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Definition df-fl 9222
Description: Define the floor (greatest integer less than or equal to) function. See flval 9224 for its value, flqlelt 9226 for its basic property, and flqcl 9225 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 10279).

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 beyond the rationals. 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 9220 . 2 class
2 vx . . 3 setvar 𝑥
3 cr 6946 . . 3 class
4 vy . . . . . . 7 setvar 𝑦
54cv 1258 . . . . . 6 class 𝑦
62cv 1258 . . . . . 6 class 𝑥
7 cle 7120 . . . . . 6 class
85, 6, 7wbr 3792 . . . . 5 wff 𝑦𝑥
9 c1 6948 . . . . . . 7 class 1
10 caddc 6950 . . . . . . 7 class +
115, 9, 10co 5540 . . . . . 6 class (𝑦 + 1)
12 clt 7119 . . . . . 6 class <
136, 11, 12wbr 3792 . . . . 5 wff 𝑥 < (𝑦 + 1)
148, 13wa 101 . . . 4 wff (𝑦𝑥𝑥 < (𝑦 + 1))
15 cz 8302 . . . 4 class
1614, 4, 15crio 5495 . . 3 class (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1)))
172, 3, 16cmpt 3846 . 2 class (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1))))
181, 17wceq 1259 1 wff ⌊ = (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1))))
Colors of variables: wff set class
This definition is referenced by:  flval  9224
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