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Theorem adddii 7094
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 7070 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
51, 2, 3, 4mp3an 1243 1 (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))
Colors of variables: wff set class
Syntax hints:   = wceq 1259  wcel 1409  (class class class)co 5539  cc 6944   + caddc 6949   · cmul 6951
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-distr 7045
This theorem depends on definitions:  df-bi 114  df-3an 898
This theorem is referenced by:  3t3e9  8139  numltc  8451  numsucc  8465  numma  8469  decmul10add  8494  4t3lem  8522  9t11e99  8555  decbin2  8566  binom2i  9520  3dec  9579  3dvds2dec  10169
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