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Definition df-fl 10055
Description: Define the floor (greatest integer less than or equal to) function. See flval 10057 for its value, flqlelt 10061 for its basic property, and flqcl 10058 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 13042).

Although we define this on real numbers so that notations are similar to the Metamath Proof Explorer, in the absence of excluded middle few theorems will be possible for all real numbers. Imagine a real number which is around 2.99995 or 3.00001 . In order to determine whether its floor is 2 or 3, it would be necessary to compute the number to arbitrary precision.

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 10053 . 2 class
2 vx . . 3 setvar 𝑥
3 cr 7631 . . 3 class
4 vy . . . . . . 7 setvar 𝑦
54cv 1330 . . . . . 6 class 𝑦
62cv 1330 . . . . . 6 class 𝑥
7 cle 7813 . . . . . 6 class
85, 6, 7wbr 3929 . . . . 5 wff 𝑦𝑥
9 c1 7633 . . . . . . 7 class 1
10 caddc 7635 . . . . . . 7 class +
115, 9, 10co 5774 . . . . . 6 class (𝑦 + 1)
12 clt 7812 . . . . . 6 class <
136, 11, 12wbr 3929 . . . . 5 wff 𝑥 < (𝑦 + 1)
148, 13wa 103 . . . 4 wff (𝑦𝑥𝑥 < (𝑦 + 1))
15 cz 9066 . . . 4 class
1614, 4, 15crio 5729 . . 3 class (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1)))
172, 3, 16cmpt 3989 . 2 class (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1))))
181, 17wceq 1331 1 wff ⌊ = (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1))))
Colors of variables: wff set class
This definition is referenced by:  flval  10057
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