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Metamath Proof Explorer

Theorem rlimre

Description: Limit of the real part of a sequence. Proposition 12-2.4(c) of Gleason p. 172. (Contributed by Mario Carneiro, 10-May-2016)

Ref Expression
Hypotheses rlimabs.1 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 )
rlimabs.2 ⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ⇝𝑟 𝐶 )
Assertion rlimre ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ ( ℜ ‘ 𝐵 ) ) ⇝𝑟 ( ℜ ‘ 𝐶 ) )

Proof

Step Hyp Ref Expression
1 rlimabs.1 ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ 𝑉 )
2 rlimabs.2 ⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ⇝𝑟 𝐶 )
3 1 2 rlimmptrcl ⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ )
4 rlimcl ⊢ ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ⇝𝑟 𝐶 → 𝐶 ∈ ℂ )
5 2 4 syl ⊢ ( 𝜑 → 𝐶 ∈ ℂ )
6 ref ⊢ ℜ : ℂ ⟶ ℝ
7 ax-resscn ⊢ ℝ ⊆ ℂ
8 fss ⊢ ( ( ℜ : ℂ ⟶ ℝ ∧ ℝ ⊆ ℂ ) → ℜ : ℂ ⟶ ℂ )
9 6 7 8 mp2an ⊢ ℜ : ℂ ⟶ ℂ
10 9 a1i ⊢ ( 𝜑 → ℜ : ℂ ⟶ ℂ )
11 recn2 ⊢ ( ( 𝐶 ∈ ℂ ∧ 𝑥 ∈ ℝ+ ) → ∃ 𝑦 ∈ ℝ+ ∀ 𝑧 ∈ ℂ ( ( abs ‘ ( 𝑧 − 𝐶 ) ) < 𝑦 → ( abs ‘ ( ( ℜ ‘ 𝑧 ) − ( ℜ ‘ 𝐶 ) ) ) < 𝑥 ) )
12 5 11 sylan ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ℝ+ ) → ∃ 𝑦 ∈ ℝ+ ∀ 𝑧 ∈ ℂ ( ( abs ‘ ( 𝑧 − 𝐶 ) ) < 𝑦 → ( abs ‘ ( ( ℜ ‘ 𝑧 ) − ( ℜ ‘ 𝐶 ) ) ) < 𝑥 ) )
13 3 5 2 10 12 rlimcn1b ⊢ ( 𝜑 → ( 𝑘 ∈ 𝐴 ↦ ( ℜ ‘ 𝐵 ) ) ⇝𝑟 ( ℜ ‘ 𝐶 ) )