Yes, indexes on constituent tables can be used in queries on views. Views usually use the MERGE
algorithm which basically merges the query on the view with the query that defines the view. You can think of it like this:
The view query:
SELECT * FROM CAR JOIN TYRES ON (CAR.ID=TYRES.CarID)
Query on view:
SELECT model, tyreSize FROM CAR_DETAILS WHERE tyreSize=19
Merged query:
SELECT model, tyreSize FROM CAR JOIN TYRES ON (CAR.ID=TYRES.CarID) WHERE tyreSize=19
The query isn't really executed like that but it gives you an idea of how it will be run. In your case, an index on tyreSize will probably be used if it exists. Depending on your other queries, an index on TYRES.CarID will also probably be helpful.