Definition df-fl 13754

Description: Define the floor (greatest integer less than or equal to) function. See flval 13756 for its value, fllelt 13759 for its basic property, and flcl 13757 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 29690).

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 13752	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 11106	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1541	. . . . . 6 class 𝑦
6	2	cv 1541	. . . . . 6 class 𝑥
7		cle 11246	. . . . . 6 class ≤
8	5, 6, 7	wbr 5148	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 11108	. . . . . . 7 class 1
10		caddc 11110	. . . . . . 7 class +
11	5, 9, 10	co 7406	. . . . . 6 class (𝑦 + 1)
12		clt 11245	. . . . . 6 class <
13	6, 11, 12	wbr 5148	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 397	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 12555	. . . 4 class ℤ
16	14, 4, 15	crio 7361	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 5231	. 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  13756