Theorem sqrtcld

Description: Closure of the square root function over the complex numbers. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypothesis	abscld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
Assertion	sqrtcld	⊢ ( 𝜑 → ( √ ‘ 𝐴 ) ∈ ℂ )

Proof

Step	Hyp	Ref	Expression
1		abscld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		sqrtcl	⊢ ( 𝐴 ∈ ℂ → ( √ ‘ 𝐴 ) ∈ ℂ )
3	1 2	syl	⊢ ( 𝜑 → ( √ ‘ 𝐴 ) ∈ ℂ )