Theorem adddii 8336

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 8311	. 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 1378	1 |- ( A x. ( B + C ) ) = ( ( A x. B ) + ( A x. C ) )

This proof depends on syntax axioms:   = wceq 1402   e. wcel 2209  (class class class)co 6085   CCcc 8177   + caddc 8182   x. cmul 8184

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-distr 8283

This proof depends on definitions:  df-bi 117  df-3an 1011

This theorem is used by:  3t3e9  9465  numltc  9811  numsucc  9825  numma  9829  decmul10add  9854  4t3lem  9882  9t11e99  9915  decbin2  9926  binom2i  11098  3dec  11166  3dvds2dec  12649  mod2xnegi  13218  decsplit  13229  log2ublem1  16140  log2ublem2  16141