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 } ) )