Theorem cjcncf

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

	Ref	Expression
Assertion	cjcncf	⊢ ∗ ∈ ( ℂ –cn→ ℂ )

Proof

Step	Hyp	Ref	Expression
1		cjf	⊢ ∗ : ℂ ⟶ ℂ
2		cjcn2	⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℝ+ ) → ∃ 𝑧 ∈ ℝ+ ∀ 𝑤 ∈ ℂ ( ( abs ‘ ( 𝑤 − 𝑥 ) ) < 𝑧 → ( abs ‘ ( ( ∗ ‘ 𝑤 ) − ( ∗ ‘ 𝑥 ) ) ) < 𝑦 ) )
3	2	rgen2	⊢ ∀ 𝑥 ∈ ℂ ∀ 𝑦 ∈ ℝ+ ∃ 𝑧 ∈ ℝ+ ∀ 𝑤 ∈ ℂ ( ( abs ‘ ( 𝑤 − 𝑥 ) ) < 𝑧 → ( abs ‘ ( ( ∗ ‘ 𝑤 ) − ( ∗ ‘ 𝑥 ) ) ) < 𝑦 )
4		ssid	⊢ ℂ ⊆ ℂ
5		elcncf2	⊢ ( ( ℂ ⊆ ℂ ∧ ℂ ⊆ ℂ ) → ( ∗ ∈ ( ℂ –cn→ ℂ ) ↔ ( ∗ : ℂ ⟶ ℂ ∧ ∀ 𝑥 ∈ ℂ ∀ 𝑦 ∈ ℝ+ ∃ 𝑧 ∈ ℝ+ ∀ 𝑤 ∈ ℂ ( ( abs ‘ ( 𝑤 − 𝑥 ) ) < 𝑧 → ( abs ‘ ( ( ∗ ‘ 𝑤 ) − ( ∗ ‘ 𝑥 ) ) ) < 𝑦 ) ) ) )
6	4 4 5	mp2an	⊢ ( ∗ ∈ ( ℂ –cn→ ℂ ) ↔ ( ∗ : ℂ ⟶ ℂ ∧ ∀ 𝑥 ∈ ℂ ∀ 𝑦 ∈ ℝ+ ∃ 𝑧 ∈ ℝ+ ∀ 𝑤 ∈ ℂ ( ( abs ‘ ( 𝑤 − 𝑥 ) ) < 𝑧 → ( abs ‘ ( ( ∗ ‘ 𝑤 ) − ( ∗ ‘ 𝑥 ) ) ) < 𝑦 ) ) )
7	1 3 6	mpbir2an	⊢ ∗ ∈ ( ℂ –cn→ ℂ )