Theorem xrsadd

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

	Ref	Expression
Assertion	xrsadd	|- +e = ( +g ` RR*s )

Proof

Step	Hyp	Ref	Expression
1		xaddf	|- +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	odrngplusg	|- ( +e e. _V -> +e = ( +g ` RR*s ) )
8	5 7	ax-mp	|- +e = ( +g ` RR*s )