Theorem elo1mpt

Description: Elementhood in the set of eventually bounded functions. (Contributed by Mario Carneiro, 21-Sep-2014) (Proof shortened by Mario Carneiro, 26-May-2016)

	Ref	Expression
Hypotheses	elo1mpt.1	⊢ ( 𝜑 → 𝐴 ⊆ ℝ )
	elo1mpt.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
Assertion	elo1mpt	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝑂(1) ↔ ∃ 𝑦 ∈ ℝ ∃ 𝑚 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑦 ≤ 𝑥 → ( abs ‘ 𝐵 ) ≤ 𝑚 ) ) )

Proof

Step	Hyp	Ref	Expression
1		elo1mpt.1	⊢ ( 𝜑 → 𝐴 ⊆ ℝ )
2		elo1mpt.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
3	2	lo1o12	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝑂(1) ↔ ( 𝑥 ∈ 𝐴 ↦ ( abs ‘ 𝐵 ) ) ∈ ≤𝑂(1) ) )
4	2	abscld	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( abs ‘ 𝐵 ) ∈ ℝ )
5	1 4	ello1mpt	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ ( abs ‘ 𝐵 ) ) ∈ ≤𝑂(1) ↔ ∃ 𝑦 ∈ ℝ ∃ 𝑚 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑦 ≤ 𝑥 → ( abs ‘ 𝐵 ) ≤ 𝑚 ) ) )
6	3 5	bitrd	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝑂(1) ↔ ∃ 𝑦 ∈ ℝ ∃ 𝑚 ∈ ℝ ∀ 𝑥 ∈ 𝐴 ( 𝑦 ≤ 𝑥 → ( abs ‘ 𝐵 ) ≤ 𝑚 ) ) )