Definition df-fl 13790

Description: Define the floor (greatest integer less than or equal to) function. See flval 13792 for its value, fllelt 13795 for its basic property, and flcl 13793 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 30270).

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 13788	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 11138	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1533	. . . . . 6 class 𝑦
6	2	cv 1533	. . . . . 6 class 𝑥
7		cle 11280	. . . . . 6 class ≤
8	5, 6, 7	wbr 5148	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 11140	. . . . . . 7 class 1
10		caddc 11142	. . . . . . 7 class +
11	5, 9, 10	co 7420	. . . . . 6 class (𝑦 + 1)
12		clt 11279	. . . . . 6 class <
13	6, 11, 12	wbr 5148	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 395	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 12589	. . . 4 class ℤ
16	14, 4, 15	crio 7375	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 5231	. 2 class (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
18	1, 17	wceq 1534	1 wff ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))

This definition is referenced by:  flval  13792