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Definition df-sgn 13621
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 * (df-xr 9934) instead of so that it can accept +∞ and -∞. Note that df-psgn 17680 defines the sign of a permutation, which is different. Value shown in sgnval 13622. (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 13620 . 2 class sgn
2 vx . . 3 setvar 𝑥
3 cxr 9929 . . 3 class *
42cv 1473 . . . . 5 class 𝑥
5 cc0 9792 . . . . 5 class 0
64, 5wceq 1474 . . . 4 wff 𝑥 = 0
7 clt 9930 . . . . . 6 class <
84, 5, 7wbr 4577 . . . . 5 wff 𝑥 < 0
9 c1 9793 . . . . . 6 class 1
109cneg 10118 . . . . 5 class -1
118, 10, 9cif 4035 . . . 4 class if(𝑥 < 0, -1, 1)
126, 5, 11cif 4035 . . 3 class if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1))
132, 3, 12cmpt 4637 . 2 class (𝑥 ∈ ℝ* ↦ if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1)))
141, 13wceq 1474 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  13622
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