Theorem adddiri 7997

Description: Distributive law (right-distributivity). (Contributed by NM, 16-Feb-1995.)

Hypotheses

Ref	Expression
axi.1	|- A e. CC
axi.2	|- B e. CC
axi.3	|- C e. CC

Assertion

Ref	Expression
adddiri	|- ( ( A + B ) x. C ) = ( ( A x. C ) + ( B x. C ) )

Proof of Theorem adddiri

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		adddir 7977	. 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	mp3an 1348	1 |- ( ( A + B ) x. C ) = ( ( A x. C ) + ( B x. C ) )

Syntax hints:   = wceq 1364   e. wcel 2160  (class class class)co 5895   CCcc 7838   + caddc 7843   x. cmul 7845

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-addcl 7936  ax-mulcom 7941  ax-distr 7944

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5196  df-fv 5243  df-ov 5898

This theorem is referenced by:  numma  9456  binom2i  10659  3dvdsdec  11901  3dvds2dec  11902  sincosq3sgn  14701  sincosq4sgn  14702  cosq23lt0  14706  2lgsoddprmlem3c  14910  2lgsoddprmlem3d  14911