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Definition df-o1 14598
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 14594 . 2 class 𝑂(1)
2 vy . . . . . . . . . 10 setvar 𝑦
32cv 1657 . . . . . . . . 9 class 𝑦
4 vf . . . . . . . . . 10 setvar 𝑓
54cv 1657 . . . . . . . . 9 class 𝑓
63, 5cfv 6123 . . . . . . . 8 class (𝑓𝑦)
7 cabs 14351 . . . . . . . 8 class abs
86, 7cfv 6123 . . . . . . 7 class (abs‘(𝑓𝑦))
9 vm . . . . . . . 8 setvar 𝑚
109cv 1657 . . . . . . 7 class 𝑚
11 cle 10392 . . . . . . 7 class
128, 10, 11wbr 4873 . . . . . 6 wff (abs‘(𝑓𝑦)) ≤ 𝑚
135cdm 5342 . . . . . . 7 class dom 𝑓
14 vx . . . . . . . . 9 setvar 𝑥
1514cv 1657 . . . . . . . 8 class 𝑥
16 cpnf 10388 . . . . . . . 8 class +∞
17 cico 12465 . . . . . . . 8 class [,)
1815, 16, 17co 6905 . . . . . . 7 class (𝑥[,)+∞)
1913, 18cin 3797 . . . . . 6 class (dom 𝑓 ∩ (𝑥[,)+∞))
2012, 2, 19wral 3117 . . . . 5 wff 𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓𝑦)) ≤ 𝑚
21 cr 10251 . . . . 5 class
2220, 9, 21wrex 3118 . . . 4 wff 𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓𝑦)) ≤ 𝑚
2322, 14, 21wrex 3118 . . 3 wff 𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓𝑦)) ≤ 𝑚
24 cc 10250 . . . 4 class
25 cpm 8123 . . . 4 class pm
2624, 21, 25co 6905 . . 3 class (ℂ ↑pm ℝ)
2723, 4, 26crab 3121 . 2 class {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓𝑦)) ≤ 𝑚}
281, 27wceq 1658 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  14634
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