Definition df-o1 15686

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‘(𝑓‘𝑦)) ≤ 𝑚}

Distinct variable group:   𝑥,𝑦,𝑓,𝑚

Detailed syntax breakdown of Definition df-o1

Step	Hyp	Ref	Expression
1		co1 15682	. 2 class 𝑂(1)
2		vy	. . . . . . . . . 10 setvar 𝑦
3	2	cv 1569	. . . . . . . . 9 class 𝑦
4		vf	. . . . . . . . . 10 setvar 𝑓
5	4	cv 1569	. . . . . . . . 9 class 𝑓
6	3, 5	cfv 6558	. . . . . . . 8 class (𝑓‘𝑦)
7		cabs 15430	. . . . . . . 8 class abs
8	6, 7	cfv 6558	. . . . . . 7 class (abs‘(𝑓‘𝑦))
9		vm	. . . . . . . 8 setvar 𝑚
10	9	cv 1569	. . . . . . 7 class 𝑚
11		cle 11371	. . . . . . 7 class ≤
12	8, 10, 11	wbr 5106	. . . . . 6 wff (abs‘(𝑓‘𝑦)) ≤ 𝑚
13	5	cdm 5655	. . . . . . 7 class dom 𝑓
14		vx	. . . . . . . . 9 setvar 𝑥
15	14	cv 1569	. . . . . . . 8 class 𝑥
16		cpnf 11367	. . . . . . . 8 class +∞
17		cico 13505	. . . . . . . 8 class [,)
18	15, 16, 17	co 7443	. . . . . . 7 class (𝑥[,)+∞)
19	13, 18	cin 4092	. . . . . 6 class (dom 𝑓 ∩ (𝑥[,)+∞))
20	12, 2, 19	wral 3080	. . . . 5 wff ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚
21		cr 11226	. . . . 5 class ℝ
22	20, 9, 21	wrex 3090	. . . 4 wff ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚
23	22, 14, 21	wrex 3090	. . 3 wff ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚
24		cc 11225	. . . 4 class ℂ
25		cpm 8876	. . . 4 class ↑pm
26	24, 21, 25	co 7443	. . 3 class (ℂ ↑pm ℝ)
27	23, 4, 26	crab 3416	. 2 class {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚}
28	1, 27	wceq 1570	1 wff 𝑂(1) = {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚}

This definition is used by:  elo1  15722