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Metamath Proof Explorer

Theorem pclun2N

Description: The projective subspace closure of the union of two subspaces equals their projective sum. (Contributed by NM, 12-Sep-2013) (New usage is discouraged.)

Ref Expression
Hypotheses pclun2.s 
|- S = ( PSubSp ` K )
pclun2.p 
|- .+ = ( +P ` K )
pclun2.c 
|- U = ( PCl ` K )
Assertion pclun2N 
|- ( ( K e. HL /\ X e. S /\ Y e. S ) -> ( U ` ( X u. Y ) ) = ( X .+ Y ) )

Proof

Step Hyp Ref Expression
1 pclun2.s 
 |-  S = ( PSubSp ` K )
2 pclun2.p 
 |-  .+ = ( +P ` K )
3 pclun2.c 
 |-  U = ( PCl ` K )
4 simp1 
 |-  ( ( K e. HL /\ X e. S /\ Y e. S ) -> K e. HL )
5 eqid 
 |-  ( Atoms ` K ) = ( Atoms ` K )
6 5 1 psubssat 
 |-  ( ( K e. HL /\ X e. S ) -> X C_ ( Atoms ` K ) )
7 6 3adant3 
 |-  ( ( K e. HL /\ X e. S /\ Y e. S ) -> X C_ ( Atoms ` K ) )
8 5 1 psubssat 
 |-  ( ( K e. HL /\ Y e. S ) -> Y C_ ( Atoms ` K ) )
9 8 3adant2 
 |-  ( ( K e. HL /\ X e. S /\ Y e. S ) -> Y C_ ( Atoms ` K ) )
10 5 2 3 pclunN 
 |-  ( ( K e. HL /\ X C_ ( Atoms ` K ) /\ Y C_ ( Atoms ` K ) ) -> ( U ` ( X u. Y ) ) = ( U ` ( X .+ Y ) ) )
11 4 7 9 10 syl3anc 
 |-  ( ( K e. HL /\ X e. S /\ Y e. S ) -> ( U ` ( X u. Y ) ) = ( U ` ( X .+ Y ) ) )
12 1 2 paddclN 
 |-  ( ( K e. HL /\ X e. S /\ Y e. S ) -> ( X .+ Y ) e. S )
13 1 3 pclidN 
 |-  ( ( K e. HL /\ ( X .+ Y ) e. S ) -> ( U ` ( X .+ Y ) ) = ( X .+ Y ) )
14 4 12 13 syl2anc 
 |-  ( ( K e. HL /\ X e. S /\ Y e. S ) -> ( U ` ( X .+ Y ) ) = ( X .+ Y ) )
15 11 14 eqtrd 
 |-  ( ( K e. HL /\ X e. S /\ Y e. S ) -> ( U ` ( X u. Y ) ) = ( X .+ Y ) )