Definition df-fl 12983

Description: Define the floor (greatest integer less than or equal to) function. See flval 12985 for its value, fllelt 12988 for its basic property, and flcl 12986 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 28019).

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 12981	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 10340	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1507	. . . . . 6 class 𝑦
6	2	cv 1507	. . . . . 6 class 𝑥
7		cle 10481	. . . . . 6 class ≤
8	5, 6, 7	wbr 4934	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 10342	. . . . . . 7 class 1
10		caddc 10344	. . . . . . 7 class +
11	5, 9, 10	co 6982	. . . . . 6 class (𝑦 + 1)
12		clt 10480	. . . . . 6 class <
13	6, 11, 12	wbr 4934	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 387	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 11799	. . . 4 class ℤ
16	14, 4, 15	crio 6942	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 5013	. 2 class (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
18	1, 17	wceq 1508	1 wff ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))

This definition is referenced by:  flval  12985