Theorem adddii 8300

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 8275	. 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 1374	1 |- ( A x. ( B + C ) ) = ( ( A x. B ) + ( A x. C ) )

Syntax hints:   = wceq 1398   e. wcel 2205  (class class class)co 6058   CCcc 8141   + caddc 8146   x. cmul 8148

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 8247

This theorem depends on definitions:  df-bi 117  df-3an 1007

This theorem is referenced by:  3t3e9  9412  numltc  9752  numsucc  9766  numma  9770  decmul10add  9795  4t3lem  9823  9t11e99  9856  decbin2  9867  binom2i  11034  3dec  11101  3dvds2dec  12577  decsplit  13152