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

Theorem adddii 7783
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 7759 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶)))
51, 2, 3, 4mp3an 1315 1 (𝐴 · (𝐵 + 𝐶)) = ((𝐴 · 𝐵) + (𝐴 · 𝐶))
Colors of variables: wff set class
Syntax hints:   = wceq 1331  wcel 1480  (class class class)co 5774  cc 7625   + caddc 7630   · cmul 7632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-distr 7731
This theorem depends on definitions:  df-bi 116  df-3an 964
This theorem is referenced by:  3t3e9  8884  numltc  9214  numsucc  9228  numma  9232  decmul10add  9257  4t3lem  9285  9t11e99  9318  decbin2  9329  binom2i  10408  3dec  10468  3dvds2dec  11569
  Copyright terms: Public domain W3C validator