Definition df-ceil 10717

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 10758 for its value, ceilqge 10762 and ceilqm1lt 10764 for its basic properties, and ceilqcl 10760 for its closure. For example, (⌈ ` ( 3 / 2 ) ) = 2 while (⌈ ` -u ( 3 / 2 ) ) = -u 1 (ex-ceil 16906).

As described in df-fl 10716 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 e. RR |-> -u ( |_ ` -u x ) )

Detailed syntax breakdown of Definition df-ceil

Step	Hyp	Ref	Expression
1		cceil 10715	. 2 class ⌈
2		vx	. . 3 setvar x
3		cr 8179	. . 3 class RR
4	2	cv 1401	. . . . . 6 class x
5	4	cneg 8500	. . . . 5 class -u x
6		cfl 10714	. . . . 5 class |_
7	5, 6	cfv 5377	. . . 4 class ( |_ ` -u x )
8	7	cneg 8500	. . 3 class -u ( |_ ` -u x )
9	2, 3, 8	cmpt 4192	. 2 class ( x e. RR |-> -u ( |_ ` -u x ) )
10	1, 9	wceq 1402	1 wff ⌈ = ( x e. RR |-> -u ( |_ ` -u x ) )

This definition is used by:  ceilqval  10758