Definition df-fl 13825

Description: Define the floor (greatest integer less than or equal to) function. See flval 13827 for its value, fllelt 13830 for its basic property, and flcl 13828 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 30739).

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 13823	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 11099	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1566	. . . . . 6 class 𝑦
6	2	cv 1566	. . . . . 6 class 𝑥
7		cle 11244	. . . . . 6 class ≤
8	5, 6, 7	wbr 5113	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 11101	. . . . . . 7 class 1
10		caddc 11103	. . . . . . 7 class +
11	5, 9, 10	co 7411	. . . . . 6 class (𝑦 + 1)
12		clt 11243	. . . . . 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 12591	. . . 4 class ℤ
16	14, 4, 15	crio 7367	. . 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  13827