Definition df-fl 10520

Description: Define the floor (greatest integer less than or equal to) function. See flval 10522 for its value, flqlelt 10526 for its basic property, and flqcl 10523 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 16257).

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 10518	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 8021	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1394	. . . . . 6 class 𝑦
6	2	cv 1394	. . . . . 6 class 𝑥
7		cle 8205	. . . . . 6 class ≤
8	5, 6, 7	wbr 4086	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 8023	. . . . . . 7 class 1
10		caddc 8025	. . . . . . 7 class +
11	5, 9, 10	co 6013	. . . . . 6 class (𝑦 + 1)
12		clt 8204	. . . . . 6 class <
13	6, 11, 12	wbr 4086	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 104	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 9469	. . . 4 class ℤ
16	14, 4, 15	crio 5965	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 4148	. 2 class (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
18	1, 17	wceq 1395	1 wff ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))

This definition is referenced by:  flval  10522