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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 w − x < z → w ‾ − x ‾ < y
3 2 rgen2 ⊢ ∀ x ∈ ℂ ∀ y ∈ ℝ + ∃ z ∈ ℝ + ∀ w ∈ ℂ w − x < z → 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 w − x < z → w ‾ − x ‾ < y
6 4 4 5 mp2an ⊢ * : ℂ ⟶cn ℂ ↔ * : ℂ ⟶ ℂ ∧ ∀ x ∈ ℂ ∀ y ∈ ℝ + ∃ z ∈ ℝ + ∀ w ∈ ℂ w − x < z → w ‾ − x ‾ < y
7 1 3 6 mpbir2an ⊢ * : ℂ ⟶cn ℂ