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	|- ( ph -> A = ( ( subringAlg ` W ) ` S ) )
	srapart.s	|- ( ph -> S C_ ( Base ` W ) )
Assertion	sratopn	|- ( ph -> ( TopOpen ` W ) = ( TopOpen ` A ) )

Proof

Step	Hyp	Ref	Expression
1		srapart.a	|- ( ph -> A = ( ( subringAlg ` W ) ` S ) )
2		srapart.s	|- ( ph -> S C_ ( Base ` W ) )
3	1 2	srabase	|- ( ph -> ( Base ` W ) = ( Base ` A ) )
4	1 2	sratset	|- ( ph -> ( TopSet ` W ) = ( TopSet ` A ) )
5	3 4	topnpropd	|- ( ph -> ( TopOpen ` W ) = ( TopOpen ` A ) )