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Definition df-fl 12455
 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.)
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 12453 . 2 class
2 vx . . 3 setvar 𝑥
3 cr 9814 . . 3 class
4 vy . . . . . . 7 setvar 𝑦
54cv 1474 . . . . . 6 class 𝑦
62cv 1474 . . . . . 6 class 𝑥
7 cle 9954 . . . . . 6 class
85, 6, 7wbr 4583 . . . . 5 wff 𝑦𝑥
9 c1 9816 . . . . . . 7 class 1
10 caddc 9818 . . . . . . 7 class +
115, 9, 10co 6549 . . . . . 6 class (𝑦 + 1)
12 clt 9953 . . . . . 6 class <
136, 11, 12wbr 4583 . . . . 5 wff 𝑥 < (𝑦 + 1)
148, 13wa 383 . . . 4 wff (𝑦𝑥𝑥 < (𝑦 + 1))
15 cz 11254 . . . 4 class
1614, 4, 15crio 6510 . . 3 class (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1)))
172, 3, 16cmpt 4643 . 2 class (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1))))
181, 17wceq 1475 1 wff ⌊ = (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1))))
 Colors of variables: wff setvar class This definition is referenced by:  flval  12457
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