Definition df-fl 13696

Description: Define the floor (greatest integer less than or equal to) function. See flval 13698 for its value, fllelt 13701 for its basic property, and flcl 13699 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 30427).

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 13694	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 11005	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1540	. . . . . 6 class 𝑦
6	2	cv 1540	. . . . . 6 class 𝑥
7		cle 11147	. . . . . 6 class ≤
8	5, 6, 7	wbr 5089	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 11007	. . . . . . 7 class 1
10		caddc 11009	. . . . . . 7 class +
11	5, 9, 10	co 7346	. . . . . 6 class (𝑦 + 1)
12		clt 11146	. . . . . 6 class <
13	6, 11, 12	wbr 5089	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 395	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 12468	. . . 4 class ℤ
16	14, 4, 15	crio 7302	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 5170	. 2 class (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
18	1, 17	wceq 1541	1 wff ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))

This definition is referenced by:  flval  13698