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Metamath Proof Explorer

Theorem climcj

Description: Limit of the complex conjugate of a sequence. Proposition 12-2.4(c) of Gleason p. 172. (Contributed by NM, 7-Jun-2006) (Revised by Mario Carneiro, 9-Feb-2014)

		Ref	Expression
	Hypotheses	climcn1lem.1	$⊢ Z = ℤ_{\geq M}$
		climcn1lem.2	$⊢ φ \to F ⇝ A$
		climcn1lem.4	$⊢ φ \to G \in W$
		climcn1lem.5	$⊢ φ \to M \in ℤ$
		climcn1lem.6	$⊢ (φ \land k \in Z) \to F (k) \in ℂ$
		climcj.7	$⊢ (φ \land k \in Z) \to G (k) = \overline{F (k)}$
	Assertion	climcj	$⊢ φ \to G ⇝ \overline{A}$

Proof

Step	Hyp	Ref	Expression
1		climcn1lem.1	$⊢ Z = ℤ_{\geq M}$
2		climcn1lem.2	$⊢ φ \to F ⇝ A$
3		climcn1lem.4	$⊢ φ \to G \in W$
4		climcn1lem.5	$⊢ φ \to M \in ℤ$
5		climcn1lem.6	$⊢ (φ \land k \in Z) \to F (k) \in ℂ$
6		climcj.7	$⊢ (φ \land k \in Z) \to G (k) = \overline{F (k)}$
7		cjf	$⊢ * : ℂ ⟶ ℂ$
8		cjcn2	$⊢ (A \in ℂ \land x \in ℝ^{+}) \to \exists y \in ℝ^{+} \forall z \in ℂ (\|z - A\| < y \to \|\overline{z} - \overline{A}\| < x)$
9	1 2 3 4 5 7 8 6	climcn1lem	$⊢ φ \to G ⇝ \overline{A}$