Theorem mulcncfOLD

Description: Obsolete version of mulcncf as of 9-Apr-2025. (Contributed by Glauco Siliprandi, 29-Jun-2017) (Proof modification is discouraged.) (New usage is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		mulcncfOLD.1	$$⊢\phi \to \left(x\in X⟼A\right):X\underset{cn}{⟶}ℂ$$
2		mulcncfOLD.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	$$⊢\phi \to \left(x\in X⟼AB\right):X\underset{cn}{⟶}ℂ$$