This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem paddvaln0N

Description: Projective subspace sum operation value for nonempty sets. (Contributed by NM, 27-Jan-2012) (New usage is discouraged.)

Ref Expression
Hypotheses paddfval.l 
|- .<_ = ( le ` K )
paddfval.j 
|- .\/ = ( join ` K )
paddfval.a 
|- A = ( Atoms ` K )
paddfval.p 
|- .+ = ( +P ` K )
Assertion paddvaln0N 
|- ( ( ( K e. Lat /\ X C_ A /\ Y C_ A ) /\ ( X =/= (/) /\ Y =/= (/) ) ) -> ( X .+ Y ) = { p e. A | E. q e. X E. r e. Y p .<_ ( q .\/ r ) } )

Proof

Step Hyp Ref Expression
1 paddfval.l 
 |-  .<_ = ( le ` K )
2 paddfval.j 
 |-  .\/ = ( join ` K )
3 paddfval.a 
 |-  A = ( Atoms ` K )
4 paddfval.p 
 |-  .+ = ( +P ` K )
5 1 2 3 4 elpaddn0 
 |-  ( ( ( K e. Lat /\ X C_ A /\ Y C_ A ) /\ ( X =/= (/) /\ Y =/= (/) ) ) -> ( s e. ( X .+ Y ) <-> ( s e. A /\ E. q e. X E. r e. Y s .<_ ( q .\/ r ) ) ) )
6 breq1 
 |-  ( p = s -> ( p .<_ ( q .\/ r ) <-> s .<_ ( q .\/ r ) ) )
7 6 2rexbidv 
 |-  ( p = s -> ( E. q e. X E. r e. Y p .<_ ( q .\/ r ) <-> E. q e. X E. r e. Y s .<_ ( q .\/ r ) ) )
8 7 elrab 
 |-  ( s e. { p e. A | E. q e. X E. r e. Y p .<_ ( q .\/ r ) } <-> ( s e. A /\ E. q e. X E. r e. Y s .<_ ( q .\/ r ) ) )
9 5 8 bitr4di 
 |-  ( ( ( K e. Lat /\ X C_ A /\ Y C_ A ) /\ ( X =/= (/) /\ Y =/= (/) ) ) -> ( s e. ( X .+ Y ) <-> s e. { p e. A | E. q e. X E. r e. Y p .<_ ( q .\/ r ) } ) )
10 9 eqrdv 
 |-  ( ( ( K e. Lat /\ X C_ A /\ Y C_ A ) /\ ( X =/= (/) /\ Y =/= (/) ) ) -> ( X .+ Y ) = { p e. A | E. q e. X E. r e. Y p .<_ ( q .\/ r ) } )