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Metamath Proof Explorer

Theorem o1add

Description: The sum 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 o1add ( ( 𝐹 ∈ 𝑂(1) ∧ 𝐺 ∈ 𝑂(1) ) → ( 𝐹 ∘f + 𝐺 ) ∈ 𝑂(1) )

Proof

Step Hyp Ref Expression
1 readdcl ⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝑥 + 𝑦 ) ∈ ℝ )
2 addcl ⊢ ( ( 𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ ) → ( 𝑚 + 𝑛 ) ∈ 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 addcld ⊢ ( ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ( 𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ ) ∧ ( ( abs ‘ 𝑚 ) ≤ 𝑥 ∧ ( abs ‘ 𝑛 ) ≤ 𝑦 ) ) → ( 𝑚 + 𝑛 ) ∈ ℂ )
6 5 abscld ⊢ ( ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ( 𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ ) ∧ ( ( abs ‘ 𝑚 ) ≤ 𝑥 ∧ ( abs ‘ 𝑛 ) ≤ 𝑦 ) ) → ( abs ‘ ( 𝑚 + 𝑛 ) ) ∈ 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 abstrid ⊢ ( ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ( 𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ ) ∧ ( ( abs ‘ 𝑚 ) ≤ 𝑥 ∧ ( abs ‘ 𝑛 ) ≤ 𝑦 ) ) → ( abs ‘ ( 𝑚 + 𝑛 ) ) ≤ ( ( 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 ‘ ( 𝑚 + 𝑛 ) ) ≤ ( 𝑥 + 𝑦 ) )
18 17 3expia ⊢ ( ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) ∧ ( 𝑚 ∈ ℂ ∧ 𝑛 ∈ ℂ ) ) → ( ( ( abs ‘ 𝑚 ) ≤ 𝑥 ∧ ( abs ‘ 𝑛 ) ≤ 𝑦 ) → ( abs ‘ ( 𝑚 + 𝑛 ) ) ≤ ( 𝑥 + 𝑦 ) ) )
19 1 2 18 o1of2 ⊢ ( ( 𝐹 ∈ 𝑂(1) ∧ 𝐺 ∈ 𝑂(1) ) → ( 𝐹 ∘f + 𝐺 ) ∈ 𝑂(1) )