MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  adddiri Structured version   Unicode version

Theorem adddiri 9643
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
StepHypRef Expression
1 axi.1 . 2  |-  A  e.  CC
2 axi.2 . 2  |-  B  e.  CC
3 axi.3 . 2  |-  C  e.  CC
4 adddir 9623 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( A  +  B
)  x.  C )  =  ( ( A  x.  C )  +  ( B  x.  C
) ) )
51, 2, 3, 4mp3an 1360 1  |-  ( ( A  +  B )  x.  C )  =  ( ( A  x.  C )  +  ( B  x.  C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437    e. wcel 1867  (class class class)co 6296   CCcc 9526    + caddc 9531    x. cmul 9533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-addcl 9588  ax-mulcom 9592  ax-distr 9595
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-iota 5556  df-fv 5600  df-ov 6299
This theorem is referenced by:  numma  11071  binom2i  12370  dec5nprm  14990  dec2nprm  14991  mod2xnegi  14995  karatsuba  15008  sincosq3sgn  23346  sincosq4sgn  23347  ang180lem2  23630  1cubrlem  23658  bposlem8  24108  normlem3  26626  problem2  30112  areaquad  35833  tgoldbachlt  38341
  Copyright terms: Public domain W3C validator