Step |
Hyp |
Ref |
Expression |
1 |
|
ielefmnd.g |
|- G = ( EndoFMnd ` A ) |
2 |
|
eqid |
|- ( Base ` G ) = ( Base ` G ) |
3 |
|
eqid |
|- ( 0g ` G ) = ( 0g ` G ) |
4 |
|
eqid |
|- ( +g ` G ) = ( +g ` G ) |
5 |
1
|
ielefmnd |
|- ( A e. V -> ( _I |` A ) e. ( Base ` G ) ) |
6 |
1 2 4
|
efmndov |
|- ( ( ( _I |` A ) e. ( Base ` G ) /\ f e. ( Base ` G ) ) -> ( ( _I |` A ) ( +g ` G ) f ) = ( ( _I |` A ) o. f ) ) |
7 |
5 6
|
sylan |
|- ( ( A e. V /\ f e. ( Base ` G ) ) -> ( ( _I |` A ) ( +g ` G ) f ) = ( ( _I |` A ) o. f ) ) |
8 |
1 2
|
efmndbasf |
|- ( f e. ( Base ` G ) -> f : A --> A ) |
9 |
8
|
adantl |
|- ( ( A e. V /\ f e. ( Base ` G ) ) -> f : A --> A ) |
10 |
|
fcoi2 |
|- ( f : A --> A -> ( ( _I |` A ) o. f ) = f ) |
11 |
9 10
|
syl |
|- ( ( A e. V /\ f e. ( Base ` G ) ) -> ( ( _I |` A ) o. f ) = f ) |
12 |
7 11
|
eqtrd |
|- ( ( A e. V /\ f e. ( Base ` G ) ) -> ( ( _I |` A ) ( +g ` G ) f ) = f ) |
13 |
5
|
anim1ci |
|- ( ( A e. V /\ f e. ( Base ` G ) ) -> ( f e. ( Base ` G ) /\ ( _I |` A ) e. ( Base ` G ) ) ) |
14 |
1 2 4
|
efmndov |
|- ( ( f e. ( Base ` G ) /\ ( _I |` A ) e. ( Base ` G ) ) -> ( f ( +g ` G ) ( _I |` A ) ) = ( f o. ( _I |` A ) ) ) |
15 |
13 14
|
syl |
|- ( ( A e. V /\ f e. ( Base ` G ) ) -> ( f ( +g ` G ) ( _I |` A ) ) = ( f o. ( _I |` A ) ) ) |
16 |
|
fcoi1 |
|- ( f : A --> A -> ( f o. ( _I |` A ) ) = f ) |
17 |
9 16
|
syl |
|- ( ( A e. V /\ f e. ( Base ` G ) ) -> ( f o. ( _I |` A ) ) = f ) |
18 |
15 17
|
eqtrd |
|- ( ( A e. V /\ f e. ( Base ` G ) ) -> ( f ( +g ` G ) ( _I |` A ) ) = f ) |
19 |
2 3 4 5 12 18
|
ismgmid2 |
|- ( A e. V -> ( _I |` A ) = ( 0g ` G ) ) |