Definition df-fl 13956

Description: Define the floor (greatest integer less than or equal to) function. See flval 13958 for its value, fllelt 13961 for its basic property, and flcl 13959 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘ -(3 / 2)) = -2 (ex-fl 31075).

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 13954	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 11223	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1569	. . . . . 6 class 𝑦
6	2	cv 1569	. . . . . 6 class 𝑥
7		cle 11368	. . . . . 6 class ≤
8	5, 6, 7	wbr 5106	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 11225	. . . . . . 7 class 1
10		caddc 11227	. . . . . . 7 class +
11	5, 9, 10	co 7440	. . . . . 6 class (𝑦 + 1)
12		clt 11367	. . . . . 6 class <
13	6, 11, 12	wbr 5106	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 401	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 12717	. . . 4 class ℤ
16	14, 4, 15	crio 7396	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 5189	. 2 class (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
18	1, 17	wceq 1570	1 wff ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))

This definition is used by:  flval  13958