ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  adddii GIF version

Theorem adddii 7561
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 7537 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
51, 2, 3, 4mp3an 1274 1 (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))
Colors of variables: wff set class
Syntax hints:   = wceq 1290  wcel 1439  (class class class)co 5668  cc 7411   + caddc 7416   · cmul 7418
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-distr 7512
This theorem depends on definitions:  df-bi 116  df-3an 927
This theorem is referenced by:  3t3e9  8636  numltc  8965  numsucc  8979  numma  8983  decmul10add  9008  4t3lem  9036  9t11e99  9069  decbin2  9080  binom2i  10126  3dec  10186  3dvds2dec  11207
  Copyright terms: Public domain W3C validator