Definition df-fl 10630

Description: Define the floor (greatest integer less than or equal to) function. See flval 10632 for its value, flqlelt 10636 for its basic property, and flqcl 10633 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 16493).

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 10628	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 8126	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1397	. . . . . 6 class 𝑦
6	2	cv 1397	. . . . . 6 class 𝑥
7		cle 8309	. . . . . 6 class ≤
8	5, 6, 7	wbr 4109	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 8128	. . . . . . 7 class 1
10		caddc 8130	. . . . . . 7 class +
11	5, 9, 10	co 6050	. . . . . 6 class (𝑦 + 1)
12		clt 8308	. . . . . 6 class <
13	6, 11, 12	wbr 4109	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 104	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 9577	. . . 4 class ℤ
16	14, 4, 15	crio 6002	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 4171	. 2 class (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
18	1, 17	wceq 1398	1 wff ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))

This definition is referenced by:  flval  10632