Definition df-rsqrt 10998

Description: Define a function whose value is the square root of a nonnegative real number.

Defining the square root for complex numbers has one difficult part: choosing between the two roots. The usual way to define a principal square root for all complex numbers relies on excluded middle or something similar. But in the case of a nonnegative real number, we don't have the complications presented for general complex numbers, and we can choose the nonnegative root.

(Contributed by Jim Kingdon, 23-Aug-2020.)

Assertion

Ref	Expression
df-rsqrt	|- sqr = ( x e. RR |-> ( iota_ y e. RR ( ( y ^ 2 ) = x /\ 0 <_ y ) ) )

Distinct variable group:   x, y

Detailed syntax breakdown of Definition df-rsqrt

Step	Hyp	Ref	Expression
1		csqrt 10996	. 2 class sqr
2		vx	. . 3 setvar x
3		cr 7805	. . 3 class RR
4		vy	. . . . . . . 8 setvar y
5	4	cv 1352	. . . . . . 7 class y
6		c2 8964	. . . . . . 7 class 2
7		cexp 10512	. . . . . . 7 class ^
8	5, 6, 7	co 5870	. . . . . 6 class ( y ^ 2 )
9	2	cv 1352	. . . . . 6 class x
10	8, 9	wceq 1353	. . . . 5 wff ( y ^ 2 ) = x
11		cc0 7806	. . . . . 6 class 0
12		cle 7987	. . . . . 6 class <_
13	11, 5, 12	wbr 4001	. . . . 5 wff 0 <_ y
14	10, 13	wa 104	. . . 4 wff ( ( y ^ 2 ) = x /\ 0 <_ y )
15	14, 4, 3	crio 5825	. . 3 class ( iota_ y e. RR ( ( y ^ 2 ) = x /\ 0 <_ y ) )
16	2, 3, 15	cmpt 4062	. 2 class ( x e. RR |-> ( iota_ y e. RR ( ( y ^ 2 ) = x /\ 0 <_ y ) ) )
17	1, 16	wceq 1353	1 wff sqr = ( x e. RR |-> ( iota_ y e. RR ( ( y ^ 2 ) = x /\ 0 <_ y ) ) )

This definition is referenced by:  sqrtrval  11000  absval  11001