Definition df-fl 13932

Description: Define the floor (greatest integer less than or equal to) function. See flval 13934 for its value, fllelt 13937 for its basic property, and flcl 13935 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘ -(3 / 2)) = -2 (ex-fl 31048).

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 13930	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 11199	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1568	. . . . . 6 class 𝑦
6	2	cv 1568	. . . . . 6 class 𝑥
7		cle 11344	. . . . . 6 class ≤
8	5, 6, 7	wbr 5102	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 11201	. . . . . . 7 class 1
10		caddc 11203	. . . . . . 7 class +
11	5, 9, 10	co 7420	. . . . . 6 class (𝑦 + 1)
12		clt 11343	. . . . . 6 class <
13	6, 11, 12	wbr 5102	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 401	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 12693	. . . 4 class ℤ
16	14, 4, 15	crio 7376	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 5185	. 2 class (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
18	1, 17	wceq 1569	1 wff ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))

This definition is used by:  flval  13934