Definition df-fl 13853

Description: Define the floor (greatest integer less than or equal to) function. See flval 13855 for its value, fllelt 13858 for its basic property, and flcl 13856 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 30911).

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 13851	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 11124	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1569	. . . . . 6 class 𝑦
6	2	cv 1569	. . . . . 6 class 𝑥
7		cle 11269	. . . . . 6 class ≤
8	5, 6, 7	wbr 5107	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 11126	. . . . . . 7 class 1
10		caddc 11128	. . . . . . 7 class +
11	5, 9, 10	co 7416	. . . . . 6 class (𝑦 + 1)
12		clt 11268	. . . . . 6 class <
13	6, 11, 12	wbr 5107	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 401	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 12616	. . . 4 class ℤ
16	14, 4, 15	crio 7372	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 5190	. 2 class (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
18	1, 17	wceq 1570	1 wff ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))

This definition is used by:  flval  13855