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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 ( ( abs ‘ ( 𝑤 − 𝑥 ) ) < 𝑧 → ( abs ‘ ( ( abs ‘ 𝑤 ) − ( abs ‘ 𝑥 ) ) ) < 𝑦 ) )
3 2 rgen2 ⊢ ∀ 𝑥 ∈ ℂ ∀ 𝑦 ∈ ℝ+ ∃ 𝑧 ∈ ℝ+ ∀ 𝑤 ∈ ℂ ( ( abs ‘ ( 𝑤 − 𝑥 ) ) < 𝑧 → ( abs ‘ ( ( abs ‘ 𝑤 ) − ( abs ‘ 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 ( ( abs ‘ ( 𝑤 − 𝑥 ) ) < 𝑧 → ( abs ‘ ( ( abs ‘ 𝑤 ) − ( abs ‘ 𝑥 ) ) ) < 𝑦 ) ) ) )
7 4 5 6 mp2an ⊢ ( abs ∈ ( ℂ –cn→ ℝ ) ↔ ( abs : ℂ ⟶ ℝ ∧ ∀ 𝑥 ∈ ℂ ∀ 𝑦 ∈ ℝ+ ∃ 𝑧 ∈ ℝ+ ∀ 𝑤 ∈ ℂ ( ( abs ‘ ( 𝑤 − 𝑥 ) ) < 𝑧 → ( abs ‘ ( ( abs ‘ 𝑤 ) − ( abs ‘ 𝑥 ) ) ) < 𝑦 ) ) )
8 1 3 7 mpbir2an ⊢ abs ∈ ( ℂ –cn→ ℝ )