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	⊢ ( 𝜑 → 𝐴 = ( ( subringAlg ‘ 𝑊 ) ‘ 𝑆 ) )
	srapart.s	⊢ ( 𝜑 → 𝑆 ⊆ ( Base ‘ 𝑊 ) )
Assertion	sratopn	⊢ ( 𝜑 → ( TopOpen ‘ 𝑊 ) = ( TopOpen ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		srapart.a	⊢ ( 𝜑 → 𝐴 = ( ( subringAlg ‘ 𝑊 ) ‘ 𝑆 ) )
2		srapart.s	⊢ ( 𝜑 → 𝑆 ⊆ ( Base ‘ 𝑊 ) )
3	1 2	srabase	⊢ ( 𝜑 → ( Base ‘ 𝑊 ) = ( Base ‘ 𝐴 ) )
4	1 2	sratset	⊢ ( 𝜑 → ( TopSet ‘ 𝑊 ) = ( TopSet ‘ 𝐴 ) )
5	3 4	topnpropd	⊢ ( 𝜑 → ( TopOpen ‘ 𝑊 ) = ( TopOpen ‘ 𝐴 ) )