Theorem cjcncf

Description: Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007) (Revised by Mario Carneiro, 28-Apr-2014)

	Ref	Expression
Assertion	cjcncf	$$⊢*:ℂ\underset{cn}{⟶}ℂ$$

Proof

Step	Hyp	Ref	Expression
1		cjf	$$⊢*:ℂ⟶ℂ$$
2		cjcn2	$$⊢\left(x\in ℂ\wedge y\in {ℝ}^{+}\right)\to \exists z\in {ℝ}^{+}\forall w\in ℂ\left(\left|w-x\right|<z\to \left|\bar{w}-\bar{x}\right|<y\right)$$
3	2	rgen2	$$⊢\forall x\in ℂ\forall y\in {ℝ}^{+}\exists z\in {ℝ}^{+}\forall w\in ℂ\left(\left|w-x\right|<z\to \left|\bar{w}-\bar{x}\right|<y\right)$$
4		ssid	$$⊢ℂ\subseteq ℂ$$
5		elcncf2	$$⊢\left(ℂ\subseteq ℂ\wedge ℂ\subseteq ℂ\right)\to \left(*:ℂ\underset{cn}{⟶}ℂ\leftrightarrow \left(*:ℂ⟶ℂ\wedge \forall x\in ℂ\forall y\in {ℝ}^{+}\exists z\in {ℝ}^{+}\forall w\in ℂ\left(\left|w-x\right|<z\to \left|\bar{w}-\bar{x}\right|<y\right)\right)\right)$$
6	4 4 5	mp2an	$$⊢*:ℂ\underset{cn}{⟶}ℂ\leftrightarrow \left(*:ℂ⟶ℂ\wedge \forall x\in ℂ\forall y\in {ℝ}^{+}\exists z\in {ℝ}^{+}\forall w\in ℂ\left(\left|w-x\right|<z\to \left|\bar{w}-\bar{x}\right|<y\right)\right)$$
7	1 3 6	mpbir2an	$$⊢*:ℂ\underset{cn}{⟶}ℂ$$