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Definition df-o1 14562
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 14558 . 2 class 𝑂(1)
2 vy . . . . . . . . . 10 setvar 𝑦
32cv 1652 . . . . . . . . 9 class 𝑦
4 vf . . . . . . . . . 10 setvar 𝑓
54cv 1652 . . . . . . . . 9 class 𝑓
63, 5cfv 6101 . . . . . . . 8 class (𝑓𝑦)
7 cabs 14315 . . . . . . . 8 class abs
86, 7cfv 6101 . . . . . . 7 class (abs‘(𝑓𝑦))
9 vm . . . . . . . 8 setvar 𝑚
109cv 1652 . . . . . . 7 class 𝑚
11 cle 10364 . . . . . . 7 class
128, 10, 11wbr 4843 . . . . . 6 wff (abs‘(𝑓𝑦)) ≤ 𝑚
135cdm 5312 . . . . . . 7 class dom 𝑓
14 vx . . . . . . . . 9 setvar 𝑥
1514cv 1652 . . . . . . . 8 class 𝑥
16 cpnf 10360 . . . . . . . 8 class +∞
17 cico 12426 . . . . . . . 8 class [,)
1815, 16, 17co 6878 . . . . . . 7 class (𝑥[,)+∞)
1913, 18cin 3768 . . . . . 6 class (dom 𝑓 ∩ (𝑥[,)+∞))
2012, 2, 19wral 3089 . . . . 5 wff 𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓𝑦)) ≤ 𝑚
21 cr 10223 . . . . 5 class
2220, 9, 21wrex 3090 . . . 4 wff 𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓𝑦)) ≤ 𝑚
2322, 14, 21wrex 3090 . . 3 wff 𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓𝑦)) ≤ 𝑚
24 cc 10222 . . . 4 class
25 cpm 8096 . . . 4 class pm
2624, 21, 25co 6878 . . 3 class (ℂ ↑pm ℝ)
2723, 4, 26crab 3093 . 2 class {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓𝑦)) ≤ 𝑚}
281, 27wceq 1653 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  14598
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