Metamath Proof Explorer


Theorem imcncf

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

Ref Expression
Assertion imcncf ℑ ∈ ( ℂ –cn→ ℝ )

Proof

Step Hyp Ref Expression
1 imf ℑ : ℂ ⟶ ℝ
2 imcn2 ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℝ+ ) → ∃ 𝑧 ∈ ℝ+𝑤 ∈ ℂ ( ( abs ‘ ( 𝑤𝑥 ) ) < 𝑧 → ( abs ‘ ( ( ℑ ‘ 𝑤 ) − ( ℑ ‘ 𝑥 ) ) ) < 𝑦 ) )
3 2 rgen2 𝑥 ∈ ℂ ∀ 𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℂ ( ( abs ‘ ( 𝑤𝑥 ) ) < 𝑧 → ( abs ‘ ( ( ℑ ‘ 𝑤 ) − ( ℑ ‘ 𝑥 ) ) ) < 𝑦 )
4 ssid ℂ ⊆ ℂ
5 ax-resscn ℝ ⊆ ℂ
6 elcncf2 ( ( ℂ ⊆ ℂ ∧ ℝ ⊆ ℂ ) → ( ℑ ∈ ( ℂ –cn→ ℝ ) ↔ ( ℑ : ℂ ⟶ ℝ ∧ ∀ 𝑥 ∈ ℂ ∀ 𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℂ ( ( abs ‘ ( 𝑤𝑥 ) ) < 𝑧 → ( abs ‘ ( ( ℑ ‘ 𝑤 ) − ( ℑ ‘ 𝑥 ) ) ) < 𝑦 ) ) ) )
7 4 5 6 mp2an ( ℑ ∈ ( ℂ –cn→ ℝ ) ↔ ( ℑ : ℂ ⟶ ℝ ∧ ∀ 𝑥 ∈ ℂ ∀ 𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤 ∈ ℂ ( ( abs ‘ ( 𝑤𝑥 ) ) < 𝑧 → ( abs ‘ ( ( ℑ ‘ 𝑤 ) − ( ℑ ‘ 𝑥 ) ) ) < 𝑦 ) ) )
8 1 3 7 mpbir2an ℑ ∈ ( ℂ –cn→ ℝ )