Definition df-fl 13758

Description: Define the floor (greatest integer less than or equal to) function. See flval 13760 for its value, fllelt 13763 for its basic property, and flcl 13761 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 30195).

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 13756	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 11106	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1532	. . . . . 6 class 𝑦
6	2	cv 1532	. . . . . 6 class 𝑥
7		cle 11248	. . . . . 6 class ≤
8	5, 6, 7	wbr 5139	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 11108	. . . . . . 7 class 1
10		caddc 11110	. . . . . . 7 class +
11	5, 9, 10	co 7402	. . . . . 6 class (𝑦 + 1)
12		clt 11247	. . . . . 6 class <
13	6, 11, 12	wbr 5139	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 395	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 12557	. . . 4 class ℤ
16	14, 4, 15	crio 7357	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 5222	. 2 class (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
18	1, 17	wceq 1533	1 wff ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))

This definition is referenced by:  flval  13760