Theorem xlimcl

Description: The limit of a sequence of extended real numbers is an extended real number. (Contributed by Glauco Siliprandi, 5-Feb-2022)

	Ref	Expression
Assertion	xlimcl	⊢ ( 𝐹 ~~>* 𝐴 → 𝐴 ∈ ℝ* )

Proof

Step	Hyp	Ref	Expression
1		letopon	⊢ ( ordTop ‘ ≤ ) ∈ ( TopOn ‘ ℝ* )
2		df-xlim	⊢ ~~>* = ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) )
3	2	breqi	⊢ ( 𝐹 ~~>* 𝐴 ↔ 𝐹 ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) ) 𝐴 )
4	3	biimpi	⊢ ( 𝐹 ~~>* 𝐴 → 𝐹 ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) ) 𝐴 )
5		lmcl	⊢ ( ( ( ordTop ‘ ≤ ) ∈ ( TopOn ‘ ℝ* ) ∧ 𝐹 ( ⇝𝑡 ‘ ( ordTop ‘ ≤ ) ) 𝐴 ) → 𝐴 ∈ ℝ* )
6	1 4 5	sylancr	⊢ ( 𝐹 ~~>* 𝐴 → 𝐴 ∈ ℝ* )