Metamath Proof Explorer

Theorem mpv

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

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

Step	Hyp	Ref	Expression
1		df-mp	\|- .P. = ( u e. P. , v e. P. \|-> { f \| E. g e. u E. h e. v f = ( g .Q h ) } )
2		mulclnq	\|- ( ( 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 ) } )

Step

Hyp

Ref

Expression

 |-  .P. = ( u e. P. , v e. P. |-> { f | E. g e. u E. h e. v f = ( g .Q h ) } )

 |-  ( ( g e. Q. /\ h e. Q. ) -> ( g .Q h ) e. Q. )

1 2

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