Theorem recncf

Description: Real part is continuous. (Contributed by Paul Chapman, 21-Oct-2007) (Revised by Mario Carneiro, 28-Apr-2014)

	Ref	Expression
Assertion	recncf	⊢ ℜ ∈ ( ℂ –cn→ ℝ )

Proof

Step	Hyp	Ref	Expression
1		ref	⊢ ℜ : ℂ ⟶ ℝ
2		recn2	⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℝ+ ) → ∃ 𝑧 ∈ ℝ+ ∀ 𝑤 ∈ ℂ ( ( abs ‘ ( 𝑤 − 𝑥 ) ) < 𝑧 → ( abs ‘ ( ( ℜ ‘ 𝑤 ) − ( ℜ ‘ 𝑥 ) ) ) < 𝑦 ) )
3	2	rgen2	⊢ ∀ 𝑥 ∈ ℂ ∀ 𝑦 ∈ ℝ+ ∃ 𝑧 ∈ ℝ+ ∀ 𝑤 ∈ ℂ ( ( abs ‘ ( 𝑤 − 𝑥 ) ) < 𝑧 → ( abs ‘ ( ( ℜ ‘ 𝑤 ) − ( ℜ ‘ 𝑥 ) ) ) < 𝑦 )
4		ssid	⊢ ℂ ⊆ ℂ
5		ax-resscn	⊢ ℝ ⊆ ℂ
6		elcncf2	⊢ ( ( ℂ ⊆ ℂ ∧ ℝ ⊆ ℂ ) → ( ℜ ∈ ( ℂ –cn→ ℝ ) ↔ ( ℜ : ℂ ⟶ ℝ ∧ ∀ 𝑥 ∈ ℂ ∀ 𝑦 ∈ ℝ+ ∃ 𝑧 ∈ ℝ+ ∀ 𝑤 ∈ ℂ ( ( abs ‘ ( 𝑤 − 𝑥 ) ) < 𝑧 → ( abs ‘ ( ( ℜ ‘ 𝑤 ) − ( ℜ ‘ 𝑥 ) ) ) < 𝑦 ) ) ) )
7	4 5 6	mp2an	⊢ ( ℜ ∈ ( ℂ –cn→ ℝ ) ↔ ( ℜ : ℂ ⟶ ℝ ∧ ∀ 𝑥 ∈ ℂ ∀ 𝑦 ∈ ℝ+ ∃ 𝑧 ∈ ℝ+ ∀ 𝑤 ∈ ℂ ( ( abs ‘ ( 𝑤 − 𝑥 ) ) < 𝑧 → ( abs ‘ ( ( ℜ ‘ 𝑤 ) − ( ℜ ‘ 𝑥 ) ) ) < 𝑦 ) ) )
8	1 3 7	mpbir2an	⊢ ℜ ∈ ( ℂ –cn→ ℝ )