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Definition df-fl 13152
Description: Define the floor (greatest integer less than or equal to) function. See flval 13154 for its value, fllelt 13157 for its basic property, and flcl 13155 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 28154).

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 13150 . 2 class
2 vx . . 3 setvar 𝑥
3 cr 10525 . . 3 class
4 vy . . . . . . 7 setvar 𝑦
54cv 1527 . . . . . 6 class 𝑦
62cv 1527 . . . . . 6 class 𝑥
7 cle 10665 . . . . . 6 class
85, 6, 7wbr 5058 . . . . 5 wff 𝑦𝑥
9 c1 10527 . . . . . . 7 class 1
10 caddc 10529 . . . . . . 7 class +
115, 9, 10co 7145 . . . . . 6 class (𝑦 + 1)
12 clt 10664 . . . . . 6 class <
136, 11, 12wbr 5058 . . . . 5 wff 𝑥 < (𝑦 + 1)
148, 13wa 396 . . . 4 wff (𝑦𝑥𝑥 < (𝑦 + 1))
15 cz 11970 . . . 4 class
1614, 4, 15crio 7102 . . 3 class (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1)))
172, 3, 16cmpt 5138 . 2 class (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1))))
181, 17wceq 1528 1 wff ⌊ = (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1))))
Colors of variables: wff setvar class
This definition is referenced by:  flval  13154
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