ]  .~  )
27 ovex 6045 . . . . . . . . . . . 12  |-  ( G ~QG  K )  e.  _V
2810, 27eqeltri 2457 . . . . . . . . . . 11  |-  .~  e.  _V
29 subgrcl 14876 . . . . . . . . . . . . . 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 14872 . . . . . . . . . . . . . . 15  |-  ( H  e.  (SubGrp `  G
)  ->  H  C_  X
)
331, 32syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  H  C_  X )
3433sselda 3291 . . . . . . . . . . . . 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 14745 . . . . . . . . . . . 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 6895 . . . . . . . . . . 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 2461 . . . . . . . . 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 6907 . . . . . . 7  |-  ( ( ( ph  /\  u  e.  H )  /\  v  e.  ( X /.  .~  ) )  ->  (
u  .x.  v )  e.  ( X /.  .~  ) )
4443ralrimiva 2732 . . . . . 6  |-  ( (
ph  /\  u  e.  H )  ->  A. v  e.  ( X /.  .~  ) ( u  .x.  v )  e.  ( X /.  .~  )
)
4544ralrimiva 2732 . . . . 5  |-  ( ph  ->  A. u  e.  H  A. v  e.  ( X /.  .~  ) ( u  .x.  v )  e.  ( X /.  .~  ) )
46 ffnov 6113 . . . . 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 14875 . . . . . . 7  |-  ( H  e.  (SubGrp `  G
)  ->  H  =  ( Base `  ( Gs  H
) ) )
491, 48syl 16 . . . . . 6  |-  ( ph  ->  H  =  ( Base `  ( Gs  H ) ) )
5049xpeq1d 4841 . . . . 5  |-  ( ph  ->  ( H  X.  ( X /.  .~  ) )  =  ( ( Base `  ( Gs  H ) )  X.  ( X /.  .~  ) ) )
5150feq2d 5521 . . . 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 6028 . . . . . . 7  |-  ( [ s ]  .~  =  u  ->  ( ( 0g
`  ( Gs  H ) )  .x.  [ s ]  .~  )  =  ( ( 0g `  ( Gs  H ) )  .x.  u ) )
54 id 20 . . . . . . 7  |-  ( [ s ]  .~  =  u  ->  [ s ]  .~  =  u )
5553, 54eqeq12d 2401 . . . . . 6  |-  ( [ s ]  .~  =  u  ->  ( ( ( 0g `  ( Gs  H ) )  .x.  [ s ]  .~  )  =  [ s ]  .~  <->  ( ( 0g `  ( Gs  H ) )  .x.  u )  =  u ) )
56 oveq2 6028 . . . . . . . 8  |-  ( [ s ]  .~  =  u  ->  ( ( a ( +g  `  ( Gs  H ) ) b )  .x.  [ s ]  .~  )  =  ( ( a ( +g  `  ( Gs  H ) ) b ) 
.x.  u ) )
57 oveq2 6028 . . . . . . . . 9  |-  ( [ s ]  .~  =  u  ->  ( b  .x.  [ s ]  .~  )  =  ( b  .x.  u ) )
5857oveq2d 6036 . . . . . . . 8  |-  ( [ s ]  .~  =  u  ->  ( a  .x.  ( b  .x.  [ s ]  .~  ) )  =  ( a  .x.  ( b  .x.  u
) ) )
5956, 58eqeq12d 2401 . . . . . . 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 2691 . . . . . 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 2387 . . . . . . . . . 10  |-  ( 0g
`  G )  =  ( 0g `  G
)
6564subg0cl 14879 . . . . . . . . 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 15181 . . . . . . . 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 14877 . . . . . . . . 9  |-  ( H  e.  (SubGrp `  G
)  ->  ( 0g `  G )  =  ( 0g `  ( Gs  H ) ) )
7163, 70syl 16 . . . . . . . 8  |-  ( (
ph  /\  s  e.  X )  ->  ( 0g `  G )  =  ( 0g `  ( Gs  H ) ) )
7271oveq1d 6035 . . . . . . 7  |-  ( (
ph  /\  s  e.  X )  ->  (
( 0g `  G
)  .x.  [ s ]  .~  )  =  ( ( 0g `  ( Gs  H ) )  .x.  [ s ]  .~  )
)
739, 25, 64grplid 14762 . . . . . . . . 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 6877 . . . . . . . 8  |-  ( ( ( 0g `  G
)  .+  s )  =  s  ->  [ ( ( 0g `  G
)  .+  s ) ]  .~  =  [ s ]  .~  )
7674, 75syl 16 . . . . . . 7  |-  ( (
ph  /\  s  e.  X )  ->  [ ( ( 0g `  G
)  .+  s ) ]  .~  =  [ s ]  .~  )
