Definition df-o1 14562

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 14558	. 2 class 𝑂(1)
2		vy	. . . . . . . . . 10 setvar 𝑦
3	2	cv 1652	. . . . . . . . 9 class 𝑦
4		vf	. . . . . . . . . 10 setvar 𝑓
5	4	cv 1652	. . . . . . . . 9 class 𝑓
6	3, 5	cfv 6101	. . . . . . . 8 class (𝑓‘𝑦)
7		cabs 14315	. . . . . . . 8 class abs
8	6, 7	cfv 6101	. . . . . . 7 class (abs‘(𝑓‘𝑦))
9		vm	. . . . . . . 8 setvar 𝑚
10	9	cv 1652	. . . . . . 7 class 𝑚
11		cle 10364	. . . . . . 7 class ≤
12	8, 10, 11	wbr 4843	. . . . . 6 wff (abs‘(𝑓‘𝑦)) ≤ 𝑚
13	5	cdm 5312	. . . . . . 7 class dom 𝑓
14		vx	. . . . . . . . 9 setvar 𝑥
15	14	cv 1652	. . . . . . . 8 class 𝑥
16		cpnf 10360	. . . . . . . 8 class +∞
17		cico 12426	. . . . . . . 8 class [,)
18	15, 16, 17	co 6878	. . . . . . 7 class (𝑥[,)+∞)
19	13, 18	cin 3768	. . . . . 6 class (dom 𝑓 ∩ (𝑥[,)+∞))
20	12, 2, 19	wral 3089	. . . . 5 wff ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚
21		cr 10223	. . . . 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 10222	. . . 4 class ℂ
25		cpm 8096	. . . 4 class ↑pm
26	24, 21, 25	co 6878	. . 3 class (ℂ ↑pm ℝ)
27	23, 4, 26	crab 3093	. 2 class {𝑓 ∈ (ℂ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(abs‘(𝑓‘𝑦)) ≤ 𝑚}
28	1, 27	wceq 1653	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  14598