Step |
Hyp |
Ref |
Expression |
1 |
|
grpissubg.b |
|- B = ( Base ` G ) |
2 |
|
grpissubg.s |
|- S = ( Base ` H ) |
3 |
|
simpl |
|- ( ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) -> S C_ B ) |
4 |
3
|
adantl |
|- ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> S C_ B ) |
5 |
2
|
grpbn0 |
|- ( H e. Grp -> S =/= (/) ) |
6 |
5
|
ad2antlr |
|- ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> S =/= (/) ) |
7 |
|
grpmnd |
|- ( G e. Grp -> G e. Mnd ) |
8 |
|
mndmgm |
|- ( G e. Mnd -> G e. Mgm ) |
9 |
7 8
|
syl |
|- ( G e. Grp -> G e. Mgm ) |
10 |
|
grpmnd |
|- ( H e. Grp -> H e. Mnd ) |
11 |
|
mndmgm |
|- ( H e. Mnd -> H e. Mgm ) |
12 |
10 11
|
syl |
|- ( H e. Grp -> H e. Mgm ) |
13 |
9 12
|
anim12i |
|- ( ( G e. Grp /\ H e. Grp ) -> ( G e. Mgm /\ H e. Mgm ) ) |
14 |
13
|
adantr |
|- ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> ( G e. Mgm /\ H e. Mgm ) ) |
15 |
14
|
ad2antrr |
|- ( ( ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) /\ a e. S ) /\ b e. S ) -> ( G e. Mgm /\ H e. Mgm ) ) |
16 |
|
simpr |
|- ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) |
17 |
16
|
ad2antrr |
|- ( ( ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) /\ a e. S ) /\ b e. S ) -> ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) |
18 |
|
simpr |
|- ( ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) /\ a e. S ) -> a e. S ) |
19 |
18
|
anim1i |
|- ( ( ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) /\ a e. S ) /\ b e. S ) -> ( a e. S /\ b e. S ) ) |
20 |
1 2
|
mgmsscl |
|- ( ( ( G e. Mgm /\ H e. Mgm ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) /\ ( a e. S /\ b e. S ) ) -> ( a ( +g ` G ) b ) e. S ) |
21 |
15 17 19 20
|
syl3anc |
|- ( ( ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) /\ a e. S ) /\ b e. S ) -> ( a ( +g ` G ) b ) e. S ) |
22 |
21
|
ralrimiva |
|- ( ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) /\ a e. S ) -> A. b e. S ( a ( +g ` G ) b ) e. S ) |
23 |
|
simpl |
|- ( ( G e. Grp /\ H e. Grp ) -> G e. Grp ) |
24 |
23
|
adantr |
|- ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> G e. Grp ) |
25 |
|
simplr |
|- ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> H e. Grp ) |
26 |
1
|
sseq2i |
|- ( S C_ B <-> S C_ ( Base ` G ) ) |
27 |
26
|
biimpi |
|- ( S C_ B -> S C_ ( Base ` G ) ) |
28 |
27
|
adantr |
|- ( ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) -> S C_ ( Base ` G ) ) |
29 |
28
|
adantl |
|- ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> S C_ ( Base ` G ) ) |
30 |
|
ovres |
|- ( ( x e. S /\ y e. S ) -> ( x ( ( +g ` G ) |` ( S X. S ) ) y ) = ( x ( +g ` G ) y ) ) |
31 |
30
|
adantl |
|- ( ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) /\ ( x e. S /\ y e. S ) ) -> ( x ( ( +g ` G ) |` ( S X. S ) ) y ) = ( x ( +g ` G ) y ) ) |
32 |
|
oveq |
|- ( ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) -> ( x ( +g ` H ) y ) = ( x ( ( +g ` G ) |` ( S X. S ) ) y ) ) |
33 |
32
|
adantl |
|- ( ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) -> ( x ( +g ` H ) y ) = ( x ( ( +g ` G ) |` ( S X. S ) ) y ) ) |
34 |
33
|
eqcomd |
|- ( ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) -> ( x ( ( +g ` G ) |` ( S X. S ) ) y ) = ( x ( +g ` H ) y ) ) |
35 |
34
|
ad2antlr |
|- ( ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) /\ ( x e. S /\ y e. S ) ) -> ( x ( ( +g ` G ) |` ( S X. S ) ) y ) = ( x ( +g ` H ) y ) ) |
36 |
31 35
|
eqtr3d |
|- ( ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) /\ ( x e. S /\ y e. S ) ) -> ( x ( +g ` G ) y ) = ( x ( +g ` H ) y ) ) |
37 |
36
|
ralrimivva |
|- ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> A. x e. S A. y e. S ( x ( +g ` G ) y ) = ( x ( +g ` H ) y ) ) |
38 |
24 25 2 29 37
|
grpinvssd |
|- ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> ( a e. S -> ( ( invg ` H ) ` a ) = ( ( invg ` G ) ` a ) ) ) |
39 |
38
|
imp |
|- ( ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) /\ a e. S ) -> ( ( invg ` H ) ` a ) = ( ( invg ` G ) ` a ) ) |
40 |
|
eqid |
|- ( invg ` H ) = ( invg ` H ) |
41 |
2 40
|
grpinvcl |
|- ( ( H e. Grp /\ a e. S ) -> ( ( invg ` H ) ` a ) e. S ) |
42 |
41
|
ad4ant24 |
|- ( ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) /\ a e. S ) -> ( ( invg ` H ) ` a ) e. S ) |
43 |
39 42
|
eqeltrrd |
|- ( ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) /\ a e. S ) -> ( ( invg ` G ) ` a ) e. S ) |
44 |
22 43
|
jca |
|- ( ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) /\ a e. S ) -> ( A. b e. S ( a ( +g ` G ) b ) e. S /\ ( ( invg ` G ) ` a ) e. S ) ) |
45 |
44
|
ralrimiva |
|- ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> A. a e. S ( A. b e. S ( a ( +g ` G ) b ) e. S /\ ( ( invg ` G ) ` a ) e. S ) ) |
46 |
|
eqid |
|- ( +g ` G ) = ( +g ` G ) |
47 |
|
eqid |
|- ( invg ` G ) = ( invg ` G ) |
48 |
1 46 47
|
issubg2 |
|- ( G e. Grp -> ( S e. ( SubGrp ` G ) <-> ( S C_ B /\ S =/= (/) /\ A. a e. S ( A. b e. S ( a ( +g ` G ) b ) e. S /\ ( ( invg ` G ) ` a ) e. S ) ) ) ) |
49 |
48
|
ad2antrr |
|- ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> ( S e. ( SubGrp ` G ) <-> ( S C_ B /\ S =/= (/) /\ A. a e. S ( A. b e. S ( a ( +g ` G ) b ) e. S /\ ( ( invg ` G ) ` a ) e. S ) ) ) ) |
50 |
4 6 45 49
|
mpbir3and |
|- ( ( ( G e. Grp /\ H e. Grp ) /\ ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) ) -> S e. ( SubGrp ` G ) ) |
51 |
50
|
ex |
|- ( ( G e. Grp /\ H e. Grp ) -> ( ( S C_ B /\ ( +g ` H ) = ( ( +g ` G ) |` ( S X. S ) ) ) -> S e. ( SubGrp ` G ) ) ) |