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Definition df-fl 12787
Description: Define the floor (greatest integer less than or equal to) function. See flval 12789 for its value, fllelt 12792 for its basic property, and flcl 12790 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 27615).

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 12785 . 2 class
2 vx . . 3 setvar 𝑥
3 cr 10127 . . 3 class
4 vy . . . . . . 7 setvar 𝑦
54cv 1631 . . . . . 6 class 𝑦
62cv 1631 . . . . . 6 class 𝑥
7 cle 10267 . . . . . 6 class
85, 6, 7wbr 4804 . . . . 5 wff 𝑦𝑥
9 c1 10129 . . . . . . 7 class 1
10 caddc 10131 . . . . . . 7 class +
115, 9, 10co 6813 . . . . . 6 class (𝑦 + 1)
12 clt 10266 . . . . . 6 class <
136, 11, 12wbr 4804 . . . . 5 wff 𝑥 < (𝑦 + 1)
148, 13wa 383 . . . 4 wff (𝑦𝑥𝑥 < (𝑦 + 1))
15 cz 11569 . . . 4 class
1614, 4, 15crio 6773 . . 3 class (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1)))
172, 3, 16cmpt 4881 . 2 class (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1))))
181, 17wceq 1632 1 wff ⌊ = (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1))))
Colors of variables: wff setvar class
This definition is referenced by:  flval  12789
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