Theorem sqrtcvallem3

Description: Equivalent to saying that the absolute value of the imaginary component of the square root of a complex number is a real number. Lemma for sqrtcval , sqrtcval2 , resqrtval , and imsqrtval . (Contributed by RP, 11-May-2024)

	Ref	Expression
Assertion	sqrtcvallem3	⊢ ( 𝐴 ∈ ℂ → ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		abscl	⊢ ( 𝐴 ∈ ℂ → ( abs ‘ 𝐴 ) ∈ ℝ )
2		recl	⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ 𝐴 ) ∈ ℝ )
3	1 2	resubcld	⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) ∈ ℝ )
4	3	rehalfcld	⊢ ( 𝐴 ∈ ℂ → ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ∈ ℝ )
5		sqrtcvallem2	⊢ ( 𝐴 ∈ ℂ → 0 ≤ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) )
6	4 5	resqrtcld	⊢ ( 𝐴 ∈ ℂ → ( √ ‘ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) ) ∈ ℝ )