Definition df-fl 13440

Description: Define the floor (greatest integer less than or equal to) function. See flval 13442 for its value, fllelt 13445 for its basic property, and flcl 13443 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 28712).

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 13438	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 10801	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1538	. . . . . 6 class 𝑦
6	2	cv 1538	. . . . . 6 class 𝑥
7		cle 10941	. . . . . 6 class ≤
8	5, 6, 7	wbr 5070	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 10803	. . . . . . 7 class 1
10		caddc 10805	. . . . . . 7 class +
11	5, 9, 10	co 7255	. . . . . 6 class (𝑦 + 1)
12		clt 10940	. . . . . 6 class <
13	6, 11, 12	wbr 5070	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 395	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 12249	. . . 4 class ℤ
16	14, 4, 15	crio 7211	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 5153	. 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  13442