Theorem adddii 7129

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 7105	. 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 1268	1 |- ( A x. ( B + C ) ) = ( ( A x. B ) + ( A x. C ) )

Syntax hints:   = wceq 1284   e. wcel 1433  (class class class)co 5532   CCcc 6979   + caddc 6984   x. cmul 6986

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-distr 7080

This theorem depends on definitions:  df-bi 115  df-3an 921

This theorem is referenced by:  3t3e9  8189  numltc  8502  numsucc  8516  numma  8520  decmul10add  8545  4t3lem  8573  9t11e99  8606  decbin2  8617  binom2i  9583  3dec  9642  3dvds2dec  10265