Step |
Hyp |
Ref |
Expression |
1 |
|
odcl.1 |
|- X = ( Base ` G ) |
2 |
|
odcl.2 |
|- O = ( od ` G ) |
3 |
|
eqid |
|- ( .g ` G ) = ( .g ` G ) |
4 |
|
eqid |
|- ( 0g ` G ) = ( 0g ` G ) |
5 |
|
eqid |
|- { y e. NN | ( y ( .g ` G ) A ) = ( 0g ` G ) } = { y e. NN | ( y ( .g ` G ) A ) = ( 0g ` G ) } |
6 |
1 3 4 2 5
|
odlem1 |
|- ( A e. X -> ( ( ( O ` A ) = 0 /\ { y e. NN | ( y ( .g ` G ) A ) = ( 0g ` G ) } = (/) ) \/ ( O ` A ) e. { y e. NN | ( y ( .g ` G ) A ) = ( 0g ` G ) } ) ) |
7 |
|
simpl |
|- ( ( ( O ` A ) = 0 /\ { y e. NN | ( y ( .g ` G ) A ) = ( 0g ` G ) } = (/) ) -> ( O ` A ) = 0 ) |
8 |
|
elrabi |
|- ( ( O ` A ) e. { y e. NN | ( y ( .g ` G ) A ) = ( 0g ` G ) } -> ( O ` A ) e. NN ) |
9 |
7 8
|
orim12i |
|- ( ( ( ( O ` A ) = 0 /\ { y e. NN | ( y ( .g ` G ) A ) = ( 0g ` G ) } = (/) ) \/ ( O ` A ) e. { y e. NN | ( y ( .g ` G ) A ) = ( 0g ` G ) } ) -> ( ( O ` A ) = 0 \/ ( O ` A ) e. NN ) ) |
10 |
6 9
|
syl |
|- ( A e. X -> ( ( O ` A ) = 0 \/ ( O ` A ) e. NN ) ) |
11 |
10
|
orcomd |
|- ( A e. X -> ( ( O ` A ) e. NN \/ ( O ` A ) = 0 ) ) |
12 |
|
elnn0 |
|- ( ( O ` A ) e. NN0 <-> ( ( O ` A ) e. NN \/ ( O ` A ) = 0 ) ) |
13 |
11 12
|
sylibr |
|- ( A e. X -> ( O ` A ) e. NN0 ) |