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Theorem adddiri 10643
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 10621 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶)))
51, 2, 3, 4mp3an 1458 1 ((𝐴 + 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (𝐵 · 𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wcel 2114  (class class class)co 7140  cc 10524   + caddc 10529   · cmul 10531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2794  ax-addcl 10586  ax-mulcom 10590  ax-distr 10593
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-v 3471  df-un 3913  df-in 3915  df-ss 3925  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-iota 6293  df-fv 6342  df-ov 7143
This theorem is referenced by:  numma  12130  binom2i  13570  3dvdsdec  15672  3dvds2dec  15673  dec5nprm  16391  dec2nprm  16392  mod2xnegi  16396  karatsuba  16409  sincosq3sgn  25091  sincosq4sgn  25092  ang180lem2  25394  1cubrlem  25425  bposlem8  25873  2lgsoddprmlem3c  25994  2lgsoddprmlem3d  25995  normlem3  28893  dpmul100  30583  dpmul1000  30585  dpadd3  30598  dpmul4  30600  problem2  32983  areaquad  40096  tgoldbachlt  44273
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