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Definition df-sgn 12664
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 9509) instead of  RR so that it can accept +oo and -oo. Note that df-psgn 16085 defines the sign of a permutation, which is different. Value shown in sgnval 12665. (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
StepHypRef Expression
1 csgn 12663 . 2  class sgn
2 vx . . 3  setvar  x
3 cxr 9504 . . 3  class  RR*
42cv 1369 . . . . 5  class  x
5 cc0 9369 . . . . 5  class  0
64, 5wceq 1370 . . . 4  wff  x  =  0
7 clt 9505 . . . . . 6  class  <
84, 5, 7wbr 4376 . . . . 5  wff  x  <  0
9 c1 9370 . . . . . 6  class  1
109cneg 9683 . . . . 5  class  -u 1
118, 10, 9cif 3875 . . . 4  class  if ( x  <  0 , 
-u 1 ,  1 )
126, 5, 11cif 3875 . . 3  class  if ( x  =  0 ,  0 ,  if ( x  <  0 , 
-u 1 ,  1 ) )
132, 3, 12cmpt 4434 . 2  class  ( x  e.  RR*  |->  if ( x  =  0 ,  0 ,  if ( x  <  0 , 
-u 1 ,  1 ) ) )
141, 13wceq 1370 1  wff sgn  =  ( x  e.  RR*  |->  if ( x  =  0 ,  0 ,  if ( x  <  0 , 
-u 1 ,  1 ) ) )
Colors of variables: wff setvar class
This definition is referenced by:  sgnval  12665
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