Definition df-fl 10043

Description: Define the floor (greatest integer less than or equal to) function. See flval 10045 for its value, flqlelt 10049 for its basic property, and flqcl 10046 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 12937).

Although we define this on real numbers so that notations are similar to the Metamath Proof Explorer, in the absence of excluded middle few theorems will be possible for all real numbers. Imagine a real number which is around 2.99995 or 3.00001 . In order to determine whether its floor is 2 or 3, it would be necessary to compute the number to arbitrary precision.

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 10041	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 7619	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1330	. . . . . 6 class 𝑦
6	2	cv 1330	. . . . . 6 class 𝑥
7		cle 7801	. . . . . 6 class ≤
8	5, 6, 7	wbr 3929	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 7621	. . . . . . 7 class 1
10		caddc 7623	. . . . . . 7 class +
11	5, 9, 10	co 5774	. . . . . 6 class (𝑦 + 1)
12		clt 7800	. . . . . 6 class <
13	6, 11, 12	wbr 3929	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 103	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 9054	. . . 4 class ℤ
16	14, 4, 15	crio 5729	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 3989	. 2 class (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
18	1, 17	wceq 1331	1 wff ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))

This definition is referenced by:  flval  10045