Metamath Proof Explorer

Theorem adddir

Description: Distributive law for complex numbers (right-distributivity). (Contributed by NM, 10-Oct-2004)

Ref Expression
Assertion adddir 
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) x. C ) = ( ( A x. C ) + ( B x. C ) ) )

Proof

Step Hyp Ref Expression
1 adddi 
 |-  ( ( C e. CC /\ A e. CC /\ B e. CC ) -> ( C x. ( A + B ) ) = ( ( C x. A ) + ( C x. B ) ) )
2 1 3coml 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( C x. ( A + B ) ) = ( ( C x. A ) + ( C x. B ) ) )
3 addcl 
 |-  ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
4 mulcom 
 |-  ( ( ( A + B ) e. CC /\ C e. CC ) -> ( ( A + B ) x. C ) = ( C x. ( A + B ) ) )
5 3 4 stoic3 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) x. C ) = ( C x. ( A + B ) ) )
6 mulcom 
 |-  ( ( A e. CC /\ C e. CC ) -> ( A x. C ) = ( C x. A ) )
7 6 3adant2 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. C ) = ( C x. A ) )
8 mulcom 
 |-  ( ( B e. CC /\ C e. CC ) -> ( B x. C ) = ( C x. B ) )
9 8 3adant1 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B x. C ) = ( C x. B ) )
10 7 9 oveq12d 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A x. C ) + ( B x. C ) ) = ( ( C x. A ) + ( C x. B ) ) )
11 2 5 10 3eqtr4d 
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) x. C ) = ( ( A x. C ) + ( B x. C ) ) )