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Mirrors > Home > MPE Home > Th. List > df-sgn | Structured version Visualization version GIF version |
Description: Signum function. We do not call it "sign", which is homophonic with "sine" (df-sin 15508). 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 15508. We define this over ℝ* (df-xr 10750) instead of ℝ so that it can accept +∞ and -∞. Note that df-psgn 18730 defines the sign of a permutation, which is different. Value shown in sgnval 14530. (Contributed by David A. Wheeler, 15-May-2015.) |
Ref | Expression |
---|---|
df-sgn | ⊢ sgn = (𝑥 ∈ ℝ* ↦ if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csgn 14528 | . 2 class sgn | |
2 | vx | . . 3 setvar 𝑥 | |
3 | cxr 10745 | . . 3 class ℝ* | |
4 | 2 | cv 1541 | . . . . 5 class 𝑥 |
5 | cc0 10608 | . . . . 5 class 0 | |
6 | 4, 5 | wceq 1542 | . . . 4 wff 𝑥 = 0 |
7 | clt 10746 | . . . . . 6 class < | |
8 | 4, 5, 7 | wbr 5027 | . . . . 5 wff 𝑥 < 0 |
9 | c1 10609 | . . . . . 6 class 1 | |
10 | 9 | cneg 10942 | . . . . 5 class -1 |
11 | 8, 10, 9 | cif 4411 | . . . 4 class if(𝑥 < 0, -1, 1) |
12 | 6, 5, 11 | cif 4411 | . . 3 class if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1)) |
13 | 2, 3, 12 | cmpt 5107 | . 2 class (𝑥 ∈ ℝ* ↦ if(𝑥 = 0, 0, if(𝑥 < 0, -1, 1))) |
14 | 1, 13 | wceq 1542 | 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 14530 |
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