Step |
Hyp |
Ref |
Expression |
1 |
|
finodsubmsubg.o |
|- O = ( od ` G ) |
2 |
|
finodsubmsubg.g |
|- ( ph -> G e. Grp ) |
3 |
|
finodsubmsubg.s |
|- ( ph -> S e. ( SubMnd ` G ) ) |
4 |
|
finodsubmsubg.1 |
|- ( ph -> A. a e. S ( O ` a ) e. NN ) |
5 |
|
eqid |
|- ( Base ` G ) = ( Base ` G ) |
6 |
|
eqid |
|- ( .g ` G ) = ( .g ` G ) |
7 |
|
eqid |
|- ( invg ` G ) = ( invg ` G ) |
8 |
2
|
adantr |
|- ( ( ph /\ a e. S ) -> G e. Grp ) |
9 |
5
|
submss |
|- ( S e. ( SubMnd ` G ) -> S C_ ( Base ` G ) ) |
10 |
3 9
|
syl |
|- ( ph -> S C_ ( Base ` G ) ) |
11 |
10
|
sselda |
|- ( ( ph /\ a e. S ) -> a e. ( Base ` G ) ) |
12 |
5 1 6 7 8 11
|
odm1inv |
|- ( ( ph /\ a e. S ) -> ( ( ( O ` a ) - 1 ) ( .g ` G ) a ) = ( ( invg ` G ) ` a ) ) |
13 |
12
|
adantr |
|- ( ( ( ph /\ a e. S ) /\ ( O ` a ) e. NN ) -> ( ( ( O ` a ) - 1 ) ( .g ` G ) a ) = ( ( invg ` G ) ` a ) ) |
14 |
|
eqid |
|- ( Base ` ( G |`s S ) ) = ( Base ` ( G |`s S ) ) |
15 |
|
eqid |
|- ( .g ` ( G |`s S ) ) = ( .g ` ( G |`s S ) ) |
16 |
|
eqid |
|- ( G |`s S ) = ( G |`s S ) |
17 |
16
|
submmnd |
|- ( S e. ( SubMnd ` G ) -> ( G |`s S ) e. Mnd ) |
18 |
3 17
|
syl |
|- ( ph -> ( G |`s S ) e. Mnd ) |
19 |
18
|
ad2antrr |
|- ( ( ( ph /\ a e. S ) /\ ( O ` a ) e. NN ) -> ( G |`s S ) e. Mnd ) |
20 |
|
nnm1nn0 |
|- ( ( O ` a ) e. NN -> ( ( O ` a ) - 1 ) e. NN0 ) |
21 |
20
|
adantl |
|- ( ( ( ph /\ a e. S ) /\ ( O ` a ) e. NN ) -> ( ( O ` a ) - 1 ) e. NN0 ) |
22 |
|
simplr |
|- ( ( ( ph /\ a e. S ) /\ ( O ` a ) e. NN ) -> a e. S ) |
23 |
16 5
|
ressbas2 |
|- ( S C_ ( Base ` G ) -> S = ( Base ` ( G |`s S ) ) ) |
24 |
10 23
|
syl |
|- ( ph -> S = ( Base ` ( G |`s S ) ) ) |
25 |
24
|
ad2antrr |
|- ( ( ( ph /\ a e. S ) /\ ( O ` a ) e. NN ) -> S = ( Base ` ( G |`s S ) ) ) |
26 |
22 25
|
eleqtrd |
|- ( ( ( ph /\ a e. S ) /\ ( O ` a ) e. NN ) -> a e. ( Base ` ( G |`s S ) ) ) |
27 |
14 15 19 21 26
|
mulgnn0cld |
|- ( ( ( ph /\ a e. S ) /\ ( O ` a ) e. NN ) -> ( ( ( O ` a ) - 1 ) ( .g ` ( G |`s S ) ) a ) e. ( Base ` ( G |`s S ) ) ) |
28 |
3
|
ad2antrr |
|- ( ( ( ph /\ a e. S ) /\ ( O ` a ) e. NN ) -> S e. ( SubMnd ` G ) ) |
29 |
6 16 15
|
submmulg |
|- ( ( S e. ( SubMnd ` G ) /\ ( ( O ` a ) - 1 ) e. NN0 /\ a e. S ) -> ( ( ( O ` a ) - 1 ) ( .g ` G ) a ) = ( ( ( O ` a ) - 1 ) ( .g ` ( G |`s S ) ) a ) ) |
30 |
28 21 22 29
|
syl3anc |
|- ( ( ( ph /\ a e. S ) /\ ( O ` a ) e. NN ) -> ( ( ( O ` a ) - 1 ) ( .g ` G ) a ) = ( ( ( O ` a ) - 1 ) ( .g ` ( G |`s S ) ) a ) ) |
31 |
27 30 25
|
3eltr4d |
|- ( ( ( ph /\ a e. S ) /\ ( O ` a ) e. NN ) -> ( ( ( O ` a ) - 1 ) ( .g ` G ) a ) e. S ) |
32 |
13 31
|
eqeltrrd |
|- ( ( ( ph /\ a e. S ) /\ ( O ` a ) e. NN ) -> ( ( invg ` G ) ` a ) e. S ) |
33 |
32
|
ex |
|- ( ( ph /\ a e. S ) -> ( ( O ` a ) e. NN -> ( ( invg ` G ) ` a ) e. S ) ) |
34 |
33
|
ralimdva |
|- ( ph -> ( A. a e. S ( O ` a ) e. NN -> A. a e. S ( ( invg ` G ) ` a ) e. S ) ) |
35 |
4 34
|
mpd |
|- ( ph -> A. a e. S ( ( invg ` G ) ` a ) e. S ) |
36 |
7
|
issubg3 |
|- ( G e. Grp -> ( S e. ( SubGrp ` G ) <-> ( S e. ( SubMnd ` G ) /\ A. a e. S ( ( invg ` G ) ` a ) e. S ) ) ) |
37 |
2 36
|
syl |
|- ( ph -> ( S e. ( SubGrp ` G ) <-> ( S e. ( SubMnd ` G ) /\ A. a e. S ( ( invg ` G ) ` a ) e. S ) ) ) |
38 |
3 35 37
|
mpbir2and |
|- ( ph -> S e. ( SubGrp ` G ) ) |