Definition df-fl 10270

Description: Define the floor (greatest integer less than or equal to) function. See flval 10272 for its value, flqlelt 10276 for its basic property, and flqcl 10273 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 14480).

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 10268	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 7810	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1352	. . . . . 6 class 𝑦
6	2	cv 1352	. . . . . 6 class 𝑥
7		cle 7993	. . . . . 6 class ≤
8	5, 6, 7	wbr 4004	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 7812	. . . . . . 7 class 1
10		caddc 7814	. . . . . . 7 class +
11	5, 9, 10	co 5875	. . . . . 6 class (𝑦 + 1)
12		clt 7992	. . . . . 6 class <
13	6, 11, 12	wbr 4004	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 104	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 9253	. . . 4 class ℤ
16	14, 4, 15	crio 5830	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 4065	. 2 class (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
18	1, 17	wceq 1353	1 wff ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))

This definition is referenced by:  flval  10272