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Theorem adddiri 10011
Description: Distributive law (right-distributivity). (Contributed by NM, 16-Feb-1995.)
Hypotheses
Ref Expression
axi.1 𝐴 ∈ ℂ
axi.2 𝐵 ∈ ℂ
axi.3 𝐶 ∈ ℂ
Assertion
Ref Expression
adddiri ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶))

Proof of Theorem adddiri
StepHypRef Expression
1 axi.1 . 2 𝐴 ∈ ℂ
2 axi.2 . 2 𝐵 ∈ ℂ
3 axi.3 . 2 𝐶 ∈ ℂ
4 adddir 9991 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))
51, 2, 3, 4mp3an 1421 1 ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  (class class class)co 6615  cc 9894   + caddc 9899   · cmul 9901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-addcl 9956  ax-mulcom 9960  ax-distr 9963
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-iota 5820  df-fv 5865  df-ov 6618
This theorem is referenced by:  numma  11517  binom2i  12930  3dvdsdec  14997  3dvdsdecOLD  14998  3dvds2dec  14999  3dvds2decOLD  15000  dec5nprm  15713  dec2nprm  15714  mod2xnegi  15718  karatsuba  15735  karatsubaOLD  15736  sincosq3sgn  24190  sincosq4sgn  24191  ang180lem2  24474  1cubrlem  24502  bposlem8  24950  2lgsoddprmlem3c  25071  2lgsoddprmlem3d  25072  normlem3  27857  problem2  31320  problem2OLD  31321  areaquad  37322  tgoldbachlt  41021  tgoldbachltOLD  41028
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