Step |
Hyp |
Ref |
Expression |
1 |
|
mulgass.b |
|- B = ( Base ` G ) |
2 |
|
mulgass.t |
|- .x. = ( .g ` G ) |
3 |
|
zcn |
|- ( N e. ZZ -> N e. CC ) |
4 |
3
|
3ad2ant2 |
|- ( ( M e. ZZ /\ N e. ZZ /\ X e. B ) -> N e. CC ) |
5 |
|
zcn |
|- ( M e. ZZ -> M e. CC ) |
6 |
5
|
3ad2ant1 |
|- ( ( M e. ZZ /\ N e. ZZ /\ X e. B ) -> M e. CC ) |
7 |
4 6
|
mulcomd |
|- ( ( M e. ZZ /\ N e. ZZ /\ X e. B ) -> ( N x. M ) = ( M x. N ) ) |
8 |
7
|
adantl |
|- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( N x. M ) = ( M x. N ) ) |
9 |
8
|
oveq1d |
|- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( ( N x. M ) .x. X ) = ( ( M x. N ) .x. X ) ) |
10 |
1 2
|
mulgass |
|- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( ( M x. N ) .x. X ) = ( M .x. ( N .x. X ) ) ) |
11 |
9 10
|
eqtrd |
|- ( ( G e. Grp /\ ( M e. ZZ /\ N e. ZZ /\ X e. B ) ) -> ( ( N x. M ) .x. X ) = ( M .x. ( N .x. X ) ) ) |