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

Definition df-fl 12406
 Description: Define the floor (greatest integer less than or equal to) function. See flval 12408 for its value, fllelt 12411 for its basic property, and flcl 12409 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 26458). 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 12404 . 2 class
2 vx . . 3 setvar 𝑥
3 cr 9787 . . 3 class
4 vy . . . . . . 7 setvar 𝑦
54cv 1473 . . . . . 6 class 𝑦
62cv 1473 . . . . . 6 class 𝑥
7 cle 9927 . . . . . 6 class
85, 6, 7wbr 4573 . . . . 5 wff 𝑦𝑥
9 c1 9789 . . . . . . 7 class 1
10 caddc 9791 . . . . . . 7 class +
115, 9, 10co 6523 . . . . . 6 class (𝑦 + 1)
12 clt 9926 . . . . . 6 class <
136, 11, 12wbr 4573 . . . . 5 wff 𝑥 < (𝑦 + 1)
148, 13wa 382 . . . 4 wff (𝑦𝑥𝑥 < (𝑦 + 1))
15 cz 11206 . . . 4 class
1614, 4, 15crio 6484 . . 3 class (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1)))
172, 3, 16cmpt 4633 . 2 class (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1))))
181, 17wceq 1474 1 wff ⌊ = (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1))))
 Colors of variables: wff setvar class This definition is referenced by:  flval  12408
 Copyright terms: Public domain W3C validator