Theorem adddiri 10051

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 10031	. 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 1424	1 |- ( ( A + B ) x. C ) = ( ( A x. C ) + ( B x. C ) )

Syntax hints:   = wceq 1483   e. wcel 1990  (class class class)co 6650   CCcc 9934   + caddc 9939   x. cmul 9941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-addcl 9996  ax-mulcom 10000  ax-distr 10003

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  numma  11557  binom2i  12974  3dvdsdec  15054  3dvdsdecOLD  15055  3dvds2dec  15056  3dvds2decOLD  15057  dec5nprm  15770  dec2nprm  15771  mod2xnegi  15775  karatsuba  15792  karatsubaOLD  15793  sincosq3sgn  24252  sincosq4sgn  24253  ang180lem2  24540  1cubrlem  24568  bposlem8  25016  2lgsoddprmlem3c  25137  2lgsoddprmlem3d  25138  normlem3  27969  dpmul100  29605  dpmul1000  29607  dpadd3  29620  dpmul4  29622  problem2  31559  problem2OLD  31560  areaquad  37802  tgoldbachlt  41704  tgoldbachltOLD  41710