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 ) ) |
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 ) ) |