Definition df-fl 13830

Description: Define the floor (greatest integer less than or equal to) function. See flval 13832 for its value, fllelt 13835 for its basic property, and flcl 13833 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 30807).

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 13828	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 11103	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1569	. . . . . 6 class 𝑦
6	2	cv 1569	. . . . . 6 class 𝑥
7		cle 11248	. . . . . 6 class ≤
8	5, 6, 7	wbr 5109	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 11105	. . . . . . 7 class 1
10		caddc 11107	. . . . . . 7 class +
11	5, 9, 10	co 7410	. . . . . 6 class (𝑦 + 1)
12		clt 11247	. . . . . 6 class <
13	6, 11, 12	wbr 5109	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 400	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 12595	. . . 4 class ℤ
16	14, 4, 15	crio 7366	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 5192	. 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  13832