7769, 72, 763eqtr3d 2427 . . . . . 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 3292 . . . . . . . . . . . 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 3292 . . . . . . . . . . . 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 14746 . . . . . . . . . . . 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 6877 . . . . . . . . . . 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 14745 . . . . . . . . . . . 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 15181 . . . . . . . . . . 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 2422 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  [ (
( a  .+  b
)  .+  s ) ]  .~  =  ( a 
.x.  [ ( b  .+  s ) ]  .~  ) )
9625subgcl 14881 . . . . . . . . . . 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 15181 . . . . . . . . . 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 15181 . . . . . . . . . . 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 6036 . . . . . . . . 9  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  ( a  .x.  ( b  .x.  [ s ]  .~  ) )  =  ( a  .x.  [ ( b  .+  s
) ]  .~  )
)
10395, 99, 1023eqtr4d 2429 . . . . . . . 8  |-  ( ( ( ph  /\  s  e.  X )  /\  (
a  e.  H  /\  b  e.  H )
)  ->  ( (
a  .+  b )  .x.  [ s ]  .~  )  =  ( a  .x.  ( b  .x.  [ s ]  .~  ) ) )
104103ralrimivva 2741 . . . . . . 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 13498 . . . . . . . . . . . . 13  |-  ( H  e.  (SubGrp `  G
)  ->  .+  =  ( +g  `  ( Gs  H ) ) )
1071, 106syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  .+  =  ( +g  `  ( Gs  H ) ) )
108107proplem3 13843 . . . . . . . . . . 11  |-  ( (
ph  /\  s  e.  X )  ->  (
a  .+  b )  =  ( a ( +g  `  ( Gs  H ) ) b ) )
109108oveq1d 6035 . . . . . . . . . 10  |-  ( (
ph  /\  s  e.  X )  ->  (
( a  .+  b
)  .x.  [ s ]  .~  )  =  ( ( a ( +g  `  ( Gs  H ) ) b )  .x.  [ s ]  .~  ) )
110109eqeq1d 2395 . . . . . . . . 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 2859 . . . . . . . 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 2859 . . . . . . 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 6907 . . . 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 2732 . . 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 2387 . . 3  |-  ( Base `  ( Gs  H ) )  =  ( Base `  ( Gs  H ) )
119 eqid 2387 . . 3  |-  ( +g  `  ( Gs  H ) )  =  ( +g  `  ( Gs  H ) )
120 eqid 2387 . . 3  |-  ( 0g
`  ( Gs  H ) )  =  ( 0g
`  ( Gs  H ) )
121118, 119, 120isga 14995 . 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 1717   A.wral 2649   _Vcvv 2899    C_ wss 3263   ~Pcpw 3742    e. cmpt 4207    X. cxp 4816   ran crn 4819    Fn wfn 5389   -->wf 5390   ` cfv 5394  (class class class)co 6020    e. cmpt2 6022    Er wer 6838   [cec 6839   /.cqs 6840   Fincfn 7045   Basecbs 13396   ↾s cress 13397   +g cplusg 13456   0gc0g 13650   Grpcgrp 14612  SubGrpcsubg 14865   ~QG cqg 14867    GrpAct cga 14993
This theorem is referenced by:  sylow2blem3  15183
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6841  df-ec 6843  df-qs 6847  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-0g 13654  df-mnd 14617  df-grp 14739  df-minusg 14740  df-sbg 14741  df-subg 14868  df-eqg 14870  df-ga 14994
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