Metamath Proof Explorer


Theorem o1sub

Description: The difference of two eventually bounded functions is eventually bounded. (Contributed by Mario Carneiro, 15-Sep-2014) (Proof shortened by Fan Zheng, 14-Jul-2016)

Ref Expression
Assertion o1sub ( ( 𝐹 ∈ 𝑂(1) ∧ 𝐺 ∈ 𝑂(1) ) → ( 𝐹f𝐺 ) ∈ 𝑂(1) )

Proof

Step Hyp Ref Expression
1 readdcl ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝑥 + 𝑦 ) ∈ ℝ )
2 subcl ( ( 𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ ) → ( 𝑚𝑛 ) ∈ ℂ )
3 simp2l ( ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ( 𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ ) ∧ ( ( abs ‘ 𝑚 ) ≤ 𝑥 ∧ ( abs ‘ 𝑛 ) ≤ 𝑦 ) ) → 𝑚 ∈ ℂ )
4 simp2r ( ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ( 𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ ) ∧ ( ( abs ‘ 𝑚 ) ≤ 𝑥 ∧ ( abs ‘ 𝑛 ) ≤ 𝑦 ) ) → 𝑛 ∈ ℂ )
5 3 4 subcld ( ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ( 𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ ) ∧ ( ( abs ‘ 𝑚 ) ≤ 𝑥 ∧ ( abs ‘ 𝑛 ) ≤ 𝑦 ) ) → ( 𝑚𝑛 ) ∈ ℂ )
6 5 abscld ( ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ( 𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ ) ∧ ( ( abs ‘ 𝑚 ) ≤ 𝑥 ∧ ( abs ‘ 𝑛 ) ≤ 𝑦 ) ) → ( abs ‘ ( 𝑚𝑛 ) ) ∈ ℝ )
7 3 abscld ( ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ( 𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ ) ∧ ( ( abs ‘ 𝑚 ) ≤ 𝑥 ∧ ( abs ‘ 𝑛 ) ≤ 𝑦 ) ) → ( abs ‘ 𝑚 ) ∈ ℝ )
8 4 abscld ( ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ( 𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ ) ∧ ( ( abs ‘ 𝑚 ) ≤ 𝑥 ∧ ( abs ‘ 𝑛 ) ≤ 𝑦 ) ) → ( abs ‘ 𝑛 ) ∈ ℝ )
9 7 8 readdcld ( ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ( 𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ ) ∧ ( ( abs ‘ 𝑚 ) ≤ 𝑥 ∧ ( abs ‘ 𝑛 ) ≤ 𝑦 ) ) → ( ( abs ‘ 𝑚 ) + ( abs ‘ 𝑛 ) ) ∈ ℝ )
10 simp1l ( ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ( 𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ ) ∧ ( ( abs ‘ 𝑚 ) ≤ 𝑥 ∧ ( abs ‘ 𝑛 ) ≤ 𝑦 ) ) → 𝑥 ∈ ℝ )
11 simp1r ( ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ( 𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ ) ∧ ( ( abs ‘ 𝑚 ) ≤ 𝑥 ∧ ( abs ‘ 𝑛 ) ≤ 𝑦 ) ) → 𝑦 ∈ ℝ )
12 10 11 readdcld ( ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ( 𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ ) ∧ ( ( abs ‘ 𝑚 ) ≤ 𝑥 ∧ ( abs ‘ 𝑛 ) ≤ 𝑦 ) ) → ( 𝑥 + 𝑦 ) ∈ ℝ )
13 3 4 abs2dif2d ( ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ( 𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ ) ∧ ( ( abs ‘ 𝑚 ) ≤ 𝑥 ∧ ( abs ‘ 𝑛 ) ≤ 𝑦 ) ) → ( abs ‘ ( 𝑚𝑛 ) ) ≤ ( ( abs ‘ 𝑚 ) + ( abs ‘ 𝑛 ) ) )
14 simp3l ( ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ( 𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ ) ∧ ( ( abs ‘ 𝑚 ) ≤ 𝑥 ∧ ( abs ‘ 𝑛 ) ≤ 𝑦 ) ) → ( abs ‘ 𝑚 ) ≤ 𝑥 )
15 simp3r ( ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ( 𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ ) ∧ ( ( abs ‘ 𝑚 ) ≤ 𝑥 ∧ ( abs ‘ 𝑛 ) ≤ 𝑦 ) ) → ( abs ‘ 𝑛 ) ≤ 𝑦 )
16 7 8 10 11 14 15 le2addd ( ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ( 𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ ) ∧ ( ( abs ‘ 𝑚 ) ≤ 𝑥 ∧ ( abs ‘ 𝑛 ) ≤ 𝑦 ) ) → ( ( abs ‘ 𝑚 ) + ( abs ‘ 𝑛 ) ) ≤ ( 𝑥 + 𝑦 ) )
17 6 9 12 13 16 letrd ( ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ( 𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ ) ∧ ( ( abs ‘ 𝑚 ) ≤ 𝑥 ∧ ( abs ‘ 𝑛 ) ≤ 𝑦 ) ) → ( abs ‘ ( 𝑚𝑛 ) ) ≤ ( 𝑥 + 𝑦 ) )
18 17 3expia ( ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ( 𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ ) ) → ( ( ( abs ‘ 𝑚 ) ≤ 𝑥 ∧ ( abs ‘ 𝑛 ) ≤ 𝑦 ) → ( abs ‘ ( 𝑚𝑛 ) ) ≤ ( 𝑥 + 𝑦 ) ) )
19 1 2 18 o1of2 ( ( 𝐹 ∈ 𝑂(1) ∧ 𝐺 ∈ 𝑂(1) ) → ( 𝐹f𝐺 ) ∈ 𝑂(1) )