Definition df-fl 10379

Description: Define the floor (greatest integer less than or equal to) function. See flval 10381 for its value, flqlelt 10385 for its basic property, and flqcl 10382 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 15479).

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 10377	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 7897	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1363	. . . . . 6 class 𝑦
6	2	cv 1363	. . . . . 6 class 𝑥
7		cle 8081	. . . . . 6 class ≤
8	5, 6, 7	wbr 4034	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 7899	. . . . . . 7 class 1
10		caddc 7901	. . . . . . 7 class +
11	5, 9, 10	co 5925	. . . . . 6 class (𝑦 + 1)
12		clt 8080	. . . . . 6 class <
13	6, 11, 12	wbr 4034	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 104	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 9345	. . . 4 class ℤ
16	14, 4, 15	crio 5879	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 4095	. 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  10381