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Definition df-sgn 14307
 Description: Signum function. We do not call it "sign", which is homophonic with "sine" (df-sin 15283). 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 ℝ* (df-xr 10478) instead of ℝ so that it can accept +∞ and -∞. Note that df-psgn 18380 defines the sign of a permutation, which is different. Value shown in sgnval 14308. (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
StepHypRef Expression
1 csgn 14306 . 2 class sgn
2 vx . . 3 setvar 𝑥
3 cxr 10473 . . 3 class *
42cv 1506 . . . . 5 class 𝑥
5 cc0 10335 . . . . 5 class 0
64, 5wceq 1507 . . . 4 wff 𝑥 = 0
7 clt 10474 . . . . . 6 class <
84, 5, 7wbr 4929 . . . . 5 wff 𝑥 < 0
9 c1 10336 . . . . . 6 class 1
109cneg 10671 . . . . 5 class -1
118, 10, 9cif 4350 . . . 4 class if(𝑥 < 0, -1, 1)
126, 5, 11cif 4350 . . 3 class if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1))
132, 3, 12cmpt 5008 . 2 class (𝑥 ∈ ℝ* ↦ if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1)))
141, 13wceq 1507 1 wff sgn = (𝑥 ∈ ℝ* ↦ if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1)))
 Colors of variables: wff setvar class This definition is referenced by:  sgnval  14308
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