Definition df-ceil 10378

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 10415 for its value, ceilqge 10419 and ceilqm1lt 10421 for its basic properties, and ceilqcl 10417 for its closure. For example, (⌈‘(3 / 2)) = 2 while (⌈‘-(3 / 2)) = -1 (ex-ceil 15456).

As described in df-fl 10377 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

Step	Hyp	Ref	Expression
1		cceil 10376	. 2 class ⌈
2		vx	. . 3 setvar 𝑥
3		cr 7895	. . 3 class ℝ
4	2	cv 1363	. . . . . 6 class 𝑥
5	4	cneg 8215	. . . . 5 class -𝑥
6		cfl 10375	. . . . 5 class ⌊
7	5, 6	cfv 5259	. . . 4 class (⌊‘-𝑥)
8	7	cneg 8215	. . 3 class -(⌊‘-𝑥)
9	2, 3, 8	cmpt 4095	. 2 class (𝑥 ∈ ℝ ↦ -(⌊‘-𝑥))
10	1, 9	wceq 1364	1 wff ⌈ = (𝑥 ∈ ℝ ↦ -(⌊‘-𝑥))

This definition is referenced by:  ceilqval  10415