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Theorem adddii 7958
Description: Distributive law (left-distributivity). (Contributed by NM, 23-Nov-1994.)
Hypotheses
Ref Expression
axi.1 𝐴 ∈ ℂ
axi.2 𝐵 ∈ ℂ
axi.3 𝐶 ∈ ℂ
Assertion
Ref Expression
adddii (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))

Proof of Theorem adddii
StepHypRef Expression
1 axi.1 . 2 𝐴 ∈ ℂ
2 axi.2 . 2 𝐵 ∈ ℂ
3 axi.3 . 2 𝐶 ∈ ℂ
4 adddi 7934 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
51, 2, 3, 4mp3an 1337 1 (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))
Colors of variables: wff set class
Syntax hints:   = wceq 1353  wcel 2148  (class class class)co 5869  cc 7800   + caddc 7805   · cmul 7807
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-distr 7906
This theorem depends on definitions:  df-bi 117  df-3an 980
This theorem is referenced by:  3t3e9  9065  numltc  9398  numsucc  9412  numma  9416  decmul10add  9441  4t3lem  9469  9t11e99  9502  decbin2  9513  binom2i  10614  3dec  10678  3dvds2dec  11854
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