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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 )