Definition df-fl 13751

Description: Define the floor (greatest integer less than or equal to) function. See flval 13753 for its value, fllelt 13756 for its basic property, and flcl 13754 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 30517).

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 13749	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 11037	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1541	. . . . . 6 class 𝑦
6	2	cv 1541	. . . . . 6 class 𝑥
7		cle 11180	. . . . . 6 class ≤
8	5, 6, 7	wbr 5085	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 11039	. . . . . . 7 class 1
10		caddc 11041	. . . . . . 7 class +
11	5, 9, 10	co 7367	. . . . . 6 class (𝑦 + 1)
12		clt 11179	. . . . . 6 class <
13	6, 11, 12	wbr 5085	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 395	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 12524	. . . 4 class ℤ
16	14, 4, 15	crio 7323	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 5166	. 2 class (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
18	1, 17	wceq 1542	1 wff ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))

This definition is referenced by:  flval  13753