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 ) )