Definition df-fl 13163

Description: Define the floor (greatest integer less than or equal to) function. See flval 13165 for its value, fllelt 13168 for its basic property, and flcl 13166 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 28226).

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 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))

Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-fl

Step	Hyp	Ref	Expression
1		cfl 13161	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 10536	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1536	. . . . . 6 class 𝑦
6	2	cv 1536	. . . . . 6 class 𝑥
7		cle 10676	. . . . . 6 class ≤
8	5, 6, 7	wbr 5066	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 10538	. . . . . . 7 class 1
10		caddc 10540	. . . . . . 7 class +
11	5, 9, 10	co 7156	. . . . . 6 class (𝑦 + 1)
12		clt 10675	. . . . . 6 class <
13	6, 11, 12	wbr 5066	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 398	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 11982	. . . 4 class ℤ
16	14, 4, 15	crio 7113	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 5146	. 2 class (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
18	1, 17	wceq 1537	1 wff ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))

This definition is referenced by:  flval  13165