Definition df-fl 13832

Description: Define the floor (greatest integer less than or equal to) function. See flval 13834 for its value, fllelt 13837 for its basic property, and flcl 13835 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 30466).

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 13830	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 11154	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1539	. . . . . 6 class 𝑦
6	2	cv 1539	. . . . . 6 class 𝑥
7		cle 11296	. . . . . 6 class ≤
8	5, 6, 7	wbr 5143	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 11156	. . . . . . 7 class 1
10		caddc 11158	. . . . . . 7 class +
11	5, 9, 10	co 7431	. . . . . 6 class (𝑦 + 1)
12		clt 11295	. . . . . 6 class <
13	6, 11, 12	wbr 5143	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 395	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 12613	. . . 4 class ℤ
16	14, 4, 15	crio 7387	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 5225	. 2 class (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
18	1, 17	wceq 1540	1 wff ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))

This definition is referenced by:  flval  13834