Theorem adddid

Description: Distributive law (left-distributivity). (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	addcld.1	|- ( ph -> A e. CC )
	addcld.2	|- ( ph -> B e. CC )
	addassd.3	|- ( ph -> C e. CC )
Assertion	adddid	|- ( ph -> ( A x. ( B + C ) ) = ( ( A x. B ) + ( A x. C ) ) )

Proof

Step	Hyp	Ref	Expression
1		addcld.1	|- ( ph -> A e. CC )
2		addcld.2	|- ( ph -> B e. CC )
3		addassd.3	|- ( ph -> C e. CC )
4		adddi	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. ( B + C ) ) = ( ( A x. B ) + ( A x. C ) ) )
5	1 2 3 4	syl3anc	|- ( ph -> ( A x. ( B + C ) ) = ( ( A x. B ) + ( A x. C ) ) )