Theorem xlimres

Description: A function converges iff its restriction to an upper integers set converges. (Contributed by Glauco Siliprandi, 5-Feb-2022)

	Ref	Expression
Hypotheses	xlimres.1	⊢ ( 𝜑 → 𝐹 ∈ ( ℝ* ↑pm ℂ ) )
	xlimres.2	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
Assertion	xlimres	⊢ ( 𝜑 → ( 𝐹 ~~>* 𝐴 ↔ ( 𝐹 ↾ ( ℤ≥ ‘ 𝑀 ) ) ~~>* 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		xlimres.1	⊢ ( 𝜑 → 𝐹 ∈ ( ℝ* ↑pm ℂ ) )
2		xlimres.2	⊢ ( 𝜑 → 𝑀 ∈ ℤ )
3		letopon	⊢ ( ordTop ‘ ≤ ) ∈ ( TopOn ‘ ℝ* )
4	3	a1i	⊢ ( 𝜑 → ( ordTop ‘ ≤ ) ∈ ( TopOn ‘ ℝ* ) )
5	4 1 2	lmres	⊢ ( 𝜑 → ( 𝐹 ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) ) 𝐴 ↔ ( 𝐹 ↾ ( ℤ≥ ‘ 𝑀 ) ) ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) ) 𝐴 ) )
6		df-xlim	⊢ ~~>* = ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) )
7	6	breqi	⊢ ( 𝐹 ~~>* 𝐴 ↔ 𝐹 ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) ) 𝐴 )
8	6	breqi	⊢ ( ( 𝐹 ↾ ( ℤ≥ ‘ 𝑀 ) ) ~~>* 𝐴 ↔ ( 𝐹 ↾ ( ℤ≥ ‘ 𝑀 ) ) ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) ) 𝐴 )
9	5 7 8	3bitr4g	⊢ ( 𝜑 → ( 𝐹 ~~>* 𝐴 ↔ ( 𝐹 ↾ ( ℤ≥ ‘ 𝑀 ) ) ~~>* 𝐴 ) )