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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 12985 for its value, fllelt 12988 for its basic property, and flcl 12986
for
its closure. For example, (⌊‘(3 / 2)) =
1 while
(⌊‘-(3 / 2)) = -2 (ex-fl 28019).
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 12981 | . 2 class ⌊ | |
2 | vx | . . 3 setvar 𝑥 | |
3 | cr 10340 | . . 3 class ℝ | |
4 | vy | . . . . . . 7 setvar 𝑦 | |
5 | 4 | cv 1507 | . . . . . 6 class 𝑦 |
6 | 2 | cv 1507 | . . . . . 6 class 𝑥 |
7 | cle 10481 | . . . . . 6 class ≤ | |
8 | 5, 6, 7 | wbr 4934 | . . . . 5 wff 𝑦 ≤ 𝑥 |
9 | c1 10342 | . . . . . . 7 class 1 | |
10 | caddc 10344 | . . . . . . 7 class + | |
11 | 5, 9, 10 | co 6982 | . . . . . 6 class (𝑦 + 1) |
12 | clt 10480 | . . . . . 6 class < | |
13 | 6, 11, 12 | wbr 4934 | . . . . 5 wff 𝑥 < (𝑦 + 1) |
14 | 8, 13 | wa 387 | . . . 4 wff (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)) |
15 | cz 11799 | . . . 4 class ℤ | |
16 | 14, 4, 15 | crio 6942 | . . 3 class (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1))) |
17 | 2, 3, 16 | cmpt 5013 | . 2 class (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))) |
18 | 1, 17 | wceq 1508 | 1 wff ⌊ = (𝑥 ∈ ℝ ↦ (℩𝑦 ∈ ℤ (𝑦 ≤ 𝑥 ∧ 𝑥 < (𝑦 + 1)))) |
Colors of variables: wff setvar class |
This definition is referenced by: flval 12985 |
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