MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-ceil Structured version   Visualization version   GIF version

Definition df-ceil 13158
Description: The ceiling (least integer greater than or equal to) function. Defined in ISO 80000-2:2009(E) operation 2-9.18 and the "NIST Digital Library of Mathematical Functions" , front introduction, "Common Notations and Definitions" section at http://dlmf.nist.gov/front/introduction#Sx4. See ceilval 13203 for its value, ceilge 13209 and ceilm1lt 13211 for its basic properties, and ceilcl 13207 for its closure. For example, (⌈‘(3 / 2)) = 2 while (⌈‘-(3 / 2)) = -1 (ex-ceil 28160).

The symbol is inspired by the gamma shaped left bracket of the usual notation. (Contributed by David A. Wheeler, 19-May-2015.)

Assertion
Ref Expression
df-ceil ⌈ = (𝑥 ∈ ℝ ↦ -(⌊‘-𝑥))

Detailed syntax breakdown of Definition df-ceil
StepHypRef Expression
1 cceil 13156 . 2 class
2 vx . . 3 setvar 𝑥
3 cr 10530 . . 3 class
42cv 1529 . . . . . 6 class 𝑥
54cneg 10865 . . . . 5 class -𝑥
6 cfl 13155 . . . . 5 class
75, 6cfv 6354 . . . 4 class (⌊‘-𝑥)
87cneg 10865 . . 3 class -(⌊‘-𝑥)
92, 3, 8cmpt 5143 . 2 class (𝑥 ∈ ℝ ↦ -(⌊‘-𝑥))
101, 9wceq 1530 1 wff ⌈ = (𝑥 ∈ ℝ ↦ -(⌊‘-𝑥))
Colors of variables: wff setvar class
This definition is referenced by:  ceilval  13203  dfceil2  13204
  Copyright terms: Public domain W3C validator