Metamath Proof Explorer

Theorem subdir

Description: Distribution of multiplication over subtraction. Theorem I.5 of Apostol p. 18. (Contributed by NM, 30-Dec-2005)

Ref Expression
Assertion subdir 
|- ( ( 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 subdi 
 |-  ( ( 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 subcl 
 |-  ( ( 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 ) ) )