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Metamath Proof Explorer

Theorem elpaddatriN

Description: Condition implying membership in a projective subspace sum with a point. (Contributed by NM, 1-Feb-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 elpaddatriN 
|- ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ ( R e. X /\ S e. A /\ S .<_ ( R .\/ Q ) ) ) -> S e. ( X .+ { 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 ) /\ ( R e. X /\ S e. A /\ S .<_ ( R .\/ Q ) ) ) -> K e. Lat )
6 simpl2 
 |-  ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ ( R e. X /\ S e. A /\ S .<_ ( R .\/ Q ) ) ) -> X C_ A )
7 simpl3 
 |-  ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ ( R e. X /\ S e. A /\ S .<_ ( R .\/ Q ) ) ) -> Q e. A )
8 7 snssd 
 |-  ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ ( R e. X /\ S e. A /\ S .<_ ( R .\/ Q ) ) ) -> { Q } C_ A )
9 simpr1 
 |-  ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ ( R e. X /\ S e. A /\ S .<_ ( R .\/ Q ) ) ) -> R e. X )
10 snidg 
 |-  ( Q e. A -> Q e. { Q } )
11 7 10 syl 
 |-  ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ ( R e. X /\ S e. A /\ S .<_ ( R .\/ Q ) ) ) -> Q e. { Q } )
12 simpr2 
 |-  ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ ( R e. X /\ S e. A /\ S .<_ ( R .\/ Q ) ) ) -> S e. A )
13 simpr3 
 |-  ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ ( R e. X /\ S e. A /\ S .<_ ( R .\/ Q ) ) ) -> S .<_ ( R .\/ Q ) )
14 1 2 3 4 elpaddri 
 |-  ( ( ( K e. Lat /\ X C_ A /\ { Q } C_ A ) /\ ( R e. X /\ Q e. { Q } ) /\ ( S e. A /\ S .<_ ( R .\/ Q ) ) ) -> S e. ( X .+ { Q } ) )
15 5 6 8 9 11 12 13 14 syl322anc 
 |-  ( ( ( K e. Lat /\ X C_ A /\ Q e. A ) /\ ( R e. X /\ S e. A /\ S .<_ ( R .\/ Q ) ) ) -> S e. ( X .+ { Q } ) )