Definition df-fl 13716

Description: Define the floor (greatest integer less than or equal to) function. See flval 13718 for its value, fllelt 13721 for its basic property, and flcl 13719 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 30526).

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 13714	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 11029	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1541	. . . . . 6 class 𝑦
6	2	cv 1541	. . . . . 6 class 𝑥
7		cle 11171	. . . . . 6 class ≤
8	5, 6, 7	wbr 5099	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 11031	. . . . . . 7 class 1
10		caddc 11033	. . . . . . 7 class +
11	5, 9, 10	co 7360	. . . . . 6 class (𝑦 + 1)
12		clt 11170	. . . . . 6 class <
13	6, 11, 12	wbr 5099	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 395	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 12492	. . . 4 class ℤ
16	14, 4, 15	crio 7316	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 5180	. 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  13718