Definition df-sgn 13827

Description: Signum function. Pronounced "signum" , otherwise it might be confused with "sine". 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. We define this over RR* (df-xr 10078) instead of RR so that it can accept +oo and -oo. Note that df-psgn 17911 defines the sign of a permutation, which is different. Value shown in sgnval 13828. (Contributed by David A. Wheeler, 15-May-2015.)

Assertion

Ref	Expression
df-sgn	|- sgn = ( x e. RR* |-> if ( x = 0 , 0 , if ( x < 0 , -u 1 , 1 ) ) )

Detailed syntax breakdown of Definition df-sgn

Step	Hyp	Ref	Expression
1		csgn 13826	. 2 class sgn
2		vx	. . 3 setvar x
3		cxr 10073	. . 3 class RR*
4	2	cv 1482	. . . . 5 class x
5		cc0 9936	. . . . 5 class 0
6	4, 5	wceq 1483	. . . 4 wff x = 0
7		clt 10074	. . . . . 6 class <
8	4, 5, 7	wbr 4653	. . . . 5 wff x < 0
9		c1 9937	. . . . . 6 class 1
10	9	cneg 10267	. . . . 5 class -u 1
11	8, 10, 9	cif 4086	. . . 4 class if ( x < 0 , -u 1 , 1 )
12	6, 5, 11	cif 4086	. . 3 class if ( x = 0 , 0 , if ( x < 0 , -u 1 , 1 ) )
13	2, 3, 12	cmpt 4729	. 2 class ( x e. RR* |-> if ( x = 0 , 0 , if ( x < 0 , -u 1 , 1 ) ) )
14	1, 13	wceq 1483	1 wff sgn = ( x e. RR* |-> if ( x = 0 , 0 , if ( x < 0 , -u 1 , 1 ) ) )

This definition is referenced by:  sgnval  13828