MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-fl Structured version   Visualization version   GIF version

Definition df-fl 12588
Description: Define the floor (greatest integer less than or equal to) function. See flval 12590 for its value, fllelt 12593 for its basic property, and flcl 12591 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 27288).

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 12586 . 2 class
2 vx . . 3 setvar 𝑥
3 cr 9932 . . 3 class
4 vy . . . . . . 7 setvar 𝑦
54cv 1481 . . . . . 6 class 𝑦
62cv 1481 . . . . . 6 class 𝑥
7 cle 10072 . . . . . 6 class
85, 6, 7wbr 4651 . . . . 5 wff 𝑦𝑥
9 c1 9934 . . . . . . 7 class 1
10 caddc 9936 . . . . . . 7 class +
115, 9, 10co 6647 . . . . . 6 class (𝑦 + 1)
12 clt 10071 . . . . . 6 class <
136, 11, 12wbr 4651 . . . . 5 wff 𝑥 < (𝑦 + 1)
148, 13wa 384 . . . 4 wff (𝑦𝑥𝑥 < (𝑦 + 1))
15 cz 11374 . . . 4 class
1614, 4, 15crio 6607 . . 3 class (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1)))
172, 3, 16cmpt 4727 . 2 class (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1))))
181, 17wceq 1482 1 wff ⌊ = (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1))))
Colors of variables: wff setvar class
This definition is referenced by:  flval  12590
  Copyright terms: Public domain W3C validator