Definition df-fl 13714

Description: Define the floor (greatest integer less than or equal to) function. See flval 13716 for its value, fllelt 13719 for its basic property, and flcl 13717 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 30409).

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 13712	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 11027	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1539	. . . . . 6 class 𝑦
6	2	cv 1539	. . . . . 6 class 𝑥
7		cle 11169	. . . . . 6 class ≤
8	5, 6, 7	wbr 5095	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 11029	. . . . . . 7 class 1
10		caddc 11031	. . . . . . 7 class +
11	5, 9, 10	co 7353	. . . . . 6 class (𝑦 + 1)
12		clt 11168	. . . . . 6 class <
13	6, 11, 12	wbr 5095	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 395	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 12489	. . . 4 class ℤ
16	14, 4, 15	crio 7309	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 5176	. 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  13716