Theorem adddii 7037

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

Hypotheses

Ref	Expression
axi.1	|- A e. CC
axi.2	|- B e. CC
axi.3	|- C e. CC

Assertion

Ref	Expression
adddii	|- ( A x. ( B + C ) ) = ( ( A x. B ) + ( A x. C ) )

Proof of Theorem adddii

Step	Hyp	Ref	Expression
1		axi.1	. 2 |- A e. CC
2		axi.2	. 2 |- B e. CC
3		axi.3	. 2 |- C e. CC
4		adddi 7013	. 2 |- ( ( 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 1232	1 |- ( A x. ( B + C ) ) = ( ( A x. B ) + ( A x. C ) )

Syntax hints:   = wceq 1243   e. wcel 1393  (class class class)co 5512   CCcc 6887   + caddc 6892   x. cmul 6894

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-distr 6988

This theorem depends on definitions:  df-bi 110  df-3an 887

This theorem is referenced by:  3t3e9  8072  numltc  8387  numsucc  8393  numma  8398  4t3lem  8438  decbin2  8471  binom2i  9360