Theorem adddii

Description: Distributive law (left-distributivity). (Contributed by NM, 23-Nov-1994)

	Ref	Expression
Hypotheses	axi.1	|- A e. CC
	axi.2	|- B e. CC
	axi.3	|- C e. CC
Assertion	adddii	|- ( A x. ( B + C ) ) = ( ( A x. B ) + ( A x. C ) )

Proof

Step	Hyp	Ref	Expression
1		axi.1	|- A e. CC
2		axi.2	|- B e. CC
3		axi.3	|- 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	mp3an	|- ( A x. ( B + C ) ) = ( ( A x. B ) + ( A x. C ) )