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Axiom ax-mulcl 7040
Description: Closure law for multiplication of complex numbers. Axiom for real and complex numbers, justified by theorem axmulcl 7000. Proofs should normally use mulcl 7066 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 6945 . . . 4 class
31, 2wcel 1409 . . 3 wff 𝐴 ∈ ℂ
4 cB . . . 4 class 𝐵
54, 2wcel 1409 . . 3 wff 𝐵 ∈ ℂ
63, 5wa 101 . 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)
7 cmul 6952 . . . 4 class ·
81, 4, 7co 5540 . . 3 class (𝐴 · 𝐵)
98, 2wcel 1409 . 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  7066
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