Metamath Proof Explorer


Theorem axmulcl

Description: Closure law for multiplication of complex numbers. Axiom 6 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulcl be used later. Instead, in most cases use mulcl . (Contributed by NM, 10-Aug-1995) (New usage is discouraged.)

Ref Expression
Assertion axmulcl ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 · 𝐵 ) ∈ ℂ )

Proof

Step Hyp Ref Expression
1 axmulf · : ( ℂ × ℂ ) ⟶ ℂ
2 1 fovcl ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 · 𝐵 ) ∈ ℂ )