Definition df-fl 10722

Description: Define the floor (greatest integer less than or equal to) function. See flval 10724 for its value, flqlelt 10729 for its basic property, and flqcl 10725 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘ -(3 / 2)) = -2 (ex-fl 16944).

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 10720	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 8184	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1401	. . . . . 6 class 𝑦
6	2	cv 1401	. . . . . 6 class 𝑥
7		cle 8367	. . . . . 6 class ≤
8	5, 6, 7	wbr 4131	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 8186	. . . . . . 7 class 1
10		caddc 8188	. . . . . . 7 class +
11	5, 9, 10	co 6088	. . . . . 6 class (𝑦 + 1)
12		clt 8366	. . . . . 6 class <
13	6, 11, 12	wbr 4131	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 104	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 9655	. . . 4 class ℤ
16	14, 4, 15	crio 6040	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 4193	. 2 class (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
18	1, 17	wceq 1402	1 wff ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))

This definition is used by:  flval  10724