Definition df-fl 10205

Description: Define the floor (greatest integer less than or equal to) function. See flval 10207 for its value, flqlelt 10211 for its basic property, and flqcl 10208 for its closure. For example, ( |_ ` ( 3 / 2 ) ) = 1 while ( |_ ` -u ( 3 / 2 ) ) = -u 2 (ex-fl 13606).

Although we define this on real numbers so that notations are similar to the Metamath Proof Explorer, in the absence of excluded middle few theorems will be possible for all real numbers. Imagine a real number which is around 2.99995 or 3.00001 . In order to determine whether its floor is 2 or 3, it would be necessary to compute the number to arbitrary precision.

The term "floor" was coined by Ken Iverson. He also invented a mathematical notation for floor, consisting of an L-shaped left bracket and its reflection as a right bracket. In APL, the left-bracket alone is used, and we borrow this idea. (Thanks to Paul Chapman for this information.) (Contributed by NM, 14-Nov-2004.)

Assertion

Ref	Expression
df-fl	|- |_ = ( x e. RR |-> ( iota_ y e. ZZ ( y <_ x /\ x < ( y + 1 ) ) ) )

Distinct variable group:   x, y

Detailed syntax breakdown of Definition df-fl

Step	Hyp	Ref	Expression
1		cfl 10203	. 2 class |_
2		vx	. . 3 setvar x
3		cr 7752	. . 3 class RR
4		vy	. . . . . . 7 setvar y
5	4	cv 1342	. . . . . 6 class y
6	2	cv 1342	. . . . . 6 class x
7		cle 7934	. . . . . 6 class <_
8	5, 6, 7	wbr 3982	. . . . 5 wff y <_ x
9		c1 7754	. . . . . . 7 class 1
10		caddc 7756	. . . . . . 7 class +
11	5, 9, 10	co 5842	. . . . . 6 class ( y + 1 )
12		clt 7933	. . . . . 6 class <
13	6, 11, 12	wbr 3982	. . . . 5 wff x < ( y + 1 )
14	8, 13	wa 103	. . . 4 wff ( y <_ x /\ x < ( y + 1 ) )
15		cz 9191	. . . 4 class ZZ
16	14, 4, 15	crio 5797	. . 3 class ( iota_ y e. ZZ ( y <_ x /\ x < ( y + 1 ) ) )
17	2, 3, 16	cmpt 4043	. 2 class ( x e. RR |-> ( iota_ y e. ZZ ( y <_ x /\ x < ( y + 1 ) ) ) )
18	1, 17	wceq 1343	1 wff |_ = ( x e. RR |-> ( iota_ y e. ZZ ( y <_ x /\ x < ( y + 1 ) ) ) )

This definition is referenced by:  flval  10207