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Definition df-o1 14069
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 14065 . 2 class 𝑂(1)
2 vy . . . . . . . . . 10 setvar 𝑦
32cv 1474 . . . . . . . . 9 class 𝑦
4 vf . . . . . . . . . 10 setvar 𝑓
54cv 1474 . . . . . . . . 9 class 𝑓
63, 5cfv 5804 . . . . . . . 8 class (𝑓𝑦)
7 cabs 13822 . . . . . . . 8 class abs
86, 7cfv 5804 . . . . . . 7 class (abs‘(𝑓𝑦))
9 vm . . . . . . . 8 setvar 𝑚
109cv 1474 . . . . . . 7 class 𝑚
11 cle 9954 . . . . . . 7 class
128, 10, 11wbr 4583 . . . . . 6 wff (abs‘(𝑓𝑦)) ≤ 𝑚
135cdm 5038 . . . . . . 7 class dom 𝑓
14 vx . . . . . . . . 9 setvar 𝑥
1514cv 1474 . . . . . . . 8 class 𝑥
16 cpnf 9950 . . . . . . . 8 class +∞
17 cico 12048 . . . . . . . 8 class [,)
1815, 16, 17co 6549 . . . . . . 7 class (𝑥[,)+∞)
1913, 18cin 3539 . . . . . 6 class (dom 𝑓 ∩ (𝑥[,)+∞))
2012, 2, 19wral 2896 . . . . 5 wff 𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓𝑦)) ≤ 𝑚
21 cr 9814 . . . . 5 class
2220, 9, 21wrex 2897 . . . 4 wff 𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓𝑦)) ≤ 𝑚
2322, 14, 21wrex 2897 . . 3 wff 𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓𝑦)) ≤ 𝑚
24 cc 9813 . . . 4 class
25 cpm 7745 . . . 4 class pm
2624, 21, 25co 6549 . . 3 class (ℂ ↑pm ℝ)
2723, 4, 26crab 2900 . 2 class {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓𝑦)) ≤ 𝑚}
281, 27wceq 1475 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  14105
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