Definition df-fl 10430

Description: Define the floor (greatest integer less than or equal to) function. See flval 10432 for its value, flqlelt 10436 for its basic property, and flqcl 10433 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 15795).

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 10428	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 7939	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1372	. . . . . 6 class 𝑦
6	2	cv 1372	. . . . . 6 class 𝑥
7		cle 8123	. . . . . 6 class ≤
8	5, 6, 7	wbr 4050	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 7941	. . . . . . 7 class 1
10		caddc 7943	. . . . . . 7 class +
11	5, 9, 10	co 5956	. . . . . 6 class (𝑦 + 1)
12		clt 8122	. . . . . 6 class <
13	6, 11, 12	wbr 4050	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 104	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 9387	. . . 4 class ℤ
16	14, 4, 15	crio 5910	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 4112	. 2 class (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
18	1, 17	wceq 1373	1 wff ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))

This definition is referenced by:  flval  10432