Definition df-fl 9605

Description: Define the floor (greatest integer less than or equal to) function. See flval 9607 for its value, flqlelt 9611 for its basic property, and flqcl 9608 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 11091).

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 9603	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 7293	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1286	. . . . . 6 class 𝑦
6	2	cv 1286	. . . . . 6 class 𝑥
7		cle 7467	. . . . . 6 class ≤
8	5, 6, 7	wbr 3820	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 7295	. . . . . . 7 class 1
10		caddc 7297	. . . . . . 7 class +
11	5, 9, 10	co 5613	. . . . . 6 class (𝑦 + 1)
12		clt 7466	. . . . . 6 class <
13	6, 11, 12	wbr 3820	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 102	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 8683	. . . 4 class ℤ
16	14, 4, 15	crio 5568	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 3874	. 2 class (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
18	1, 17	wceq 1287	1 wff ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))

This definition is referenced by:  flval  9607