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Mirrors > Home > ILE Home > Th. List > df-fl | GIF version |
Description: Define the floor
(greatest integer less than or equal to) function. See
flval 9116 for its value, flqlelt 9118 for its basic property, and flqcl 9117 for
its closure. For example, (⌊‘(3 / 2)) =
1 while
(⌊‘-(3 / 2)) = -2 (ex-fl 9895).
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.) |
Ref | Expression |
---|---|
df-fl | ⊢ ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cfl 9112 | . 2 class ⌊ | |
2 | vx | . . 3 setvar 𝑥 | |
3 | cr 6888 | . . 3 class ℝ | |
4 | vy | . . . . . . 7 setvar 𝑦 | |
5 | 4 | cv 1242 | . . . . . 6 class 𝑦 |
6 | 2 | cv 1242 | . . . . . 6 class 𝑥 |
7 | cle 7061 | . . . . . 6 class ≤ | |
8 | 5, 6, 7 | wbr 3764 | . . . . 5 wff 𝑦 ≤ 𝑥 |
9 | c1 6890 | . . . . . . 7 class 1 | |
10 | caddc 6892 | . . . . . . 7 class + | |
11 | 5, 9, 10 | co 5512 | . . . . . 6 class (𝑦 + 1) |
12 | clt 7060 | . . . . . 6 class < | |
13 | 6, 11, 12 | wbr 3764 | . . . . 5 wff 𝑥 < (𝑦 + 1) |
14 | 8, 13 | wa 97 | . . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)) |
15 | cz 8245 | . . . 4 class ℤ | |
16 | 14, 4, 15 | crio 5467 | . . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))) |
17 | 2, 3, 16 | cmpt 3818 | . 2 class (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))) |
18 | 1, 17 | wceq 1243 | 1 wff ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))) |
Colors of variables: wff set class |
This definition is referenced by: flval 9116 |
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