Metamath Proof Explorer


Definition df-lo1

Description: Define the set of eventually upper bounded real functions. This fills a gap in O(1) coverage, to express statements like f ( x ) <_ g ( x ) + O ( x ) via ( x e. RR+ |-> ( f ( x ) - g ( x ) ) / x ) e. <_O(1) . (Contributed by Mario Carneiro, 25-May-2016)

Ref Expression
Assertion df-lo1 ≤𝑂(1) = { 𝑓 ∈ ( ℝ ↑pm ℝ ) ∣ ∃ 𝑥 ∈ ℝ ∃ 𝑚 ∈ ℝ ∀ 𝑦 ∈ ( dom 𝑓 ∩ ( 𝑥 [,) +∞ ) ) ( 𝑓𝑦 ) ≤ 𝑚 }

Detailed syntax breakdown

Step Hyp Ref Expression
0 clo1 ≤𝑂(1)
1 vf 𝑓
2 cr
3 cpm pm
4 2 2 3 co ( ℝ ↑pm ℝ )
5 vx 𝑥
6 vm 𝑚
7 vy 𝑦
8 1 cv 𝑓
9 8 cdm dom 𝑓
10 5 cv 𝑥
11 cico [,)
12 cpnf +∞
13 10 12 11 co ( 𝑥 [,) +∞ )
14 9 13 cin ( dom 𝑓 ∩ ( 𝑥 [,) +∞ ) )
15 7 cv 𝑦
16 15 8 cfv ( 𝑓𝑦 )
17 cle
18 6 cv 𝑚
19 16 18 17 wbr ( 𝑓𝑦 ) ≤ 𝑚
20 19 7 14 wral 𝑦 ∈ ( dom 𝑓 ∩ ( 𝑥 [,) +∞ ) ) ( 𝑓𝑦 ) ≤ 𝑚
21 20 6 2 wrex 𝑚 ∈ ℝ ∀ 𝑦 ∈ ( dom 𝑓 ∩ ( 𝑥 [,) +∞ ) ) ( 𝑓𝑦 ) ≤ 𝑚
22 21 5 2 wrex 𝑥 ∈ ℝ ∃ 𝑚 ∈ ℝ ∀ 𝑦 ∈ ( dom 𝑓 ∩ ( 𝑥 [,) +∞ ) ) ( 𝑓𝑦 ) ≤ 𝑚
23 22 1 4 crab { 𝑓 ∈ ( ℝ ↑pm ℝ ) ∣ ∃ 𝑥 ∈ ℝ ∃ 𝑚 ∈ ℝ ∀ 𝑦 ∈ ( dom 𝑓 ∩ ( 𝑥 [,) +∞ ) ) ( 𝑓𝑦 ) ≤ 𝑚 }
24 0 23 wceq ≤𝑂(1) = { 𝑓 ∈ ( ℝ ↑pm ℝ ) ∣ ∃ 𝑥 ∈ ℝ ∃ 𝑚 ∈ ℝ ∀ 𝑦 ∈ ( dom 𝑓 ∩ ( 𝑥 [,) +∞ ) ) ( 𝑓𝑦 ) ≤ 𝑚 }