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Definition df-fl 9114
 Description: Define the floor (greatest integer less than or equal to) function. See flval 9116 for its value, flqlelt 9118 for its basic property, and flqcl 9117 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 9895). Although we define this on real numbers so that notations are similar to the Metamath Proof Explorer, in the absence of excluded middle few theorems will be possible beyond the rationals. Imagine a real number which is around 2.99995 or 3.00001 . In order to determine whether its floor is 2 or 3, it would be necessary to compute the number to arbitrary precision. 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 9112 . 2 class
2 vx . . 3 setvar 𝑥
3 cr 6888 . . 3 class
4 vy . . . . . . 7 setvar 𝑦
54cv 1242 . . . . . 6 class 𝑦
62cv 1242 . . . . . 6 class 𝑥
7 cle 7061 . . . . . 6 class
85, 6, 7wbr 3764 . . . . 5 wff 𝑦𝑥
9 c1 6890 . . . . . . 7 class 1
10 caddc 6892 . . . . . . 7 class +
115, 9, 10co 5512 . . . . . 6 class (𝑦 + 1)
12 clt 7060 . . . . . 6 class <
136, 11, 12wbr 3764 . . . . 5 wff 𝑥 < (𝑦 + 1)
148, 13wa 97 . . . 4 wff (𝑦𝑥𝑥 < (𝑦 + 1))
15 cz 8245 . . . 4 class
1614, 4, 15crio 5467 . . 3 class (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1)))
172, 3, 16cmpt 3818 . 2 class (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1))))
181, 17wceq 1243 1 wff ⌊ = (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1))))
 Colors of variables: wff set class This definition is referenced by:  flval  9116
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