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Definition df-ceil 9188
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 ceilqval 9221 for its value, ceilqge 9225 and ceilqm1lt 9227 for its basic properties, and ceilqcl 9223 for its closure. For example, (⌈‘(3 / 2)) = 2 while (⌈‘-(3 / 2)) = -1 (ex-ceil 10223).

As described in df-fl 9187 most theorems are only for rationals, not reals.

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 9186 . 2 class
2 vx . . 3 setvar 𝑥
3 cr 6916 . . 3 class
42cv 1256 . . . . . 6 class 𝑥
54cneg 7216 . . . . 5 class -𝑥
6 cfl 9185 . . . . 5 class
75, 6cfv 4927 . . . 4 class (⌊‘-𝑥)
87cneg 7216 . . 3 class -(⌊‘-𝑥)
92, 3, 8cmpt 3843 . 2 class (𝑥 ∈ ℝ ↦ -(⌊‘-𝑥))
101, 9wceq 1257 1 wff ⌈ = (𝑥 ∈ ℝ ↦ -(⌊‘-𝑥))
Colors of variables: wff set class
This definition is referenced by:  ceilqval  9221
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