Definition df-fl 13332

Description: Define the floor (greatest integer less than or equal to) function. See flval 13334 for its value, fllelt 13337 for its basic property, and flcl 13335 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 28484).

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 13330	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 10693	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1542	. . . . . 6 class 𝑦
6	2	cv 1542	. . . . . 6 class 𝑥
7		cle 10833	. . . . . 6 class ≤
8	5, 6, 7	wbr 5039	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 10695	. . . . . . 7 class 1
10		caddc 10697	. . . . . . 7 class +
11	5, 9, 10	co 7191	. . . . . 6 class (𝑦 + 1)
12		clt 10832	. . . . . 6 class <
13	6, 11, 12	wbr 5039	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 399	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 12141	. . . 4 class ℤ
16	14, 4, 15	crio 7147	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 5120	. 2 class (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))
18	1, 17	wceq 1543	1 wff ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))))

This definition is referenced by:  flval  13334