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Metamath Proof Explorer

Theorem lo1o12

Description: A function is eventually bounded iff its absolute value is eventually upper bounded. (This function is useful for converting theorems about <_O(1) to O(1) .) (Contributed by Mario Carneiro, 26-May-2016)

Ref Expression
Hypothesis lo1o12.1 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
Assertion lo1o12 ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝑂(1) ↔ ( 𝑥 ∈ 𝐴 ↦ ( abs ‘ 𝐵 ) ) ∈ ≤𝑂(1) ) )

Proof

Step Hyp Ref Expression
1 lo1o12.1 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
2 1 fmpttd ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ℂ )
3 lo1o1 ⊢ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) : 𝐴 ⟶ ℂ → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝑂(1) ↔ ( abs ∘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ∈ ≤𝑂(1) ) )
4 2 3 syl ⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝑂(1) ↔ ( abs ∘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ∈ ≤𝑂(1) ) )
5 absf ⊢ abs : ℂ ⟶ ℝ
6 5 a1i ⊢ ( 𝜑 → abs : ℂ ⟶ ℝ )
7 6 1 cofmpt ⊢ ( 𝜑 → ( abs ∘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) = ( 𝑥 ∈ 𝐴 ↦ ( abs ‘ 𝐵 ) ) )
8 7 eleq1d ⊢ ( 𝜑 → ( ( abs ∘ ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) ∈ ≤𝑂(1) ↔ ( 𝑥 ∈ 𝐴 ↦ ( abs ‘ 𝐵 ) ) ∈ ≤𝑂(1) ) )
9 4 8 bitrd ⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ∈ 𝑂(1) ↔ ( 𝑥 ∈ 𝐴 ↦ ( abs ‘ 𝐵 ) ) ∈ ≤𝑂(1) ) )