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Axiom ax-mulcl 7730
Description: Closure law for multiplication of complex numbers. Axiom for real and complex numbers, justified by theorem axmulcl 7686. Proofs should normally use mulcl 7759 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 7630 . . . 4 class
31, 2wcel 1480 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 1480 . . 3 wff 𝐵 ∈ ℂ
63, 5wa 103 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)
7 cmul 7637 . . . 4 class ·
81, 4, 7co 5774 . . 3 class (𝐴 · 𝐵)
98, 2wcel 1480 . 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  7759
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