Theorem adddird 7144

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 7110	. 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 1169	1 |- ( ph -> ( ( A + B ) x. C ) = ( ( A x. C ) + ( B x. C ) ) )

Syntax hints:   -> wi 4   = 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-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-addcl 7072  ax-mulcom 7077  ax-distr 7080

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-rex 2354  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-iota 4887  df-fv 4930  df-ov 5535

This theorem is referenced by:  adddirp1d  7145  joinlmuladdmuld  7146  1p1times  7242  recextlem1  7741  divdirap  7785  subsq  9581  subsq2  9582  binom2  9585  binom3  9590  remullem  9758  resqrexlemover  9896  resqrexlemcalc1  9900