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Mirrors > Home > MPE Home > Th. List > df-fl | Structured version Visualization version GIF version |
Description: Define the floor
(greatest integer less than or equal to) function. See
flval 12457 for its value, fllelt 12460 for its basic property, and flcl 12458
for
its closure. For example, (⌊‘(3 / 2)) =
1 while
(⌊‘-(3 / 2)) = -2 (ex-fl 26696).
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 12453 | . 2 class ⌊ | |
2 | vx | . . 3 setvar 𝑥 | |
3 | cr 9814 | . . 3 class ℝ | |
4 | vy | . . . . . . 7 setvar 𝑦 | |
5 | 4 | cv 1474 | . . . . . 6 class 𝑦 |
6 | 2 | cv 1474 | . . . . . 6 class 𝑥 |
7 | cle 9954 | . . . . . 6 class ≤ | |
8 | 5, 6, 7 | wbr 4583 | . . . . 5 wff 𝑦 ≤ 𝑥 |
9 | c1 9816 | . . . . . . 7 class 1 | |
10 | caddc 9818 | . . . . . . 7 class + | |
11 | 5, 9, 10 | co 6549 | . . . . . 6 class (𝑦 + 1) |
12 | clt 9953 | . . . . . 6 class < | |
13 | 6, 11, 12 | wbr 4583 | . . . . 5 wff 𝑥 < (𝑦 + 1) |
14 | 8, 13 | wa 383 | . . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)) |
15 | cz 11254 | . . . 4 class ℤ | |
16 | 14, 4, 15 | crio 6510 | . . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))) |
17 | 2, 3, 16 | cmpt 4643 | . 2 class (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))) |
18 | 1, 17 | wceq 1475 | 1 wff ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))) |
Colors of variables: wff setvar class |
This definition is referenced by: flval 12457 |
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