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Axiom ax-mulcl 7686
Description: Closure law for multiplication of complex numbers. Axiom for real and complex numbers, justified by theorem axmulcl 7642. Proofs should normally use mulcl 7715 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 7586 . . . 4 class
31, 2wcel 1465 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 1465 . . 3 wff 𝐵 ∈ ℂ
63, 5wa 103 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)
7 cmul 7593 . . . 4 class ·
81, 4, 7co 5742 . . 3 class (𝐴 · 𝐵)
98, 2wcel 1465 . 2 wff (𝐴 · 𝐵) ∈ ℂ
106, 9wi 4 1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
Colors of variables: wff set class
This axiom is referenced by:  mulcl  7715
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