Definition df-fl 10339

Description: Define the floor (greatest integer less than or equal to) function. See flval 10341 for its value, flqlelt 10345 for its basic property, and flqcl 10342 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 15217).

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 10337	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 7871	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1363	. . . . . 6 class 𝑦
6	2	cv 1363	. . . . . 6 class 𝑥
7		cle 8055	. . . . . 6 class ≤
8	5, 6, 7	wbr 4029	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 7873	. . . . . . 7 class 1
10		caddc 7875	. . . . . . 7 class +
11	5, 9, 10	co 5918	. . . . . 6 class (𝑦 + 1)
12		clt 8054	. . . . . 6 class <
13	6, 11, 12	wbr 4029	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 104	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 9317	. . . 4 class ℤ
16	14, 4, 15	crio 5872	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 4090	. 2 class (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
18	1, 17	wceq 1364	1 wff ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))

This definition is referenced by:  flval  10341