Definition df-fl 13613

Description: Define the floor (greatest integer less than or equal to) function. See flval 13615 for its value, fllelt 13618 for its basic property, and flcl 13616 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 29099).

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 13611	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 10971	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1539	. . . . . 6 class 𝑦
6	2	cv 1539	. . . . . 6 class 𝑥
7		cle 11111	. . . . . 6 class ≤
8	5, 6, 7	wbr 5092	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 10973	. . . . . . 7 class 1
10		caddc 10975	. . . . . . 7 class +
11	5, 9, 10	co 7337	. . . . . 6 class (𝑦 + 1)
12		clt 11110	. . . . . 6 class <
13	6, 11, 12	wbr 5092	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 396	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 12420	. . . 4 class ℤ
16	14, 4, 15	crio 7292	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 5175	. 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  13615