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) ) → ( 𝐹 ∘f − 𝐺 ) ∈ 𝑂(1) )

Proof

Step	Hyp	Ref	Expression
1		readdcl	⊢ ( ( 𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝑥 + 𝑦 ) ∈ ℝ )
2		subcl	⊢ ( ( 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	subcld	⊢ ( ( ( 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	abs2dif2d	⊢ ( ( ( 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) )