Definition df-fl 12593

Description: Define the floor (greatest integer less than or equal to) function. See flval 12595 for its value, fllelt 12598 for its basic property, and flcl 12596 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 27304).

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 12591	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 9935	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1482	. . . . . 6 class 𝑦
6	2	cv 1482	. . . . . 6 class 𝑥
7		cle 10075	. . . . . 6 class ≤
8	5, 6, 7	wbr 4653	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 9937	. . . . . . 7 class 1
10		caddc 9939	. . . . . . 7 class +
11	5, 9, 10	co 6650	. . . . . 6 class (𝑦 + 1)
12		clt 10074	. . . . . 6 class <
13	6, 11, 12	wbr 4653	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 384	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 11377	. . . 4 class ℤ
16	14, 4, 15	crio 6610	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 4729	. 2 class (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
18	1, 17	wceq 1483	1 wff ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))

This definition is referenced by:  flval  12595