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Metamath Proof Explorer

Theorem abscncf

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

Ref Expression
Assertion abscncf ⊢ abs : ℂ ⟶cn ℝ

Proof

Step Hyp Ref Expression
1 absf ⊢ abs : ℂ ⟶ ℝ
2 abscn2 ⊢ 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 ax-resscn ⊢ ℝ ⊆ ℂ
6 elcncf2 ⊢ ℂ ⊆ ℂ ∧ ℝ ⊆ ℂ → abs : ℂ ⟶cn ℝ ↔ abs : ℂ ⟶ ℝ ∧ ∀ x ∈ ℂ ∀ y ∈ ℝ + ∃ z ∈ ℝ + ∀ w ∈ ℂ w − x < z → w − x < y
7 4 5 6 mp2an ⊢ abs : ℂ ⟶cn ℝ ↔ abs : ℂ ⟶ ℝ ∧ ∀ x ∈ ℂ ∀ y ∈ ℝ + ∃ z ∈ ℝ + ∀ w ∈ ℂ w − x < z → w − x < y
8 1 3 7 mpbir2an ⊢ abs : ℂ ⟶cn ℝ