Theorem adddii 8081

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 8056	. 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 1349	1 |- ( A x. ( B + C ) ) = ( ( A x. B ) + ( A x. C ) )

Syntax hints:   = wceq 1372   e. wcel 2175  (class class class)co 5943   CCcc 7922   + caddc 7927   x. cmul 7929

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 8028

This theorem depends on definitions:  df-bi 117  df-3an 982

This theorem is referenced by:  3t3e9  9193  numltc  9528  numsucc  9542  numma  9546  decmul10add  9571  4t3lem  9599  9t11e99  9632  decbin2  9643  binom2i  10791  3dec  10857  3dvds2dec  12148  decsplit  12723