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Definition df-fl 9605
Description: Define the floor (greatest integer less than or equal to) function. See flval 9607 for its value, flqlelt 9611 for its basic property, and flqcl 9608 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 11091).

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 9603 . 2 class
2 vx . . 3 setvar 𝑥
3 cr 7293 . . 3 class
4 vy . . . . . . 7 setvar 𝑦
54cv 1286 . . . . . 6 class 𝑦
62cv 1286 . . . . . 6 class 𝑥
7 cle 7467 . . . . . 6 class
85, 6, 7wbr 3820 . . . . 5 wff 𝑦𝑥
9 c1 7295 . . . . . . 7 class 1
10 caddc 7297 . . . . . . 7 class +
115, 9, 10co 5613 . . . . . 6 class (𝑦 + 1)
12 clt 7466 . . . . . 6 class <
136, 11, 12wbr 3820 . . . . 5 wff 𝑥 < (𝑦 + 1)
148, 13wa 102 . . . 4 wff (𝑦𝑥𝑥 < (𝑦 + 1))
15 cz 8683 . . . 4 class
1614, 4, 15crio 5568 . . 3 class (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1)))
172, 3, 16cmpt 3874 . 2 class (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1))))
181, 17wceq 1287 1 wff ⌊ = (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1))))
Colors of variables: wff set class
This definition is referenced by:  flval  9607
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