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Metamath Proof Explorer

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 )