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 Description: Distributive law (left-distributivity). (Contributed by Mario Carneiro, 27-May-2016.)
Hypotheses
Ref Expression
Assertion
Ref Expression
adddid (𝜑 → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))

StepHypRef Expression
1 addcld.1 . 2 (𝜑𝐴 ∈ ℂ)
2 addcld.2 . 2 (𝜑𝐵 ∈ ℂ)
3 addassd.3 . 2 (𝜑𝐶 ∈ ℂ)
4 adddi 7013 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
51, 2, 3, 4syl3anc 1135 1 (𝜑 → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393  (class class class)co 5512  ℂcc 6887   + caddc 6892   · cmul 6894 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-distr 6988 This theorem depends on definitions:  df-bi 110  df-3an 887 This theorem is referenced by:  subdi  7382  mulreim  7595  apadd1  7599  conjmulap  7705  cju  7913  flhalf  9144  binom2  9362  binom3  9366  remim  9460  mulreap  9464  readd  9469  remullem  9471  imadd  9477  cjadd  9484
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