Theorem sqrtrege0d

Description: The real part of the square root function is nonnegative. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypothesis	abscld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
Assertion	sqrtrege0d	⊢ ( 𝜑 → 0 ≤ ( ℜ ‘ ( √ ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		abscld.1	⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2		sqrtrege0	⊢ ( 𝐴 ∈ ℂ → 0 ≤ ( ℜ ‘ ( √ ‘ 𝐴 ) ) )
3	1 2	syl	⊢ ( 𝜑 → 0 ≤ ( ℜ ‘ ( √ ‘ 𝐴 ) ) )