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Definition df-fl 13832
Description: Define the floor (greatest integer less than or equal to) function. See flval 13834 for its value, fllelt 13837 for its basic property, and flcl 13835 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 30466).

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 (𝑦𝑥𝑥 < (𝑦 + 1))))
Distinct variable group:   𝑥,𝑦

Detailed syntax breakdown of Definition df-fl
StepHypRef Expression
1 cfl 13830 . 2 class
2 vx . . 3 setvar 𝑥
3 cr 11154 . . 3 class
4 vy . . . . . . 7 setvar 𝑦
54cv 1539 . . . . . 6 class 𝑦
62cv 1539 . . . . . 6 class 𝑥
7 cle 11296 . . . . . 6 class
85, 6, 7wbr 5143 . . . . 5 wff 𝑦𝑥
9 c1 11156 . . . . . . 7 class 1
10 caddc 11158 . . . . . . 7 class +
115, 9, 10co 7431 . . . . . 6 class (𝑦 + 1)
12 clt 11295 . . . . . 6 class <
136, 11, 12wbr 5143 . . . . 5 wff 𝑥 < (𝑦 + 1)
148, 13wa 395 . . . 4 wff (𝑦𝑥𝑥 < (𝑦 + 1))
15 cz 12613 . . . 4 class
1614, 4, 15crio 7387 . . 3 class (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1)))
172, 3, 16cmpt 5225 . 2 class (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1))))
181, 17wceq 1540 1 wff ⌊ = (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1))))
Colors of variables: wff setvar class
This definition is referenced by:  flval  13834
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