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 ) )