Definition df-sgn 15127

Description: Signum function. We do not call it "sign", which is homophonic with "sine" (df-sin 16106). Defined as "sgn" in ISO 80000-2:2009(E) operation 2-9.13. It is named "sign" (with the same definition) in the "NIST Digital Library of Mathematical Functions" , front introduction, "Common Notations and Definitions" section at http://dlmf.nist.gov/front/introduction#Sx4 16106. We define this over ℝ^* (df-xr 11300) instead of ℝ so that it can accept +∞ and -∞. Note that df-psgn 19510 defines the sign of a permutation, which is different. Value shown in sgnval 15128. (Contributed by David A. Wheeler, 15-May-2015.)

Assertion

Ref	Expression
df-sgn	⊢ sgn = (𝑥 ∈ ℝ* ↦ if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1)))

Detailed syntax breakdown of Definition df-sgn

Step	Hyp	Ref	Expression
1		csgn 15126	. 2 class sgn
2		vx	. . 3 setvar 𝑥
3		cxr 11295	. . 3 class ℝ*
4	2	cv 1538	. . . . 5 class 𝑥
5		cc0 11156	. . . . 5 class 0
6	4, 5	wceq 1539	. . . 4 wff 𝑥 = 0
7		clt 11296	. . . . . 6 class <
8	4, 5, 7	wbr 5142	. . . . 5 wff 𝑥 < 0
9		c1 11157	. . . . . 6 class 1
10	9	cneg 11494	. . . . 5 class -1
11	8, 10, 9	cif 4524	. . . 4 class if(𝑥 < 0, -1, 1)
12	6, 5, 11	cif 4524	. . 3 class if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1))
13	2, 3, 12	cmpt 5224	. 2 class (𝑥 ∈ ℝ* ↦ if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1)))
14	1, 13	wceq 1539	1 wff sgn = (𝑥 ∈ ℝ* ↦ if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1)))

This definition is referenced by:  sgnval  15128