Definition df-fl 10531

Description: Define the floor (greatest integer less than or equal to) function. See flval 10533 for its value, flqlelt 10537 for its basic property, and flqcl 10534 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 16343).

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 10529	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 8031	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1396	. . . . . 6 class 𝑦
6	2	cv 1396	. . . . . 6 class 𝑥
7		cle 8215	. . . . . 6 class ≤
8	5, 6, 7	wbr 4088	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 8033	. . . . . . 7 class 1
10		caddc 8035	. . . . . . 7 class +
11	5, 9, 10	co 6018	. . . . . 6 class (𝑦 + 1)
12		clt 8214	. . . . . 6 class <
13	6, 11, 12	wbr 4088	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 104	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 9479	. . . 4 class ℤ
16	14, 4, 15	crio 5970	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 4150	. 2 class (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
18	1, 17	wceq 1397	1 wff ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))

This definition is referenced by:  flval  10533