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Theorem adddid 7415
Description: Distributive law (left-distributivity). (Contributed by Mario Carneiro, 27-May-2016.)
Hypotheses
Ref Expression
addcld.1 (𝜑𝐴 ∈ ℂ)
addcld.2 (𝜑𝐵 ∈ ℂ)
addassd.3 (𝜑𝐶 ∈ ℂ)
Assertion
Ref Expression
adddid (𝜑 → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))

Proof of Theorem adddid
StepHypRef Expression
1 addcld.1 . 2 (𝜑𝐴 ∈ ℂ)
2 addcld.2 . 2 (𝜑𝐵 ∈ ℂ)
3 addassd.3 . 2 (𝜑𝐶 ∈ ℂ)
4 adddi 7377 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
51, 2, 3, 4syl3anc 1170 1 (𝜑 → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  wcel 1434  (class class class)co 5591  cc 7251   + caddc 7256   · cmul 7258
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-distr 7352
This theorem depends on definitions:  df-bi 115  df-3an 922
This theorem is referenced by:  subdi  7766  mulreim  7981  apadd1  7985  conjmulap  8094  cju  8315  flhalf  9598  modqcyc  9655  addmodlteq  9694  binom2  9901  binom3  9906  sqoddm1div8  9941  bcpasc  10009  remim  10121  mulreap  10125  readd  10130  remullem  10132  imadd  10138  cjadd  10145  bezoutlemnewy  10765  dvdsmulgcd  10794  lcmgcdlem  10839
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