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Definition df-fl 13161
Description: Define the floor (greatest integer less than or equal to) function. See flval 13163 for its value, fllelt 13166 for its basic property, and flcl 13164 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 28235).

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 13159 . 2 class
2 vx . . 3 setvar 𝑥
3 cr 10529 . . 3 class
4 vy . . . . . . 7 setvar 𝑦
54cv 1537 . . . . . 6 class 𝑦
62cv 1537 . . . . . 6 class 𝑥
7 cle 10669 . . . . . 6 class
85, 6, 7wbr 5033 . . . . 5 wff 𝑦𝑥
9 c1 10531 . . . . . . 7 class 1
10 caddc 10533 . . . . . . 7 class +
115, 9, 10co 7139 . . . . . 6 class (𝑦 + 1)
12 clt 10668 . . . . . 6 class <
136, 11, 12wbr 5033 . . . . 5 wff 𝑥 < (𝑦 + 1)
148, 13wa 399 . . . 4 wff (𝑦𝑥𝑥 < (𝑦 + 1))
15 cz 11973 . . . 4 class
1614, 4, 15crio 7096 . . 3 class (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1)))
172, 3, 16cmpt 5113 . 2 class (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1))))
181, 17wceq 1538 1 wff ⌊ = (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1))))
Colors of variables: wff setvar class
This definition is referenced by:  flval  13163
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