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Metamath Proof Explorer

Theorem liminfgelimsupuz

Description: The inferior limit is greater than or equal to the superior limit if and only if they are equal. (Contributed by Glauco Siliprandi, 2-Jan-2022)

Ref Expression
Hypotheses liminfgelimsupuz.1 ⊢ ( 𝜑 → 𝑀 ∈ ℤ )
liminfgelimsupuz.2 ⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
liminfgelimsupuz.3 ⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ℝ* )
Assertion liminfgelimsupuz ( 𝜑 → ( ( lim sup ‘ 𝐹 ) ≤ ( lim inf ‘ 𝐹 ) ↔ ( lim inf ‘ 𝐹 ) = ( lim sup ‘ 𝐹 ) ) )

Proof

Step Hyp Ref Expression
1 liminfgelimsupuz.1 ⊢ ( 𝜑 → 𝑀 ∈ ℤ )
2 liminfgelimsupuz.2 ⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
3 liminfgelimsupuz.3 ⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ℝ* )
4 2 fvexi ⊢ 𝑍 ∈ V
5 4 a1i ⊢ ( 𝜑 → 𝑍 ∈ V )
6 3 5 fexd ⊢ ( 𝜑 → 𝐹 ∈ V )
7 6 liminfcld ⊢ ( 𝜑 → ( lim inf ‘ 𝐹 ) ∈ ℝ* )
8 7 adantr ⊢ ( ( 𝜑 ∧ ( lim sup ‘ 𝐹 ) ≤ ( lim inf ‘ 𝐹 ) ) → ( lim inf ‘ 𝐹 ) ∈ ℝ* )
9 6 limsupcld ⊢ ( 𝜑 → ( lim sup ‘ 𝐹 ) ∈ ℝ* )
10 9 adantr ⊢ ( ( 𝜑 ∧ ( lim sup ‘ 𝐹 ) ≤ ( lim inf ‘ 𝐹 ) ) → ( lim sup ‘ 𝐹 ) ∈ ℝ* )
11 1 2 3 liminflelimsupuz ⊢ ( 𝜑 → ( lim inf ‘ 𝐹 ) ≤ ( lim sup ‘ 𝐹 ) )
12 11 adantr ⊢ ( ( 𝜑 ∧ ( lim sup ‘ 𝐹 ) ≤ ( lim inf ‘ 𝐹 ) ) → ( lim inf ‘ 𝐹 ) ≤ ( lim sup ‘ 𝐹 ) )
13 simpr ⊢ ( ( 𝜑 ∧ ( lim sup ‘ 𝐹 ) ≤ ( lim inf ‘ 𝐹 ) ) → ( lim sup ‘ 𝐹 ) ≤ ( lim inf ‘ 𝐹 ) )
14 8 10 12 13 xrletrid ⊢ ( ( 𝜑 ∧ ( lim sup ‘ 𝐹 ) ≤ ( lim inf ‘ 𝐹 ) ) → ( lim inf ‘ 𝐹 ) = ( lim sup ‘ 𝐹 ) )
15 9 adantr ⊢ ( ( 𝜑 ∧ ( lim inf ‘ 𝐹 ) = ( lim sup ‘ 𝐹 ) ) → ( lim sup ‘ 𝐹 ) ∈ ℝ* )
16 id ⊢ ( ( lim inf ‘ 𝐹 ) = ( lim sup ‘ 𝐹 ) → ( lim inf ‘ 𝐹 ) = ( lim sup ‘ 𝐹 ) )
17 16 eqcomd ⊢ ( ( lim inf ‘ 𝐹 ) = ( lim sup ‘ 𝐹 ) → ( lim sup ‘ 𝐹 ) = ( lim inf ‘ 𝐹 ) )
18 17 adantl ⊢ ( ( 𝜑 ∧ ( lim inf ‘ 𝐹 ) = ( lim sup ‘ 𝐹 ) ) → ( lim sup ‘ 𝐹 ) = ( lim inf ‘ 𝐹 ) )
19 15 18 xreqled ⊢ ( ( 𝜑 ∧ ( lim inf ‘ 𝐹 ) = ( lim sup ‘ 𝐹 ) ) → ( lim sup ‘ 𝐹 ) ≤ ( lim inf ‘ 𝐹 ) )
20 14 19 impbida ⊢ ( 𝜑 → ( ( lim sup ‘ 𝐹 ) ≤ ( lim inf ‘ 𝐹 ) ↔ ( lim inf ‘ 𝐹 ) = ( lim sup ‘ 𝐹 ) ) )