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Metamath Proof Explorer

Theorem climabs

Description: Limit of the absolute value of a sequence. Proposition 12-2.4(c) of Gleason p. 172. (Contributed by NM, 7-Jun-2006) (Revised by Mario Carneiro, 9-Feb-2014)

Ref Expression
Hypotheses climcn1lem.1 ⊢ Z = ℤ ≥ M
climcn1lem.2 ⊢ φ → F ⇝ A
climcn1lem.4 ⊢ φ → G ∈ W
climcn1lem.5 ⊢ φ → M ∈ ℤ
climcn1lem.6 ⊢ φ ∧ k ∈ Z → F ⁡ k ∈ ℂ
climabs.7 ⊢ φ ∧ k ∈ Z → G ⁡ k = F ⁡ k
Assertion climabs ⊢ φ → G ⇝ A

Proof

Step Hyp Ref Expression
1 climcn1lem.1 ⊢ Z = ℤ ≥ M
2 climcn1lem.2 ⊢ φ → F ⇝ A
3 climcn1lem.4 ⊢ φ → G ∈ W
4 climcn1lem.5 ⊢ φ → M ∈ ℤ
5 climcn1lem.6 ⊢ φ ∧ k ∈ Z → F ⁡ k ∈ ℂ
6 climabs.7 ⊢ φ ∧ k ∈ Z → G ⁡ k = F ⁡ k
7 absf ⊢ abs : ℂ ⟶ ℝ
8 ax-resscn ⊢ ℝ ⊆ ℂ
9 fss ⊢ abs : ℂ ⟶ ℝ ∧ ℝ ⊆ ℂ → abs : ℂ ⟶ ℂ
10 7 8 9 mp2an ⊢ abs : ℂ ⟶ ℂ
11 abscn2 ⊢ A ∈ ℂ ∧ x ∈ ℝ + → ∃ y ∈ ℝ + ∀ z ∈ ℂ z − A < y → z − A < x
12 1 2 3 4 5 10 11 6 climcn1lem ⊢ φ → G ⇝ A