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 e. CC -> 0 <_ ( ( ( abs ` A ) - ( Re ` A ) ) / 2 ) )

Proof

Step	Hyp	Ref	Expression
1		abscl	|- ( A e. CC -> ( abs ` A ) e. RR )
2		recl	|- ( A e. CC -> ( Re ` A ) e. RR )
3	1 2	resubcld	|- ( A e. CC -> ( ( abs ` A ) - ( Re ` A ) ) e. RR )
4		2rp	|- 2 e. RR+
5	4	a1i	|- ( A e. CC -> 2 e. RR+ )
6		releabs	|- ( A e. CC -> ( Re ` A ) <_ ( abs ` A ) )
7	1 2	subge0d	|- ( A e. CC -> ( 0 <_ ( ( abs ` A ) - ( Re ` A ) ) <-> ( Re ` A ) <_ ( abs ` A ) ) )
8	6 7	mpbird	|- ( A e. CC -> 0 <_ ( ( abs ` A ) - ( Re ` A ) ) )
9	3 5 8	divge0d	|- ( A e. CC -> 0 <_ ( ( ( abs ` A ) - ( Re ` A ) ) / 2 ) )