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Definition df-bigo 44599
Description: Define the function "big-O", mapping a real function g to the set of real functions "of order g(x)". Definition in section 1.1 of [AhoHopUll] p. 2. This is a generalization of "big-O of one", see df-o1 14839 and df-lo1 14840. As explained in the comment of df-o1 , any big-O can be represented in terms of 𝑂(1) and division, see elbigolo1 44608. (Contributed by AV, 15-May-2020.)
Assertion
Ref Expression
df-bigo Ο = (𝑔 ∈ (ℝ ↑pm ℝ) ↦ {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ (𝑚 · (𝑔𝑦))})
Distinct variable group:   𝑓,𝑔,𝑥,𝑚,𝑦

Detailed syntax breakdown of Definition df-bigo
StepHypRef Expression
1 cbigo 44598 . 2 class Ο
2 vg . . 3 setvar 𝑔
3 cr 10528 . . . 4 class
4 cpm 8399 . . . 4 class pm
53, 3, 4co 7148 . . 3 class (ℝ ↑pm ℝ)
6 vy . . . . . . . . . 10 setvar 𝑦
76cv 1530 . . . . . . . . 9 class 𝑦
8 vf . . . . . . . . . 10 setvar 𝑓
98cv 1530 . . . . . . . . 9 class 𝑓
107, 9cfv 6348 . . . . . . . 8 class (𝑓𝑦)
11 vm . . . . . . . . . 10 setvar 𝑚
1211cv 1530 . . . . . . . . 9 class 𝑚
132cv 1530 . . . . . . . . . 10 class 𝑔
147, 13cfv 6348 . . . . . . . . 9 class (𝑔𝑦)
15 cmul 10534 . . . . . . . . 9 class ·
1612, 14, 15co 7148 . . . . . . . 8 class (𝑚 · (𝑔𝑦))
17 cle 10668 . . . . . . . 8 class
1810, 16, 17wbr 5057 . . . . . . 7 wff (𝑓𝑦) ≤ (𝑚 · (𝑔𝑦))
199cdm 5548 . . . . . . . 8 class dom 𝑓
20 vx . . . . . . . . . 10 setvar 𝑥
2120cv 1530 . . . . . . . . 9 class 𝑥
22 cpnf 10664 . . . . . . . . 9 class +∞
23 cico 12732 . . . . . . . . 9 class [,)
2421, 22, 23co 7148 . . . . . . . 8 class (𝑥[,)+∞)
2519, 24cin 3933 . . . . . . 7 class (dom 𝑓 ∩ (𝑥[,)+∞))
2618, 6, 25wral 3136 . . . . . 6 wff 𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ (𝑚 · (𝑔𝑦))
2726, 11, 3wrex 3137 . . . . 5 wff 𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ (𝑚 · (𝑔𝑦))
2827, 20, 3wrex 3137 . . . 4 wff 𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ (𝑚 · (𝑔𝑦))
2928, 8, 5crab 3140 . . 3 class {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ (𝑚 · (𝑔𝑦))}
302, 5, 29cmpt 5137 . 2 class (𝑔 ∈ (ℝ ↑pm ℝ) ↦ {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ (𝑚 · (𝑔𝑦))})
311, 30wceq 1531 1 wff Ο = (𝑔 ∈ (ℝ ↑pm ℝ) ↦ {𝑓 ∈ (ℝ ↑pm ℝ) ∣ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝑓 ∩ (𝑥[,)+∞))(𝑓𝑦) ≤ (𝑚 · (𝑔𝑦))})
Colors of variables: wff setvar class
This definition is referenced by:  bigoval  44600  elbigofrcl  44601
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