Definition df-fl 13761

Description: Define the floor (greatest integer less than or equal to) function. See flval 13763 for its value, fllelt 13766 for its basic property, and flcl 13764 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 30383).

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 13759	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 11074	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1539	. . . . . 6 class 𝑦
6	2	cv 1539	. . . . . 6 class 𝑥
7		cle 11216	. . . . . 6 class ≤
8	5, 6, 7	wbr 5110	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 11076	. . . . . . 7 class 1
10		caddc 11078	. . . . . . 7 class +
11	5, 9, 10	co 7390	. . . . . 6 class (𝑦 + 1)
12		clt 11215	. . . . . 6 class <
13	6, 11, 12	wbr 5110	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 395	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 12536	. . . 4 class ℤ
16	14, 4, 15	crio 7346	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 5191	. 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  13763