Metamath Proof Explorer


Theorem crngocom

Description: The multiplication operation of a commutative ring is commutative. (Contributed by Jeff Madsen, 8-Jun-2010)

Ref Expression
Hypotheses crngocom.1
|- G = ( 1st ` R )
crngocom.2
|- H = ( 2nd ` R )
crngocom.3
|- X = ran G
Assertion crngocom
|- ( ( R e. CRingOps /\ A e. X /\ B e. X ) -> ( A H B ) = ( B H A ) )

Proof

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 ) )