Step |
Hyp |
Ref |
Expression |
1 |
|
0subm.z |
|- .0. = ( 0g ` G ) |
2 |
|
eqid |
|- ( Base ` G ) = ( Base ` G ) |
3 |
2 1
|
mndidcl |
|- ( G e. Mnd -> .0. e. ( Base ` G ) ) |
4 |
3
|
snssd |
|- ( G e. Mnd -> { .0. } C_ ( Base ` G ) ) |
5 |
1
|
fvexi |
|- .0. e. _V |
6 |
5
|
snid |
|- .0. e. { .0. } |
7 |
6
|
a1i |
|- ( G e. Mnd -> .0. e. { .0. } ) |
8 |
|
velsn |
|- ( a e. { .0. } <-> a = .0. ) |
9 |
|
velsn |
|- ( b e. { .0. } <-> b = .0. ) |
10 |
8 9
|
anbi12i |
|- ( ( a e. { .0. } /\ b e. { .0. } ) <-> ( a = .0. /\ b = .0. ) ) |
11 |
|
eqid |
|- ( +g ` G ) = ( +g ` G ) |
12 |
2 11 1
|
mndlid |
|- ( ( G e. Mnd /\ .0. e. ( Base ` G ) ) -> ( .0. ( +g ` G ) .0. ) = .0. ) |
13 |
3 12
|
mpdan |
|- ( G e. Mnd -> ( .0. ( +g ` G ) .0. ) = .0. ) |
14 |
|
ovex |
|- ( .0. ( +g ` G ) .0. ) e. _V |
15 |
14
|
elsn |
|- ( ( .0. ( +g ` G ) .0. ) e. { .0. } <-> ( .0. ( +g ` G ) .0. ) = .0. ) |
16 |
13 15
|
sylibr |
|- ( G e. Mnd -> ( .0. ( +g ` G ) .0. ) e. { .0. } ) |
17 |
|
oveq12 |
|- ( ( a = .0. /\ b = .0. ) -> ( a ( +g ` G ) b ) = ( .0. ( +g ` G ) .0. ) ) |
18 |
17
|
eleq1d |
|- ( ( a = .0. /\ b = .0. ) -> ( ( a ( +g ` G ) b ) e. { .0. } <-> ( .0. ( +g ` G ) .0. ) e. { .0. } ) ) |
19 |
16 18
|
syl5ibrcom |
|- ( G e. Mnd -> ( ( a = .0. /\ b = .0. ) -> ( a ( +g ` G ) b ) e. { .0. } ) ) |
20 |
10 19
|
syl5bi |
|- ( G e. Mnd -> ( ( a e. { .0. } /\ b e. { .0. } ) -> ( a ( +g ` G ) b ) e. { .0. } ) ) |
21 |
20
|
ralrimivv |
|- ( G e. Mnd -> A. a e. { .0. } A. b e. { .0. } ( a ( +g ` G ) b ) e. { .0. } ) |
22 |
2 1 11
|
issubm |
|- ( G e. Mnd -> ( { .0. } e. ( SubMnd ` G ) <-> ( { .0. } C_ ( Base ` G ) /\ .0. e. { .0. } /\ A. a e. { .0. } A. b e. { .0. } ( a ( +g ` G ) b ) e. { .0. } ) ) ) |
23 |
4 7 21 22
|
mpbir3and |
|- ( G e. Mnd -> { .0. } e. ( SubMnd ` G ) ) |