Definition df-fl 13828

Description: Define the floor (greatest integer less than or equal to) function. See flval 13830 for its value, fllelt 13833 for its basic property, and flcl 13831 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 30475).

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 13826	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 11151	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1535	. . . . . 6 class 𝑦
6	2	cv 1535	. . . . . 6 class 𝑥
7		cle 11293	. . . . . 6 class ≤
8	5, 6, 7	wbr 5147	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 11153	. . . . . . 7 class 1
10		caddc 11155	. . . . . . 7 class +
11	5, 9, 10	co 7430	. . . . . 6 class (𝑦 + 1)
12		clt 11292	. . . . . 6 class <
13	6, 11, 12	wbr 5147	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 395	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 12610	. . . 4 class ℤ
16	14, 4, 15	crio 7386	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 5230	. 2 class (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
18	1, 17	wceq 1536	1 wff ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))

This definition is referenced by:  flval  13830