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Metamath Proof Explorer

Theorem recn2

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

Ref Expression
Assertion recn2 ( ( 𝐴 ∈ ℂ ∧ 𝑥 ∈ ℝ+ ) → ∃ 𝑦 ∈ ℝ+ ∀ 𝑧 ∈ ℂ ( ( abs ‘ ( 𝑧 − 𝐴 ) ) < 𝑦 → ( abs ‘ ( ( ℜ ‘ 𝑧 ) − ( ℜ ‘ 𝐴 ) ) ) < 𝑥 ) )

Proof

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