Theorem hlimconvi

Description: Convergence of a sequence on a Hilbert space. (Contributed by NM, 16-Aug-1999) (Revised by Mario Carneiro, 14-May-2014) (New usage is discouraged.)

	Ref	Expression
Hypothesis	hlim.1	⊢ 𝐴 ∈ V
Assertion	hlimconvi	⊢ ( ( 𝐹 ⇝𝑣 𝐴 ∧ 𝐵 ∈ ℝ+ ) → ∃ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ( ℤ≥ ‘ 𝑦 ) ( normℎ ‘ ( ( 𝐹 ‘ 𝑧 ) −ℎ 𝐴 ) ) < 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		hlim.1	⊢ 𝐴 ∈ V
2	1	hlimi	⊢ ( 𝐹 ⇝𝑣 𝐴 ↔ ( ( 𝐹 : ℕ ⟶ ℋ ∧ 𝐴 ∈ ℋ ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ( ℤ≥ ‘ 𝑦 ) ( normℎ ‘ ( ( 𝐹 ‘ 𝑧 ) −ℎ 𝐴 ) ) < 𝑥 ) )
3	2	simprbi	⊢ ( 𝐹 ⇝𝑣 𝐴 → ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ( ℤ≥ ‘ 𝑦 ) ( normℎ ‘ ( ( 𝐹 ‘ 𝑧 ) −ℎ 𝐴 ) ) < 𝑥 )
4		breq2	⊢ ( 𝑥 = 𝐵 → ( ( normℎ ‘ ( ( 𝐹 ‘ 𝑧 ) −ℎ 𝐴 ) ) < 𝑥 ↔ ( normℎ ‘ ( ( 𝐹 ‘ 𝑧 ) −ℎ 𝐴 ) ) < 𝐵 ) )
5	4	rexralbidv	⊢ ( 𝑥 = 𝐵 → ( ∃ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ( ℤ≥ ‘ 𝑦 ) ( normℎ ‘ ( ( 𝐹 ‘ 𝑧 ) −ℎ 𝐴 ) ) < 𝑥 ↔ ∃ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ( ℤ≥ ‘ 𝑦 ) ( normℎ ‘ ( ( 𝐹 ‘ 𝑧 ) −ℎ 𝐴 ) ) < 𝐵 ) )
6	5	rspccva	⊢ ( ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ( ℤ≥ ‘ 𝑦 ) ( normℎ ‘ ( ( 𝐹 ‘ 𝑧 ) −ℎ 𝐴 ) ) < 𝑥 ∧ 𝐵 ∈ ℝ+ ) → ∃ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ( ℤ≥ ‘ 𝑦 ) ( normℎ ‘ ( ( 𝐹 ‘ 𝑧 ) −ℎ 𝐴 ) ) < 𝐵 )
7	3 6	sylan	⊢ ( ( 𝐹 ⇝𝑣 𝐴 ∧ 𝐵 ∈ ℝ+ ) → ∃ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ( ℤ≥ ‘ 𝑦 ) ( normℎ ‘ ( ( 𝐹 ‘ 𝑧 ) −ℎ 𝐴 ) ) < 𝐵 )