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 = ( ℤ𝑀 )
climcn1lem.2 ( 𝜑𝐹𝐴 )
climcn1lem.4 ( 𝜑𝐺𝑊 )
climcn1lem.5 ( 𝜑𝑀 ∈ ℤ )
climcn1lem.6 ( ( 𝜑𝑘𝑍 ) → ( 𝐹𝑘 ) ∈ ℂ )
climre.7 ( ( 𝜑𝑘𝑍 ) → ( 𝐺𝑘 ) = ( ℜ ‘ ( 𝐹𝑘 ) ) )
Assertion climre ( 𝜑𝐺 ⇝ ( ℜ ‘ 𝐴 ) )

Proof

Step Hyp Ref Expression
1 climcn1lem.1 𝑍 = ( ℤ𝑀 )
2 climcn1lem.2 ( 𝜑𝐹𝐴 )
3 climcn1lem.4 ( 𝜑𝐺𝑊 )
4 climcn1lem.5 ( 𝜑𝑀 ∈ ℤ )
5 climcn1lem.6 ( ( 𝜑𝑘𝑍 ) → ( 𝐹𝑘 ) ∈ ℂ )
6 climre.7 ( ( 𝜑𝑘𝑍 ) → ( 𝐺𝑘 ) = ( ℜ ‘ ( 𝐹𝑘 ) ) )
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 ‘ ( 𝑧𝐴 ) ) < 𝑦 → ( abs ‘ ( ( ℜ ‘ 𝑧 ) − ( ℜ ‘ 𝐴 ) ) ) < 𝑥 ) )
12 1 2 3 4 5 10 11 6 climcn1lem ( 𝜑𝐺 ⇝ ( ℜ ‘ 𝐴 ) )