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Definition df-fl 12549
Description: Define the floor (greatest integer less than or equal to) function. See flval 12551 for its value, fllelt 12554 for its basic property, and flcl 12552 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 27192).

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 12547 . 2 class
2 vx . . 3 setvar 𝑥
3 cr 9895 . . 3 class
4 vy . . . . . . 7 setvar 𝑦
54cv 1479 . . . . . 6 class 𝑦
62cv 1479 . . . . . 6 class 𝑥
7 cle 10035 . . . . . 6 class
85, 6, 7wbr 4623 . . . . 5 wff 𝑦𝑥
9 c1 9897 . . . . . . 7 class 1
10 caddc 9899 . . . . . . 7 class +
115, 9, 10co 6615 . . . . . 6 class (𝑦 + 1)
12 clt 10034 . . . . . 6 class <
136, 11, 12wbr 4623 . . . . 5 wff 𝑥 < (𝑦 + 1)
148, 13wa 384 . . . 4 wff (𝑦𝑥𝑥 < (𝑦 + 1))
15 cz 11337 . . . 4 class
1614, 4, 15crio 6575 . . 3 class (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1)))
172, 3, 16cmpt 4683 . 2 class (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1))))
181, 17wceq 1480 1 wff ⌊ = (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1))))
Colors of variables: wff setvar class
This definition is referenced by:  flval  12551
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