Theorem sratset

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	sratset	$$⊢\phi \to \mathrm{TopSet}\left(W\right)=\mathrm{TopSet}\left(A\right)$$

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		df-tset	$$⊢\mathrm{TopSet}=\mathrm{Slot}9$$
4		9nn	$$⊢9\in \mathbb{N}$$
5		8lt9	$$⊢8<9$$
6	5	olci	$$⊢\left(9<5\vee 8<9\right)$$
7	1 2 3 4 6	sralem	$$⊢\phi \to \mathrm{TopSet}\left(W\right)=\mathrm{TopSet}\left(A\right)$$