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Metamath Proof Explorer

Theorem climre

Description: Limit of the real part 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 )
climre.7 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐺 ‘ 𝑘 ) = ( ℜ ‘ ( 𝐹 ‘ 𝑘 ) ) )
Assertion climre ( 𝜑 → 𝐺 ⇝ ( ℜ ‘ 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 climre.7 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐺 ‘ 𝑘 ) = ( ℜ ‘ ( 𝐹 ‘ 𝑘 ) ) )
7 ref ⊢ ℜ : ℂ ⟶ ℝ
8 ax-resscn ⊢ ℝ ⊆ ℂ
9 fss ⊢ ( ( ℜ : ℂ ⟶ ℝ ∧ ℝ ⊆ ℂ ) → ℜ : ℂ ⟶ ℂ )
10 7 8 9 mp2an ⊢ ℜ : ℂ ⟶ ℂ
11 recn2 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+ ) → ∃ 𝑦 ∈ ℝ+ ∀ 𝑧 ∈ ℂ ( ( abs ‘ ( 𝑧 − 𝐴 ) ) < 𝑦 → ( abs ‘ ( ( ℜ ‘ 𝑧 ) − ( ℜ ‘ 𝐴 ) ) ) < 𝑥 ) )
12 1 2 3 4 5 10 11 6 climcn1lem ⊢ ( 𝜑 → 𝐺 ⇝ ( ℜ ‘ 𝐴 ) )