Definition df-o1 15452

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 15448	. 2 class 𝑂(1)
2		vy	. . . . . . . . . 10 setvar 𝑦
3	2	cv 1541	. . . . . . . . 9 class 𝑦
4		vf	. . . . . . . . . 10 setvar 𝑓
5	4	cv 1541	. . . . . . . . 9 class 𝑓
6	3, 5	cfv 6498	. . . . . . . 8 class (𝑓‘𝑦)
7		cabs 15196	. . . . . . . 8 class abs
8	6, 7	cfv 6498	. . . . . . 7 class (abs‘(𝑓‘𝑦))
9		vm	. . . . . . . 8 setvar 𝑚
10	9	cv 1541	. . . . . . 7 class 𝑚
11		cle 11180	. . . . . . 7 class ≤
12	8, 10, 11	wbr 5085	. . . . . 6 wff (abs‘(𝑓‘𝑦)) ≤ 𝑚
13	5	cdm 5631	. . . . . . 7 class dom 𝑓
14		vx	. . . . . . . . 9 setvar 𝑥
15	14	cv 1541	. . . . . . . 8 class 𝑥
16		cpnf 11176	. . . . . . . 8 class +∞
17		cico 13300	. . . . . . . 8 class [,)
18	15, 16, 17	co 7367	. . . . . . 7 class (𝑥[,)+∞)
19	13, 18	cin 3888	. . . . . 6 class (dom 𝑓 ∩ (𝑥[,)+∞))
20	12, 2, 19	wral 3051	. . . . 5 wff ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚
21		cr 11037	. . . . 5 class ℝ
22	20, 9, 21	wrex 3061	. . . 4 wff ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚
23	22, 14, 21	wrex 3061	. . 3 wff ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚
24		cc 11036	. . . 4 class ℂ
25		cpm 8774	. . . 4 class ↑pm
26	24, 21, 25	co 7367	. . 3 class (ℂ ↑pm ℝ)
27	23, 4, 26	crab 3389	. 2 class {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚}
28	1, 27	wceq 1542	1 wff 𝑂(1) = {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚}

This definition is referenced by:  elo1  15488