Metamath Proof Explorer

Theorem elpaddat

Description: Membership in a projective subspace sum with a point. (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 elpaddat 
|- ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ X =/= (/) ) -> ( S e. ( X .+ { Q } ) <-> ( S e. A /\ E. p e. X S .<_ ( p .\/ Q ) ) ) )

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 simpl1 
 |-  ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ X =/= (/) ) -> K e. Lat )
6 simpl2 
 |-  ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ X =/= (/) ) -> X C_ A )
7 simpl3 
 |-  ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ X =/= (/) ) -> Q e. A )
8 7 snssd 
 |-  ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ X =/= (/) ) -> { Q } C_ A )
9 simpr 
 |-  ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ X =/= (/) ) -> X =/= (/) )
10 snnzg 
 |-  ( Q e. A -> { Q } =/= (/) )
11 7 10 syl 
 |-  ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ X =/= (/) ) -> { Q } =/= (/) )
12 1 2 3 4 elpaddn0 
 |-  ( ( ( K e. Lat /\ X C_ A /\ { Q } C_ A ) /\ ( X =/= (/) /\ { Q } =/= (/) ) ) -> ( S e. ( X .+ { Q } ) <-> ( S e. A /\ E. p e. X E. r e. { Q } S .<_ ( p .\/ r ) ) ) )
13 5 6 8 9 11 12 syl32anc 
 |-  ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ X =/= (/) ) -> ( S e. ( X .+ { Q } ) <-> ( S e. A /\ E. p e. X E. r e. { Q } S .<_ ( p .\/ r ) ) ) )
14 oveq2 
 |-  ( r = Q -> ( p .\/ r ) = ( p .\/ Q ) )
15 14 breq2d 
 |-  ( r = Q -> ( S .<_ ( p .\/ r ) <-> S .<_ ( p .\/ Q ) ) )
16 15 rexsng 
 |-  ( Q e. A -> ( E. r e. { Q } S .<_ ( p .\/ r ) <-> S .<_ ( p .\/ Q ) ) )
17 7 16 syl 
 |-  ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ X =/= (/) ) -> ( E. r e. { Q } S .<_ ( p .\/ r ) <-> S .<_ ( p .\/ Q ) ) )
18 17 rexbidv 
 |-  ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ X =/= (/) ) -> ( E. p e. X E. r e. { Q } S .<_ ( p .\/ r ) <-> E. p e. X S .<_ ( p .\/ Q ) ) )
19 18 anbi2d 
 |-  ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ X =/= (/) ) -> ( ( S e. A /\ E. p e. X E. r e. { Q } S .<_ ( p .\/ r ) ) <-> ( S e. A /\ E. p e. X S .<_ ( p .\/ Q ) ) ) )
20 13 19 bitrd 
 |-  ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ X =/= (/) ) -> ( S e. ( X .+ { Q } ) <-> ( S e. A /\ E. p e. X S .<_ ( p .\/ Q ) ) ) )