MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax-mulass Structured version   Visualization version   GIF version

Axiom ax-mulass 10456
Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass 10432. Proofs should normally use mulass 10478 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
Assertion
Ref Expression
ax-mulass ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

Detailed syntax breakdown of Axiom ax-mulass
StepHypRef Expression
1 cA . . . 4 class 𝐴
2 cc 10388 . . . 4 class
31, 2wcel 2083 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 2083 . . 3 wff 𝐵 ∈ ℂ
6 cC . . . 4 class 𝐶
76, 2wcel 2083 . . 3 wff 𝐶 ∈ ℂ
83, 5, 7w3a 1080 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9 cmul 10395 . . . . 5 class ·
101, 4, 9co 7023 . . . 4 class (𝐴 · 𝐵)
1110, 6, 9co 7023 . . 3 class ((𝐴 · 𝐵) · 𝐶)
124, 6, 9co 7023 . . . 4 class (𝐵 · 𝐶)
131, 12, 9co 7023 . . 3 class (𝐴 · (𝐵 · 𝐶))
1411, 13wceq 1525 . 2 wff ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
158, 14wi 4 1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))
Colors of variables: wff setvar class
This axiom is referenced by:  mulass  10478
  Copyright terms: Public domain W3C validator