Definition df-fl 10074

Description: Define the floor (greatest integer less than or equal to) function. See flval 10076 for its value, flqlelt 10080 for its basic property, and flqcl 10077 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 13108).

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 10072	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 7643	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1331	. . . . . 6 class 𝑦
6	2	cv 1331	. . . . . 6 class 𝑥
7		cle 7825	. . . . . 6 class ≤
8	5, 6, 7	wbr 3937	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 7645	. . . . . . 7 class 1
10		caddc 7647	. . . . . . 7 class +
11	5, 9, 10	co 5782	. . . . . 6 class (𝑦 + 1)
12		clt 7824	. . . . . 6 class <
13	6, 11, 12	wbr 3937	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 103	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 9078	. . . 4 class ℤ
16	14, 4, 15	crio 5737	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 3997	. 2 class (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
18	1, 17	wceq 1332	1 wff ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))

This definition is referenced by:  flval  10076