Theorem o1mul

Description: The product 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	o1mul	⊢ ( ( 𝐹 ∈ 𝑂(1) ∧ 𝐺 ∈ 𝑂(1) ) → ( 𝐹 ∘f · 𝐺 ) ∈ 𝑂(1) )

Proof

Step	Hyp	Ref	Expression
1		remulcl	⊢ ( ( 𝑚 ∈ ℝ ∧ 𝑛 ∈ ℝ ) → ( 𝑚 · 𝑛 ) ∈ ℝ )
2		mulcl	⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) → ( 𝑥 · 𝑦 ) ∈ ℂ )
3		simp2l	⊢ ( ( ( 𝑚 ∈ ℝ ∧ 𝑛 ∈ ℝ ) ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ∧ ( ( abs ‘ 𝑥 ) ≤ 𝑚 ∧ ( abs ‘ 𝑦 ) ≤ 𝑛 ) ) → 𝑥 ∈ ℂ )
4		simp2r	⊢ ( ( ( 𝑚 ∈ ℝ ∧ 𝑛 ∈ ℝ ) ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ∧ ( ( abs ‘ 𝑥 ) ≤ 𝑚 ∧ ( abs ‘ 𝑦 ) ≤ 𝑛 ) ) → 𝑦 ∈ ℂ )
5	3 4	absmuld	⊢ ( ( ( 𝑚 ∈ ℝ ∧ 𝑛 ∈ ℝ ) ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ∧ ( ( abs ‘ 𝑥 ) ≤ 𝑚 ∧ ( abs ‘ 𝑦 ) ≤ 𝑛 ) ) → ( abs ‘ ( 𝑥 · 𝑦 ) ) = ( ( abs ‘ 𝑥 ) · ( abs ‘ 𝑦 ) ) )
6	3	abscld	⊢ ( ( ( 𝑚 ∈ ℝ ∧ 𝑛 ∈ ℝ ) ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ∧ ( ( abs ‘ 𝑥 ) ≤ 𝑚 ∧ ( abs ‘ 𝑦 ) ≤ 𝑛 ) ) → ( abs ‘ 𝑥 ) ∈ ℝ )
7		simp1l	⊢ ( ( ( 𝑚 ∈ ℝ ∧ 𝑛 ∈ ℝ ) ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ∧ ( ( abs ‘ 𝑥 ) ≤ 𝑚 ∧ ( abs ‘ 𝑦 ) ≤ 𝑛 ) ) → 𝑚 ∈ ℝ )
8	4	abscld	⊢ ( ( ( 𝑚 ∈ ℝ ∧ 𝑛 ∈ ℝ ) ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ∧ ( ( abs ‘ 𝑥 ) ≤ 𝑚 ∧ ( abs ‘ 𝑦 ) ≤ 𝑛 ) ) → ( abs ‘ 𝑦 ) ∈ ℝ )
9		simp1r	⊢ ( ( ( 𝑚 ∈ ℝ ∧ 𝑛 ∈ ℝ ) ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ∧ ( ( abs ‘ 𝑥 ) ≤ 𝑚 ∧ ( abs ‘ 𝑦 ) ≤ 𝑛 ) ) → 𝑛 ∈ ℝ )
10	3	absge0d	⊢ ( ( ( 𝑚 ∈ ℝ ∧ 𝑛 ∈ ℝ ) ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ∧ ( ( abs ‘ 𝑥 ) ≤ 𝑚 ∧ ( abs ‘ 𝑦 ) ≤ 𝑛 ) ) → 0 ≤ ( abs ‘ 𝑥 ) )
11	4	absge0d	⊢ ( ( ( 𝑚 ∈ ℝ ∧ 𝑛 ∈ ℝ ) ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ∧ ( ( abs ‘ 𝑥 ) ≤ 𝑚 ∧ ( abs ‘ 𝑦 ) ≤ 𝑛 ) ) → 0 ≤ ( abs ‘ 𝑦 ) )
12		simp3l	⊢ ( ( ( 𝑚 ∈ ℝ ∧ 𝑛 ∈ ℝ ) ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ∧ ( ( abs ‘ 𝑥 ) ≤ 𝑚 ∧ ( abs ‘ 𝑦 ) ≤ 𝑛 ) ) → ( abs ‘ 𝑥 ) ≤ 𝑚 )
13		simp3r	⊢ ( ( ( 𝑚 ∈ ℝ ∧ 𝑛 ∈ ℝ ) ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ∧ ( ( abs ‘ 𝑥 ) ≤ 𝑚 ∧ ( abs ‘ 𝑦 ) ≤ 𝑛 ) ) → ( abs ‘ 𝑦 ) ≤ 𝑛 )
14	6 7 8 9 10 11 12 13	lemul12ad	⊢ ( ( ( 𝑚 ∈ ℝ ∧ 𝑛 ∈ ℝ ) ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ∧ ( ( abs ‘ 𝑥 ) ≤ 𝑚 ∧ ( abs ‘ 𝑦 ) ≤ 𝑛 ) ) → ( ( abs ‘ 𝑥 ) · ( abs ‘ 𝑦 ) ) ≤ ( 𝑚 · 𝑛 ) )
15	5 14	eqbrtrd	⊢ ( ( ( 𝑚 ∈ ℝ ∧ 𝑛 ∈ ℝ ) ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ∧ ( ( abs ‘ 𝑥 ) ≤ 𝑚 ∧ ( abs ‘ 𝑦 ) ≤ 𝑛 ) ) → ( abs ‘ ( 𝑥 · 𝑦 ) ) ≤ ( 𝑚 · 𝑛 ) )
16	15	3expia	⊢ ( ( ( 𝑚 ∈ ℝ ∧ 𝑛 ∈ ℝ ) ∧ ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ) ) → ( ( ( abs ‘ 𝑥 ) ≤ 𝑚 ∧ ( abs ‘ 𝑦 ) ≤ 𝑛 ) → ( abs ‘ ( 𝑥 · 𝑦 ) ) ≤ ( 𝑚 · 𝑛 ) ) )
17	1 2 16	o1of2	⊢ ( ( 𝐹 ∈ 𝑂(1) ∧ 𝐺 ∈ 𝑂(1) ) → ( 𝐹 ∘f · 𝐺 ) ∈ 𝑂(1) )