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Axiom ax-mulcl 9854
Description: Closure law for multiplication of complex numbers. Axiom 6 of 22 for real and complex numbers, justified by theorem axmulcl 9830. Proofs should normally use mulcl 9876 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
Assertion
Ref Expression
ax-mulcl ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)

Detailed syntax breakdown of Axiom ax-mulcl
StepHypRef Expression
1 cA . . . 4 class 𝐴
2 cc 9790 . . . 4 class
31, 2wcel 1976 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 1976 . . 3 wff 𝐵 ∈ ℂ
63, 5wa 382 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)
7 cmul 9797 . . . 4 class ·
81, 4, 7co 6526 . . 3 class (𝐴 · 𝐵)
98, 2wcel 1976 . 2 wff (𝐴 · 𝐵) ∈ ℂ
106, 9wi 4 1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
Colors of variables: wff setvar class
This axiom is referenced by:  mulcl  9876
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