Theorem xrsmul

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

	Ref	Expression
Assertion	xrsmul	|- *e = ( .r ` RR*s )

Proof

Step	Hyp	Ref	Expression
1		xmulf	|- *e : ( RR* X. RR* ) --> RR*
2		xrex	|- RR* e. _V
3	2 2	xpex	|- ( RR* X. RR* ) e. _V
4		fex2	|- ( ( *e : ( RR* X. RR* ) --> RR* /\ ( RR* X. RR* ) e. _V /\ RR* e. _V ) -> *e e. _V )
5	1 3 2 4	mp3an	|- *e e. _V
6		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 ) ) ) >. } )
7	6	odrngmulr	|- ( *e e. _V -> *e = ( .r ` RR*s ) )
8	5 7	ax-mp	|- *e = ( .r ` RR*s )