Definition df-fl 13827

Description: Define the floor (greatest integer less than or equal to) function. See flval 13829 for its value, fllelt 13832 for its basic property, and flcl 13830 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 30741).

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 13825	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 11101	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1566	. . . . . 6 class 𝑦
6	2	cv 1566	. . . . . 6 class 𝑥
7		cle 11246	. . . . . 6 class ≤
8	5, 6, 7	wbr 5113	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 11103	. . . . . . 7 class 1
10		caddc 11105	. . . . . . 7 class +
11	5, 9, 10	co 7413	. . . . . 6 class (𝑦 + 1)
12		clt 11245	. . . . . 6 class <
13	6, 11, 12	wbr 5113	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 400	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 12593	. . . 4 class ℤ
16	14, 4, 15	crio 7369	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 5196	. 2 class (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
18	1, 17	wceq 1567	1 wff ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))

This definition is referenced by:  flval  13829