Theorem imcncf

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

	Ref	Expression
Assertion	imcncf	$$⊢\Im :ℂ\underset{cn}{⟶}ℝ$$

Proof

Step	Hyp	Ref	Expression
1		imf	$$⊢\Im :ℂ⟶ℝ$$
2		imcn2	$$⊢\left(x\in ℂ\wedge y\in {ℝ}^{+}\right)\to \exists z\in {ℝ}^{+}\forall w\in ℂ\left(\left|w-x\right|<z\to \left|\Im \left(w\right)-\Im \left(x\right)\right|<y\right)$$
3	2	rgen2	$$⊢\forall x\in ℂ\forall y\in {ℝ}^{+}\exists z\in {ℝ}^{+}\forall w\in ℂ\left(\left|w-x\right|<z\to \left|\Im \left(w\right)-\Im \left(x\right)\right|<y\right)$$
4		ssid	$$⊢ℂ\subseteq ℂ$$
5		ax-resscn	$$⊢ℝ\subseteq ℂ$$
6		elcncf2	$$⊢\left(ℂ\subseteq ℂ\wedge ℝ\subseteq ℂ\right)\to \left(\Im :ℂ\underset{cn}{⟶}ℝ\leftrightarrow \left(\Im :ℂ⟶ℝ\wedge \forall x\in ℂ\forall y\in {ℝ}^{+}\exists z\in {ℝ}^{+}\forall w\in ℂ\left(\left|w-x\right|<z\to \left|\Im \left(w\right)-\Im \left(x\right)\right|<y\right)\right)\right)$$
7	4 5 6	mp2an	$$⊢\Im :ℂ\underset{cn}{⟶}ℝ\leftrightarrow \left(\Im :ℂ⟶ℝ\wedge \forall x\in ℂ\forall y\in {ℝ}^{+}\exists z\in {ℝ}^{+}\forall w\in ℂ\left(\left|w-x\right|<z\to \left|\Im \left(w\right)-\Im \left(x\right)\right|<y\right)\right)$$
8	1 3 7	mpbir2an	$$⊢\Im :ℂ\underset{cn}{⟶}ℝ$$