Theorem adddiri 8190

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 8170	. 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 1373	1 |- ( ( A + B ) x. C ) = ( ( A x. C ) + ( B x. C ) )

Syntax hints:   = wceq 1397   e. wcel 2202  (class class class)co 6018   CCcc 8030   + caddc 8035   x. cmul 8037

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-addcl 8128  ax-mulcom 8133  ax-distr 8136

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-iota 5286  df-fv 5334  df-ov 6021

This theorem is referenced by:  numma  9654  binom2i  10911  3dvdsdec  12444  3dvds2dec  12445  dec5nprm  13005  dec2nprm  13006  karatsuba  13021  sincosq3sgn  15571  sincosq4sgn  15572  cosq23lt0  15576  2lgsoddprmlem3c  15857  2lgsoddprmlem3d  15858