Theorem sqrtcvallem4

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

	Ref	Expression
Assertion	sqrtcvallem4	⊢ ( 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	readdcld	⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) ∈ ℝ )
4		2rp	⊢ 2 ∈ ℝ+
5	4	a1i	⊢ ( 𝐴 ∈ ℂ → 2 ∈ ℝ+ )
6		negcl	⊢ ( 𝐴 ∈ ℂ → - 𝐴 ∈ ℂ )
7	6	releabsd	⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ - 𝐴 ) ≤ ( abs ‘ - 𝐴 ) )
8	6	abscld	⊢ ( 𝐴 ∈ ℂ → ( abs ‘ - 𝐴 ) ∈ ℝ )
9	6	recld	⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ - 𝐴 ) ∈ ℝ )
10	8 9	subge0d	⊢ ( 𝐴 ∈ ℂ → ( 0 ≤ ( ( abs ‘ - 𝐴 ) − ( ℜ ‘ - 𝐴 ) ) ↔ ( ℜ ‘ - 𝐴 ) ≤ ( abs ‘ - 𝐴 ) ) )
11	7 10	mpbird	⊢ ( 𝐴 ∈ ℂ → 0 ≤ ( ( abs ‘ - 𝐴 ) − ( ℜ ‘ - 𝐴 ) ) )
12		absneg	⊢ ( 𝐴 ∈ ℂ → ( abs ‘ - 𝐴 ) = ( abs ‘ 𝐴 ) )
13		reneg	⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ - 𝐴 ) = - ( ℜ ‘ 𝐴 ) )
14	12 13	oveq12d	⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ - 𝐴 ) − ( ℜ ‘ - 𝐴 ) ) = ( ( abs ‘ 𝐴 ) − - ( ℜ ‘ 𝐴 ) ) )
15	1	recnd	⊢ ( 𝐴 ∈ ℂ → ( abs ‘ 𝐴 ) ∈ ℂ )
16	2	recnd	⊢ ( 𝐴 ∈ ℂ → ( ℜ ‘ 𝐴 ) ∈ ℂ )
17	15 16	subnegd	⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ 𝐴 ) − - ( ℜ ‘ 𝐴 ) ) = ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) )
18	14 17	eqtrd	⊢ ( 𝐴 ∈ ℂ → ( ( abs ‘ - 𝐴 ) − ( ℜ ‘ - 𝐴 ) ) = ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) )
19	11 18	breqtrd	⊢ ( 𝐴 ∈ ℂ → 0 ≤ ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) )
20	3 5 19	divge0d	⊢ ( 𝐴 ∈ ℂ → 0 ≤ ( ( ( abs ‘ 𝐴 ) + ( ℜ ‘ 𝐴 ) ) / 2 ) )