Theorem adddiri 8327

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 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	mp3an 1378	1 |- ( ( A + B ) x. C ) = ( ( A x. C ) + ( B x. C ) )

Syntax hints:   = 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:  numma  9799  binom2i  11063  3dvdsdec  12610  3dvds2dec  12611  dec5nprm  13171  dec2nprm  13172  karatsuba  13187  sincosq3sgn  15852  sincosq4sgn  15853  cosq23lt0  15857  2lgsoddprmlem3c  16142  2lgsoddprmlem3d  16143