Definition df-fl 13843

Description: Define the floor (greatest integer less than or equal to) function. See flval 13845 for its value, fllelt 13848 for its basic property, and flcl 13846 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 30479).

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 13841	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 11183	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1536	. . . . . 6 class 𝑦
6	2	cv 1536	. . . . . 6 class 𝑥
7		cle 11325	. . . . . 6 class ≤
8	5, 6, 7	wbr 5166	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 11185	. . . . . . 7 class 1
10		caddc 11187	. . . . . . 7 class +
11	5, 9, 10	co 7448	. . . . . 6 class (𝑦 + 1)
12		clt 11324	. . . . . 6 class <
13	6, 11, 12	wbr 5166	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 395	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 12639	. . . 4 class ℤ
16	14, 4, 15	crio 7403	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 5249	. 2 class (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
18	1, 17	wceq 1537	1 wff ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))

This definition is referenced by:  flval  13845