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Axiom ax-mulcom 10590
Description: Multiplication of complex numbers is commutative. Axiom 8 of 22 for real and complex numbers, justified by theorem axmulcom 10566. Proofs should normally use mulcom 10612 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 10524 . . . 4 class
31, 2wcel 2111 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 2111 . . 3 wff 𝐵 ∈ ℂ
63, 5wa 399 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)
7 cmul 10531 . . . 4 class ·
81, 4, 7co 7135 . . 3 class (𝐴 · 𝐵)
94, 1, 7co 7135 . . 3 class (𝐵 · 𝐴)
108, 9wceq 1538 . 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  10612
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