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	$$⊢Z={\mathbb{Z}}_{\ge M}$$
	climcn1lem.2	$$⊢\phi \to F⇝A$$
	climcn1lem.4	$$⊢\phi \to G\in W$$
	climcn1lem.5	$$⊢\phi \to M\in \mathbb{Z}$$
	climcn1lem.6	$$⊢\left(\phi \wedge k\in Z\right)\to F\left(k\right)\in ℂ$$
	climcj.7	$$⊢\left(\phi \wedge k\in Z\right)\to G\left(k\right)=\overline{F\left(k\right)}$$
Assertion	climcj	$$⊢\phi \to G⇝\bar{A}$$

Proof

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