Theorem sqrtcvallem2

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

	Ref	Expression
Assertion	sqrtcvallem2	⊢ ( 𝐴 ∈ ℂ → 0 ≤ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) )

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		2rp	⊢ 2 ∈ ℝ+
5	4	a1i	⊢ ( 𝐴 ∈ ℂ → 2 ∈ ℝ+ )
6		releabs	⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ 𝐴 ) ≤ ( abs ‘ 𝐴 ) )
7	1 2	subge0d	⊢ ( 𝐴 ∈ ℂ → ( 0 ≤ ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) ↔ ( ℜ ‘ 𝐴 ) ≤ ( abs ‘ 𝐴 ) ) )
8	6 7	mpbird	⊢ ( 𝐴 ∈ ℂ → 0 ≤ ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) )
9	3 5 8	divge0d	⊢ ( 𝐴 ∈ ℂ → 0 ≤ ( ( ( abs ‘ 𝐴 ) − ( ℜ ‘ 𝐴 ) ) / 2 ) )