Definition df-fl 13804

Description: Define the floor (greatest integer less than or equal to) function. See flval 13806 for its value, fllelt 13809 for its basic property, and flcl 13807 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 30651).

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 13802	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 11074	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1561	. . . . . 6 class 𝑦
6	2	cv 1561	. . . . . 6 class 𝑥
7		cle 11219	. . . . . 6 class ≤
8	5, 6, 7	wbr 5102	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 11076	. . . . . . 7 class 1
10		caddc 11078	. . . . . . 7 class +
11	5, 9, 10	co 7398	. . . . . 6 class (𝑦 + 1)
12		clt 11218	. . . . . 6 class <
13	6, 11, 12	wbr 5102	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 399	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 12570	. . . 4 class ℤ
16	14, 4, 15	crio 7354	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 5183	. 2 class (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
18	1, 17	wceq 1562	1 wff ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))

This definition is referenced by:  flval  13806