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Metamath Proof Explorer

Theorem elpadd2at2

Description: Membership in a projective subspace sum of two points. (Contributed by NM, 8-Mar-2012)

Ref Expression
Hypotheses paddfval.l 
|- .<_ = ( le ` K )
paddfval.j 
|- .\/ = ( join ` K )
paddfval.a 
|- A = ( Atoms ` K )
paddfval.p 
|- .+ = ( +P ` K )
Assertion elpadd2at2 
|- ( ( K e. Lat /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( S e. ( { Q } .+ { R } ) <-> S .<_ ( 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 elpadd2at 
 |-  ( ( K e. Lat /\ Q e. A /\ R e. A ) -> ( S e. ( { Q } .+ { R } ) <-> ( S e. A /\ S .<_ ( Q .\/ R ) ) ) )
6 5 3adant3r3 
 |-  ( ( K e. Lat /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( S e. ( { Q } .+ { R } ) <-> ( S e. A /\ S .<_ ( Q .\/ R ) ) ) )
7 simpr3 
 |-  ( ( K e. Lat /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> S e. A )
8 7 biantrurd 
 |-  ( ( K e. Lat /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( S .<_ ( Q .\/ R ) <-> ( S e. A /\ S .<_ ( Q .\/ R ) ) ) )
9 6 8 bitr4d 
 |-  ( ( K e. Lat /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( S e. ( { Q } .+ { R } ) <-> S .<_ ( Q .\/ R ) ) )