Metamath Proof Explorer


Theorem imcn2

Description: The imaginary part function is continuous. (Contributed by Mario Carneiro, 9-Feb-2014)

Ref Expression
Assertion imcn2 ( ( 𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+ ) → ∃ 𝑦 ∈ ℝ+𝑧 ∈ ℂ ( ( abs ‘ ( 𝑧𝐴 ) ) < 𝑦 → ( abs ‘ ( ( ℑ ‘ 𝑧 ) − ( ℑ ‘ 𝐴 ) ) ) < 𝑥 ) )

Proof

Step Hyp Ref Expression
1 imf ℑ : ℂ ⟶ ℝ
2 ax-resscn ℝ ⊆ ℂ
3 fss ( ( ℑ : ℂ ⟶ ℝ ∧ ℝ ⊆ ℂ ) → ℑ : ℂ ⟶ ℂ )
4 1 2 3 mp2an ℑ : ℂ ⟶ ℂ
5 imsub ( ( 𝑧 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( ℑ ‘ ( 𝑧𝐴 ) ) = ( ( ℑ ‘ 𝑧 ) − ( ℑ ‘ 𝐴 ) ) )
6 5 fveq2d ( ( 𝑧 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( abs ‘ ( ℑ ‘ ( 𝑧𝐴 ) ) ) = ( abs ‘ ( ( ℑ ‘ 𝑧 ) − ( ℑ ‘ 𝐴 ) ) ) )
7 subcl ( ( 𝑧 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( 𝑧𝐴 ) ∈ ℂ )
8 absimle ( ( 𝑧𝐴 ) ∈ ℂ → ( abs ‘ ( ℑ ‘ ( 𝑧𝐴 ) ) ) ≤ ( abs ‘ ( 𝑧𝐴 ) ) )
9 7 8 syl ( ( 𝑧 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( abs ‘ ( ℑ ‘ ( 𝑧𝐴 ) ) ) ≤ ( abs ‘ ( 𝑧𝐴 ) ) )
10 6 9 eqbrtrrd ( ( 𝑧 ∈ ℂ ∧ 𝐴 ∈ ℂ ) → ( abs ‘ ( ( ℑ ‘ 𝑧 ) − ( ℑ ‘ 𝐴 ) ) ) ≤ ( abs ‘ ( 𝑧𝐴 ) ) )
11 4 10 cn1lem ( ( 𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+ ) → ∃ 𝑦 ∈ ℝ+𝑧 ∈ ℂ ( ( abs ‘ ( 𝑧𝐴 ) ) < 𝑦 → ( abs ‘ ( ( ℑ ‘ 𝑧 ) − ( ℑ ‘ 𝐴 ) ) ) < 𝑥 ) )