Step |
Hyp |
Ref |
Expression |
1 |
|
crngocom.1 |
|- G = ( 1st ` R ) |
2 |
|
crngocom.2 |
|- H = ( 2nd ` R ) |
3 |
|
crngocom.3 |
|- X = ran G |
4 |
1 2 3
|
iscrngo2 |
|- ( R e. CRingOps <-> ( R e. RingOps /\ A. x e. X A. y e. X ( x H y ) = ( y H x ) ) ) |
5 |
4
|
simprbi |
|- ( R e. CRingOps -> A. x e. X A. y e. X ( x H y ) = ( y H x ) ) |
6 |
|
oveq1 |
|- ( x = A -> ( x H y ) = ( A H y ) ) |
7 |
|
oveq2 |
|- ( x = A -> ( y H x ) = ( y H A ) ) |
8 |
6 7
|
eqeq12d |
|- ( x = A -> ( ( x H y ) = ( y H x ) <-> ( A H y ) = ( y H A ) ) ) |
9 |
|
oveq2 |
|- ( y = B -> ( A H y ) = ( A H B ) ) |
10 |
|
oveq1 |
|- ( y = B -> ( y H A ) = ( B H A ) ) |
11 |
9 10
|
eqeq12d |
|- ( y = B -> ( ( A H y ) = ( y H A ) <-> ( A H B ) = ( B H A ) ) ) |
12 |
8 11
|
rspc2v |
|- ( ( A e. X /\ B e. X ) -> ( A. x e. X A. y e. X ( x H y ) = ( y H x ) -> ( A H B ) = ( B H A ) ) ) |
13 |
5 12
|
mpan9 |
|- ( ( R e. CRingOps /\ ( A e. X /\ B e. X ) ) -> ( A H B ) = ( B H A ) ) |
14 |
13
|
3impb |
|- ( ( R e. CRingOps /\ A e. X /\ B e. X ) -> ( A H B ) = ( B H A ) ) |