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 11206
Description: Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by Theorem axmulass 11182. Proofs should normally use mulass 11228 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 11138 . . . 4 class
31, 2wcel 2098 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 2098 . . 3 wff 𝐵 ∈ ℂ
6 cC . . . 4 class 𝐶
76, 2wcel 2098 . . 3 wff 𝐶 ∈ ℂ
83, 5, 7w3a 1084 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9 cmul 11145 . . . . 5 class ·
101, 4, 9co 7419 . . . 4 class (𝐴 · 𝐵)
1110, 6, 9co 7419 . . 3 class ((𝐴 · 𝐵) · 𝐶)
124, 6, 9co 7419 . . . 4 class (𝐵 · 𝐶)
131, 12, 9co 7419 . . 3 class (𝐴 · (𝐵 · 𝐶))
1411, 13wceq 1533 . 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  11228
  Copyright terms: Public domain W3C validator