Definition df-fl 9274

Description: Define the floor (greatest integer less than or equal to) function. See flval 9276 for its value, flqlelt 9278 for its basic property, and flqcl 9277 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 10563).

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 beyond the rationals. 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 9272	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 6980	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1283	. . . . . 6 class 𝑦
6	2	cv 1283	. . . . . 6 class 𝑥
7		cle 7154	. . . . . 6 class ≤
8	5, 6, 7	wbr 3785	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 6982	. . . . . . 7 class 1
10		caddc 6984	. . . . . . 7 class +
11	5, 9, 10	co 5532	. . . . . 6 class (𝑦 + 1)
12		clt 7153	. . . . . 6 class <
13	6, 11, 12	wbr 3785	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 102	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 8351	. . . 4 class ℤ
16	14, 4, 15	crio 5487	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 3839	. 2 class (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
18	1, 17	wceq 1284	1 wff ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))

This definition is referenced by:  flval  9276