Theorem lo1mul2

Description: The product of an eventually upper bounded function and a positive eventually upper bounded function is eventually upper bounded. (Contributed by Mario Carneiro, 26-May-2016)

	Ref	Expression
Hypotheses	o1add2.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 )
	o1add2.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐶 ∈ 𝑉 )
	lo1add.3	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ ≤𝑂(1) )
	lo1add.4	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ∈ ≤𝑂(1) )
	lo1mul.5	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 0 ≤ 𝐵 )
Assertion	lo1mul2	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( 𝐶 · 𝐵 ) ) ∈ ≤𝑂(1) )

Proof

Step	Hyp	Ref	Expression
1		o1add2.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 )
2		o1add2.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐶 ∈ 𝑉 )
3		lo1add.3	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ ≤𝑂(1) )
4		lo1add.4	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ∈ ≤𝑂(1) )
5		lo1mul.5	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 0 ≤ 𝐵 )
6	2 4	lo1mptrcl	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐶 ∈ ℝ )
7	6	recnd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐶 ∈ ℂ )
8	1 3	lo1mptrcl	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ )
9	8	recnd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
10	7 9	mulcomd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐶 · 𝐵 ) = ( 𝐵 · 𝐶 ) )
11	10	mpteq2dva	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( 𝐶 · 𝐵 ) ) = ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 · 𝐶 ) ) )
12	1 2 3 4 5	lo1mul	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 · 𝐶 ) ) ∈ ≤𝑂(1) )
13	11 12	eqeltrd	⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ ( 𝐶 · 𝐵 ) ) ∈ ≤𝑂(1) )