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

Proof

Step	Hyp	Ref	Expression
1		srapart.a	⊢ ( 𝜑 → 𝐴 = ( ( subringAlg ‘ 𝑊 ) ‘ 𝑆 ) )
2		srapart.s	⊢ ( 𝜑 → 𝑆 ⊆ ( Base ‘ 𝑊 ) )
3		df-tset	⊢ TopSet = Slot 9
4		9nn	⊢ 9 ∈ ℕ
5		8lt9	⊢ 8 < 9
6	5	olci	⊢ ( 9 < 5 ∨ 8 < 9 )
7	1 2 3 4 6	sralem	⊢ ( 𝜑 → ( TopSet ‘ 𝑊 ) = ( TopSet ‘ 𝐴 ) )