Definition df-fl 13807

Description: Define the floor (greatest integer less than or equal to) function. See flval 13809 for its value, fllelt 13812 for its basic property, and flcl 13810 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 30374).

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 13805	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 11126	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1539	. . . . . 6 class 𝑦
6	2	cv 1539	. . . . . 6 class 𝑥
7		cle 11268	. . . . . 6 class ≤
8	5, 6, 7	wbr 5119	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 11128	. . . . . . 7 class 1
10		caddc 11130	. . . . . . 7 class +
11	5, 9, 10	co 7403	. . . . . 6 class (𝑦 + 1)
12		clt 11267	. . . . . 6 class <
13	6, 11, 12	wbr 5119	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 395	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 12586	. . . 4 class ℤ
16	14, 4, 15	crio 7359	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 5201	. 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  13809