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Definition df-o1 15199
Description: Define the set of eventually bounded functions. We don't bother to build the full conception of big-O notation, because we can represent any big-O in terms of 𝑂(1) and division, and any little-O in terms of a limit and division. We could also use limsup for this, but it only works on integer sequences, while this will work for real sequences or integer sequences. (Contributed by Mario Carneiro, 15-Sep-2014.)
Assertion
Ref Expression
df-o1 𝑂(1) = {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓𝑦)) ≤ 𝑚}
Distinct variable group:   𝑥,𝑦,𝑓,𝑚

Detailed syntax breakdown of Definition df-o1
StepHypRef Expression
1 co1 15195 . 2 class 𝑂(1)
2 vy . . . . . . . . . 10 setvar 𝑦
32cv 1538 . . . . . . . . 9 class 𝑦
4 vf . . . . . . . . . 10 setvar 𝑓
54cv 1538 . . . . . . . . 9 class 𝑓
63, 5cfv 6433 . . . . . . . 8 class (𝑓𝑦)
7 cabs 14945 . . . . . . . 8 class abs
86, 7cfv 6433 . . . . . . 7 class (abs‘(𝑓𝑦))
9 vm . . . . . . . 8 setvar 𝑚
109cv 1538 . . . . . . 7 class 𝑚
11 cle 11010 . . . . . . 7 class
128, 10, 11wbr 5074 . . . . . 6 wff (abs‘(𝑓𝑦)) ≤ 𝑚
135cdm 5589 . . . . . . 7 class dom 𝑓
14 vx . . . . . . . . 9 setvar 𝑥
1514cv 1538 . . . . . . . 8 class 𝑥
16 cpnf 11006 . . . . . . . 8 class +∞
17 cico 13081 . . . . . . . 8 class [,)
1815, 16, 17co 7275 . . . . . . 7 class (𝑥[,)+∞)
1913, 18cin 3886 . . . . . 6 class (dom 𝑓 ∩ (𝑥[,)+∞))
2012, 2, 19wral 3064 . . . . 5 wff 𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓𝑦)) ≤ 𝑚
21 cr 10870 . . . . 5 class
2220, 9, 21wrex 3065 . . . 4 wff 𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓𝑦)) ≤ 𝑚
2322, 14, 21wrex 3065 . . 3 wff 𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓𝑦)) ≤ 𝑚
24 cc 10869 . . . 4 class
25 cpm 8616 . . . 4 class pm
2624, 21, 25co 7275 . . 3 class (ℂ ↑pm ℝ)
2723, 4, 26crab 3068 . 2 class {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓𝑦)) ≤ 𝑚}
281, 27wceq 1539 1 wff 𝑂(1) = {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓𝑦)) ≤ 𝑚}
Colors of variables: wff setvar class
This definition is referenced by:  elo1  15235
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