Metamath Proof Explorer


Theorem subdiri

Description: Distribution of multiplication over subtraction. Theorem I.5 of Apostol p. 18. (Contributed by NM, 8-May-1999)

Ref Expression
Hypotheses mulm1.1
|- A e. CC
mulneg.2
|- B e. CC
subdi.3
|- C e. CC
Assertion subdiri
|- ( ( A - B ) x. C ) = ( ( A x. C ) - ( B x. C ) )

Proof

Step Hyp Ref Expression
1 mulm1.1
 |-  A e. CC
2 mulneg.2
 |-  B e. CC
3 subdi.3
 |-  C e. CC
4 subdir
 |-  ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) x. C ) = ( ( A x. C ) - ( B x. C ) ) )
5 1 2 3 4 mp3an
 |-  ( ( A - B ) x. C ) = ( ( A x. C ) - ( B x. C ) )