Definition df-fl 13746

Description: Define the floor (greatest integer less than or equal to) function. See flval 13748 for its value, fllelt 13751 for its basic property, and flcl 13749 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 30536).

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 13744	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 11032	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1541	. . . . . 6 class 𝑦
6	2	cv 1541	. . . . . 6 class 𝑥
7		cle 11175	. . . . . 6 class ≤
8	5, 6, 7	wbr 5086	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 11034	. . . . . . 7 class 1
10		caddc 11036	. . . . . . 7 class +
11	5, 9, 10	co 7362	. . . . . 6 class (𝑦 + 1)
12		clt 11174	. . . . . 6 class <
13	6, 11, 12	wbr 5086	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 395	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 12519	. . . 4 class ℤ
16	14, 4, 15	crio 7318	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 5167	. 2 class (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
18	1, 17	wceq 1542	1 wff ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))

This definition is referenced by:  flval  13748