Metamath Proof Explorer

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 w − x < z → ℜ ⁡ w − ℜ ⁡ x < y
3 2 rgen2 ⊢ ∀ x ∈ ℂ ∀ y ∈ ℝ + ∃ z ∈ ℝ + ∀ w ∈ ℂ w − x < z → 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 w − x < z → 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 w − x < z → ℜ ⁡ w − ℜ ⁡ x < y
8 1 3 7 mpbir2an ⊢ ℜ : ℂ ⟶cn ℝ