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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 ) )