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

Theorem mulassi 7768
Description: Associative law for multiplication. (Contributed by NM, 23-Nov-1994.)
Hypotheses
Ref Expression
axi.1 𝐴 ∈ ℂ
axi.2 𝐵 ∈ ℂ
axi.3 𝐶 ∈ ℂ
Assertion
Ref Expression
mulassi ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))

Proof of Theorem mulassi
StepHypRef Expression
1 axi.1 . 2 𝐴 ∈ ℂ
2 axi.2 . 2 𝐵 ∈ ℂ
3 axi.3 . 2 𝐶 ∈ ℂ
4 mulass 7744 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
51, 2, 3, 4mp3an 1315 1 ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
Colors of variables: wff set class
Syntax hints:   = wceq 1331  wcel 1480  (class class class)co 5767  cc 7611   · cmul 7618
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-mulass 7716
This theorem depends on definitions:  df-bi 116  df-3an 964
This theorem is referenced by:  8th4div3  8932  numma  9218  decbin0  9314  sq4e2t8  10383  3dec  10454  ef01bndlem  11452  3dvdsdec  11551  3dvds2dec  11552  sincos4thpi  12910  sincos6thpi  12912
  Copyright terms: Public domain W3C validator