Definition df-fl 13812

Description: Define the floor (greatest integer less than or equal to) function. See flval 13814 for its value, fllelt 13817 for its basic property, and flcl 13815 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 30380).

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 13810	. 2 class ⌊
2		vx	. . 3 setvar 𝑥
3		cr 11157	. . 3 class ℝ
4		vy	. . . . . . 7 setvar 𝑦
5	4	cv 1533	. . . . . 6 class 𝑦
6	2	cv 1533	. . . . . 6 class 𝑥
7		cle 11299	. . . . . 6 class ≤
8	5, 6, 7	wbr 5153	. . . . 5 wff 𝑦 ≤ 𝑥
9		c1 11159	. . . . . . 7 class 1
10		caddc 11161	. . . . . . 7 class +
11	5, 9, 10	co 7424	. . . . . 6 class (𝑦 + 1)
12		clt 11298	. . . . . 6 class <
13	6, 11, 12	wbr 5153	. . . . 5 wff 𝑥 < (𝑦 + 1)
14	8, 13	wa 394	. . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))
15		cz 12610	. . . 4 class ℤ
16	14, 4, 15	crio 7379	. . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))
17	2, 3, 16	cmpt 5236	. 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  13814

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