Definition df-fl 13703

Description: Define the floor (greatest integer less than or equal to) function. See flval 13705 for its value, fllelt 13708 for its basic property, and flcl 13706 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 30448).

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 13701	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 11016	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1540	. . . . . 6 class 𝑦
6	2	cv 1540	. . . . . 6 class 𝑥
7		cle 11158	. . . . . 6 class ≤
8	5, 6, 7	wbr 5095	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 11018	. . . . . . 7 class 1
10		caddc 11020	. . . . . . 7 class +
11	5, 9, 10	co 7355	. . . . . 6 class (𝑦 + 1)
12		clt 11157	. . . . . 6 class <
13	6, 11, 12	wbr 5095	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 395	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 12479	. . . 4 class ℤ
16	14, 4, 15	crio 7311	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 5176	. 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  13705