Theorem adddird 8341

Description: Distributive law (right-distributivity). (Contributed by Mario Carneiro, 27-May-2016.)

Hypotheses

Ref	Expression
addcld.1	|- ( ph -> A e. CC )
addcld.2	|- ( ph -> B e. CC )
addassd.3	|- ( ph -> C e. CC )

Assertion

Ref	Expression
adddird	|- ( ph -> ( ( A + B ) x. C ) = ( ( A x. C ) + ( B x. C ) ) )

Proof of Theorem adddird

Step	Hyp	Ref	Expression
1		addcld.1	. 2 |- ( ph -> A e. CC )
2		addcld.2	. 2 |- ( ph -> B e. CC )
3		addassd.3	. 2 |- ( ph -> C e. CC )
4		adddir 8307	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A + B ) x. C ) = ( ( A x. C ) + ( B x. C ) ) )
5	1, 2, 3, 4	syl3anc 1278	1 |- ( ph -> ( ( A + B ) x. C ) = ( ( A x. C ) + ( B x. C ) ) )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209  (class class class)co 6075   CCcc 8167   + caddc 8172   x. cmul 8174

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-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-addcl 8265  ax-mulcom 8270  ax-distr 8273

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-iota 5332  df-fv 5380  df-ov 6078

This theorem is referenced by:  adddirp1d  8342  joinlmuladdmuld  8343  1p1times  8450  recextlem1  8969  divdirap  9017  lincmble  10385  subsq  11061  subsq2  11062  binom2  11066  binom3  11072  remullem  11614  resqrexlemover  11754  resqrexlemcalc1  11758  bdtrilem  11983  binomlem  12228  mul4sqlem  13150  dvexp  15735  plyaddlem1  15771  rpcxpadd  15930  binom4  16004  lgsquad2lem1  16114  2sqlem4  16151