Theorem mulcncf

Description: The multiplication of two continuous complex functions is continuous. (Contributed by Glauco Siliprandi, 29-Jun-2017)

	Ref	Expression
Hypotheses	mulcncf.1	$$⊢\phi \to \left(x\in X⟼A\right):X\underset{cn}{⟶}ℂ$$
	mulcncf.2	$$⊢\phi \to \left(x\in X⟼B\right):X\underset{cn}{⟶}ℂ$$
Assertion	mulcncf	$$⊢\phi \to \left(x\in X⟼AB\right):X\underset{cn}{⟶}ℂ$$

Proof

Step	Hyp	Ref	Expression
1		mulcncf.1	$$⊢\phi \to \left(x\in X⟼A\right):X\underset{cn}{⟶}ℂ$$
2		mulcncf.2	$$⊢\phi \to \left(x\in X⟼B\right):X\underset{cn}{⟶}ℂ$$
3		eqid	$$⊢\mathrm{TopOpen}\left({ℂ}_{\mathrm{fld}}\right)=\mathrm{TopOpen}\left({ℂ}_{\mathrm{fld}}\right)$$
4	3	mulcn	$$⊢\times \in \left(\left(\mathrm{TopOpen}\left({ℂ}_{\mathrm{fld}}\right){\times }_t\mathrm{TopOpen}\left({ℂ}_{\mathrm{fld}}\right)\right)\mathrm{Cn}\mathrm{TopOpen}\left({ℂ}_{\mathrm{fld}}\right)\right)$$
5	4	a1i	$$⊢\phi \to \times \in \left(\left(\mathrm{TopOpen}\left({ℂ}_{\mathrm{fld}}\right){\times }_t\mathrm{TopOpen}\left({ℂ}_{\mathrm{fld}}\right)\right)\mathrm{Cn}\mathrm{TopOpen}\left({ℂ}_{\mathrm{fld}}\right)\right)$$
6	3 5 1 2	cncfmpt2f	$$⊢\phi \to \left(x\in X⟼AB\right):X\underset{cn}{⟶}ℂ$$