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Definition df-fl 12813
Description: Define the floor (greatest integer less than or equal to) function. See flval 12815 for its value, fllelt 12818 for its basic property, and flcl 12816 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 27634).

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 12811 . 2 class
2 vx . . 3 setvar 𝑥
3 cr 10216 . . 3 class
4 vy . . . . . . 7 setvar 𝑦
54cv 1636 . . . . . 6 class 𝑦
62cv 1636 . . . . . 6 class 𝑥
7 cle 10356 . . . . . 6 class
85, 6, 7wbr 4844 . . . . 5 wff 𝑦𝑥
9 c1 10218 . . . . . . 7 class 1
10 caddc 10220 . . . . . . 7 class +
115, 9, 10co 6870 . . . . . 6 class (𝑦 + 1)
12 clt 10355 . . . . . 6 class <
136, 11, 12wbr 4844 . . . . 5 wff 𝑥 < (𝑦 + 1)
148, 13wa 384 . . . 4 wff (𝑦𝑥𝑥 < (𝑦 + 1))
15 cz 11639 . . . 4 class
1614, 4, 15crio 6830 . . 3 class (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1)))
172, 3, 16cmpt 4923 . 2 class (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1))))
181, 17wceq 1637 1 wff ⌊ = (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1))))
Colors of variables: wff setvar class
This definition is referenced by:  flval  12815
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