Theorem subdird

Description: Distribution of multiplication over subtraction. Theorem I.5 of Apostol p. 18. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	mulm1d.1	|- ( ph -> A e. CC )
	mulnegd.2	|- ( ph -> B e. CC )
	subdid.3	|- ( ph -> C e. CC )
Assertion	subdird	|- ( ph -> ( ( A - B ) x. C ) = ( ( A x. C ) - ( B x. C ) ) )

Proof

Step	Hyp	Ref	Expression
1		mulm1d.1	|- ( ph -> A e. CC )
2		mulnegd.2	|- ( ph -> B e. CC )
3		subdid.3	|- ( ph -> 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	syl3anc	|- ( ph -> ( ( A - B ) x. C ) = ( ( A x. C ) - ( B x. C ) ) )