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Metamath Proof Explorer

Theorem elpadd2at

Description: Membership in a projective subspace sum of two points. (Contributed by NM, 29-Jan-2012)

Ref Expression
Hypotheses paddfval.l 
|- .<_ = ( le ` K )
paddfval.j 
|- .\/ = ( join ` K )
paddfval.a 
|- A = ( Atoms ` K )
paddfval.p 
|- .+ = ( +P ` K )
Assertion elpadd2at 
|- ( ( K e. Lat /\ Q e. A /\ R e. A ) -> ( S e. ( { Q } .+ { R } ) <-> ( S e. A /\ 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 simp1 
 |-  ( ( K e. Lat /\ Q e. A /\ R e. A ) -> K e. Lat )
6 simp2 
 |-  ( ( K e. Lat /\ Q e. A /\ R e. A ) -> Q e. A )
7 6 snssd 
 |-  ( ( K e. Lat /\ Q e. A /\ R e. A ) -> { Q } C_ A )
8 simp3 
 |-  ( ( K e. Lat /\ Q e. A /\ R e. A ) -> R e. A )
9 snnzg 
 |-  ( Q e. A -> { Q } =/= (/) )
10 9 3ad2ant2 
 |-  ( ( K e. Lat /\ Q e. A /\ R e. A ) -> { Q } =/= (/) )
11 1 2 3 4 elpaddat 
 |-  ( ( ( K e. Lat /\ { Q } C_ A /\ R e. A ) /\ { Q } =/= (/) ) -> ( S e. ( { Q } .+ { R } ) <-> ( S e. A /\ E. r e. { Q } S .<_ ( r .\/ R ) ) ) )
12 5 7 8 10 11 syl31anc 
 |-  ( ( K e. Lat /\ Q e. A /\ R e. A ) -> ( S e. ( { Q } .+ { R } ) <-> ( S e. A /\ E. r e. { Q } S .<_ ( r .\/ R ) ) ) )
13 oveq1 
 |-  ( r = Q -> ( r .\/ R ) = ( Q .\/ R ) )
14 13 breq2d 
 |-  ( r = Q -> ( S .<_ ( r .\/ R ) <-> S .<_ ( Q .\/ R ) ) )
15 14 rexsng 
 |-  ( Q e. A -> ( E. r e. { Q } S .<_ ( r .\/ R ) <-> S .<_ ( Q .\/ R ) ) )
16 15 3ad2ant2 
 |-  ( ( K e. Lat /\ Q e. A /\ R e. A ) -> ( E. r e. { Q } S .<_ ( r .\/ R ) <-> S .<_ ( Q .\/ R ) ) )
17 16 anbi2d 
 |-  ( ( K e. Lat /\ Q e. A /\ R e. A ) -> ( ( S e. A /\ E. r e. { Q } S .<_ ( r .\/ R ) ) <-> ( S e. A /\ S .<_ ( Q .\/ R ) ) ) )
18 12 17 bitrd 
 |-  ( ( K e. Lat /\ Q e. A /\ R e. A ) -> ( S e. ( { Q } .+ { R } ) <-> ( S e. A /\ S .<_ ( Q .\/ R ) ) ) )