Definition df-fl 13521

Description: Define the floor (greatest integer less than or equal to) function. See flval 13523 for its value, fllelt 13526 for its basic property, and flcl 13524 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 28820).

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 13519	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 10879	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1538	. . . . . 6 class 𝑦
6	2	cv 1538	. . . . . 6 class 𝑥
7		cle 11019	. . . . . 6 class ≤
8	5, 6, 7	wbr 5075	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 10881	. . . . . . 7 class 1
10		caddc 10883	. . . . . . 7 class +
11	5, 9, 10	co 7284	. . . . . 6 class (𝑦 + 1)
12		clt 11018	. . . . . 6 class <
13	6, 11, 12	wbr 5075	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 396	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 12328	. . . 4 class ℤ
16	14, 4, 15	crio 7240	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 5158	. 2 class (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
18	1, 17	wceq 1539	1 wff ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))

This definition is referenced by:  flval  13523