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Definition df-o1 14420
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 14416 . 2 class 𝑂(1)
2 vy . . . . . . . . . 10 setvar 𝑦
32cv 1631 . . . . . . . . 9 class 𝑦
4 vf . . . . . . . . . 10 setvar 𝑓
54cv 1631 . . . . . . . . 9 class 𝑓
63, 5cfv 6049 . . . . . . . 8 class (𝑓𝑦)
7 cabs 14173 . . . . . . . 8 class abs
86, 7cfv 6049 . . . . . . 7 class (abs‘(𝑓𝑦))
9 vm . . . . . . . 8 setvar 𝑚
109cv 1631 . . . . . . 7 class 𝑚
11 cle 10267 . . . . . . 7 class
128, 10, 11wbr 4804 . . . . . 6 wff (abs‘(𝑓𝑦)) ≤ 𝑚
135cdm 5266 . . . . . . 7 class dom 𝑓
14 vx . . . . . . . . 9 setvar 𝑥
1514cv 1631 . . . . . . . 8 class 𝑥
16 cpnf 10263 . . . . . . . 8 class +∞
17 cico 12370 . . . . . . . 8 class [,)
1815, 16, 17co 6813 . . . . . . 7 class (𝑥[,)+∞)
1913, 18cin 3714 . . . . . 6 class (dom 𝑓 ∩ (𝑥[,)+∞))
2012, 2, 19wral 3050 . . . . 5 wff 𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓𝑦)) ≤ 𝑚
21 cr 10127 . . . . 5 class
2220, 9, 21wrex 3051 . . . 4 wff 𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓𝑦)) ≤ 𝑚
2322, 14, 21wrex 3051 . . 3 wff 𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓𝑦)) ≤ 𝑚
24 cc 10126 . . . 4 class
25 cpm 8024 . . . 4 class pm
2624, 21, 25co 6813 . . 3 class (ℂ ↑pm ℝ)
2723, 4, 26crab 3054 . 2 class {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓𝑦)) ≤ 𝑚}
281, 27wceq 1632 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  14456
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