Metamath Proof Explorer


Theorem plpv

Description: Value of addition on positive reals. (Contributed by NM, 28-Feb-1996) (New usage is discouraged.)

Ref Expression
Assertion plpv
|- ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) = { x | E. y e. A E. z e. B x = ( y +Q z ) } )

Proof

Step Hyp Ref Expression
1 df-plp
 |-  +P. = ( u e. P. , v e. P. |-> { f | E. g e. u E. h e. v f = ( g +Q h ) } )
2 addclnq
 |-  ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
3 1 2 genpv
 |-  ( ( A e. P. /\ B e. P. ) -> ( A +P. B ) = { x | E. y e. A E. z e. B x = ( y +Q z ) } )