Theorem adddii 8189

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 8164	. 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 1373	1 |- ( A x. ( B + C ) ) = ( ( A x. B ) + ( A x. C ) )

Syntax hints:   = wceq 1397   e. wcel 2202  (class class class)co 6018   CCcc 8030   + caddc 8035   x. cmul 8037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-distr 8136

This theorem depends on definitions:  df-bi 117  df-3an 1006

This theorem is referenced by:  3t3e9  9301  numltc  9636  numsucc  9650  numma  9654  decmul10add  9679  4t3lem  9707  9t11e99  9740  decbin2  9751  binom2i  10911  3dec  10977  3dvds2dec  12445  decsplit  13020