Definition df-sgn

Description: Signum function. We do not call it "sign", which is homophonic with "sine" ( df-sin ). 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 ) instead of RR so that it can accept +oo and -oo . Note that df-psgn defines the sign of a permutation, which is different. Value shown in sgnval . (Contributed by David A. Wheeler, 15-May-2015)

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

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		csgn	|- sgn
1		vx	|- x
2		cxr	|- RR*
3	1	cv	|- x
4		cc0	|- 0
5	3 4	wceq	|- x = 0
6		clt	|- <
7	3 4 6	wbr	|- x < 0
8		c1	|- 1
9	8	cneg	|- -u 1
10	7 9 8	cif	|- if ( x < 0 , -u 1 , 1 )
11	5 4 10	cif	|- if ( x = 0 , 0 , if ( x < 0 , -u 1 , 1 ) )
12	1 2 11	cmpt	|- ( x e. RR* |-> if ( x = 0 , 0 , if ( x < 0 , -u 1 , 1 ) ) )
13	0 12	wceq	|- sgn = ( x e. RR* |-> if ( x = 0 , 0 , if ( x < 0 , -u 1 , 1 ) ) )