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Definition df-fl 9826
 Description: Define the floor (greatest integer less than or equal to) function. See flval 9828 for its value, flqlelt 9832 for its basic property, and flqcl 9829 for its closure. For example, (⌊‘(3 / 2)) = 1 while (⌊‘-(3 / 2)) = -2 (ex-fl 12376). 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 for all real numbers. 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 9824 . 2 class
2 vx . . 3 setvar 𝑥
3 cr 7446 . . 3 class
4 vy . . . . . . 7 setvar 𝑦
54cv 1295 . . . . . 6 class 𝑦
62cv 1295 . . . . . 6 class 𝑥
7 cle 7620 . . . . . 6 class
85, 6, 7wbr 3867 . . . . 5 wff 𝑦𝑥
9 c1 7448 . . . . . . 7 class 1
10 caddc 7450 . . . . . . 7 class +
115, 9, 10co 5690 . . . . . 6 class (𝑦 + 1)
12 clt 7619 . . . . . 6 class <
136, 11, 12wbr 3867 . . . . 5 wff 𝑥 < (𝑦 + 1)
148, 13wa 103 . . . 4 wff (𝑦𝑥𝑥 < (𝑦 + 1))
15 cz 8848 . . . 4 class
1614, 4, 15crio 5645 . . 3 class (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1)))
172, 3, 16cmpt 3921 . 2 class (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1))))
181, 17wceq 1296 1 wff ⌊ = (𝑥 ∈ ℝ ↦ (𝑦 ∈ ℤ (𝑦𝑥𝑥 < (𝑦 + 1))))
 Colors of variables: wff set class This definition is referenced by:  flval  9828
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