Theorem xrstset

Description: The topology component of the extended real number structure. (Contributed by Mario Carneiro, 21-Aug-2015)

	Ref	Expression
Assertion	xrstset	|- ( ordTop ` <_ ) = ( TopSet ` RR*s )

Proof

Step	Hyp	Ref	Expression
1		fvex	|- ( ordTop ` <_ ) e. _V
2		df-xrs	|- RR*s = ( { <. ( Base ` ndx ) , RR* >. , <. ( +g ` ndx ) , +e >. , <. ( .r ` ndx ) , *e >. } u. { <. ( TopSet ` ndx ) , ( ordTop ` <_ ) >. , <. ( le ` ndx ) , <_ >. , <. ( dist ` ndx ) , ( x e. RR* , y e. RR* |-> if ( x <_ y , ( y +e -e x ) , ( x +e -e y ) ) ) >. } )
3	2	odrngtset	|- ( ( ordTop ` <_ ) e. _V -> ( ordTop ` <_ ) = ( TopSet ` RR*s ) )
4	1 3	ax-mp	|- ( ordTop ` <_ ) = ( TopSet ` RR*s )