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

Axiom ax-mulcom 9879
 Description: Multiplication of complex numbers is commutative. Axiom 8 of 22 for real and complex numbers, justified by theorem axmulcom 9855. Proofs should normally use mulcom 9901 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
Assertion
Ref Expression
ax-mulcom ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))

Detailed syntax breakdown of Axiom ax-mulcom
StepHypRef Expression
1 cA . . . 4 class 𝐴
2 cc 9813 . . . 4 class
31, 2wcel 1977 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 1977 . . 3 wff 𝐵 ∈ ℂ
63, 5wa 383 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)
7 cmul 9820 . . . 4 class ·
81, 4, 7co 6549 . . 3 class (𝐴 · 𝐵)
94, 1, 7co 6549 . . 3 class (𝐵 · 𝐴)
108, 9wceq 1475 . 2 wff (𝐴 · 𝐵) = (𝐵 · 𝐴)
116, 10wi 4 1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) = (𝐵 · 𝐴))
 Colors of variables: wff setvar class This axiom is referenced by:  mulcom  9901
 Copyright terms: Public domain W3C validator