Theorem sratopn

Description: Topology component of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015) (Revised by Thierry Arnoux, 16-Jun-2019)

	Ref	Expression
Hypotheses	srapart.a	$$⊢\phi \to A=\mathrm{subringAlg}\left(W\right)\left(S\right)$$
	srapart.s	$$⊢\phi \to S\subseteq {\mathrm{Base}}_W$$
Assertion	sratopn	$$⊢\phi \to \mathrm{TopOpen}\left(W\right)=\mathrm{TopOpen}\left(A\right)$$

Proof

Step	Hyp	Ref	Expression
1		srapart.a	$$⊢\phi \to A=\mathrm{subringAlg}\left(W\right)\left(S\right)$$
2		srapart.s	$$⊢\phi \to S\subseteq {\mathrm{Base}}_W$$
3	1 2	srabase	$$⊢\phi \to {\mathrm{Base}}_W={\mathrm{Base}}_A$$
4	1 2	sratset	$$⊢\phi \to \mathrm{TopSet}\left(W\right)=\mathrm{TopSet}\left(A\right)$$
5	3 4	topnpropd	$$⊢\phi \to \mathrm{TopOpen}\left(W\right)=\mathrm{TopOpen}\left(A\right)$$