Theorem mulcld

Description: Closure law for multiplication. (Contributed by Mario Carneiro, 27-May-2016)

	Ref	Expression
Hypotheses	addcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
	addcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
Assertion	mulcld	⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) ∈ ℂ )

Proof

Step	Hyp	Ref	Expression
1		addcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		addcld.2	⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3		mulcl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 · 𝐵 ) ∈ ℂ )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 · 𝐵 ) ∈ ℂ )