Theorem addclpr

Description: Closure of addition on positive reals. First statement of Proposition 9-3.5 of Gleason p. 123. (Contributed by NM, 13-Mar-1996) (New usage is discouraged.)

	Ref	Expression
Assertion	addclpr	|- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. P. )

Proof

Step	Hyp	Ref	Expression
1		df-plp	|- +P. = ( w e. P. , v e. P. |-> { x | E. y e. w E. z e. v x = ( y +Q z ) } )
2		addclnq	|- ( ( y e. Q. /\ z e. Q. ) -> ( y +Q z ) e. Q. )
3		ltanq	|- ( h e. Q. -> ( f ( h +Q f )
4		addcomnq	|- ( x +Q y ) = ( y +Q x )
5		addclprlem2	|- ( ( ( ( A e. P. /\ g e. A ) /\ ( B e. P. /\ h e. B ) ) /\ x e. Q. ) -> ( x x e. ( A +P. B ) ) )
6	1 2 3 4 5	genpcl	|- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) e. P. )