Step |
Hyp |
Ref |
Expression |
1 |
|
mulgnndir.b |
|- B = ( Base ` G ) |
2 |
|
mulgnndir.t |
|- .x. = ( .g ` G ) |
3 |
|
mulgnndir.p |
|- .+ = ( +g ` G ) |
4 |
|
1z |
|- 1 e. ZZ |
5 |
1 2 3
|
mulgdir |
|- ( ( G e. Grp /\ ( N e. ZZ /\ 1 e. ZZ /\ X e. B ) ) -> ( ( N + 1 ) .x. X ) = ( ( N .x. X ) .+ ( 1 .x. X ) ) ) |
6 |
4 5
|
mp3anr2 |
|- ( ( G e. Grp /\ ( N e. ZZ /\ X e. B ) ) -> ( ( N + 1 ) .x. X ) = ( ( N .x. X ) .+ ( 1 .x. X ) ) ) |
7 |
6
|
3impb |
|- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> ( ( N + 1 ) .x. X ) = ( ( N .x. X ) .+ ( 1 .x. X ) ) ) |
8 |
1 2
|
mulg1 |
|- ( X e. B -> ( 1 .x. X ) = X ) |
9 |
8
|
3ad2ant3 |
|- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> ( 1 .x. X ) = X ) |
10 |
9
|
oveq2d |
|- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> ( ( N .x. X ) .+ ( 1 .x. X ) ) = ( ( N .x. X ) .+ X ) ) |
11 |
7 10
|
eqtrd |
|- ( ( G e. Grp /\ N e. ZZ /\ X e. B ) -> ( ( N + 1 ) .x. X ) = ( ( N .x. X ) .+ X ) ) |