Definition df-ceil 10724

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 10765 for its value, ceilqge 10769 and ceilqm1lt 10771 for its basic properties, and ceilqcl 10767 for its closure. For example, (⌈‘(3 / 2)) = 2 while (⌈‘ -(3 / 2)) = -1 (ex-ceil 16957).

As described in df-fl 10723 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 10722	. 2 class ⌈
2		vx	. . 3 setvar 𝑥
3		cr 8185	. . 3 class ℝ
4	2	cv 1401	. . . . . 6 class 𝑥
5	4	cneg 8506	. . . . 5 class -𝑥
6		cfl 10721	. . . . 5 class ⌊
7	5, 6	cfv 5380	. . . 4 class (⌊‘ -𝑥)
8	7	cneg 8506	. . 3 class -(⌊‘ -𝑥)
9	2, 3, 8	cmpt 4193	. 2 class (𝑥 ∈ ℝ ↦ -(⌊‘ -𝑥))
10	1, 9	wceq 1402	1 wff ⌈ = (𝑥 ∈ ℝ ↦ -(⌊‘ -𝑥))

This definition is used by:  ceilqval  10765