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Metamath Proof Explorer

Theorem pclbtwnN

Description: A projective subspace sandwiched between a set of atoms and the set's projective subspace closure equals the closure. (Contributed by NM, 8-Sep-2013) (New usage is discouraged.)

Ref Expression
Hypotheses pclid.s 
|- S = ( PSubSp ` K )
pclid.c 
|- U = ( PCl ` K )
Assertion pclbtwnN 
|- ( ( ( K e. V /\ X e. S ) /\ ( Y C_ X /\ X C_ ( U ` Y ) ) ) -> X = ( U ` Y ) )

Proof

Step Hyp Ref Expression
1 pclid.s 
 |-  S = ( PSubSp ` K )
2 pclid.c 
 |-  U = ( PCl ` K )
3 simprr 
 |-  ( ( ( K e. V /\ X e. S ) /\ ( Y C_ X /\ X C_ ( U ` Y ) ) ) -> X C_ ( U ` Y ) )
4 simpll 
 |-  ( ( ( K e. V /\ X e. S ) /\ ( Y C_ X /\ X C_ ( U ` Y ) ) ) -> K e. V )
5 simprl 
 |-  ( ( ( K e. V /\ X e. S ) /\ ( Y C_ X /\ X C_ ( U ` Y ) ) ) -> Y C_ X )
6 eqid 
 |-  ( Atoms ` K ) = ( Atoms ` K )
7 6 1 psubssat 
 |-  ( ( K e. V /\ X e. S ) -> X C_ ( Atoms ` K ) )
8 7 adantr 
 |-  ( ( ( K e. V /\ X e. S ) /\ ( Y C_ X /\ X C_ ( U ` Y ) ) ) -> X C_ ( Atoms ` K ) )
9 6 2 pclssN 
 |-  ( ( K e. V /\ Y C_ X /\ X C_ ( Atoms ` K ) ) -> ( U ` Y ) C_ ( U ` X ) )
10 4 5 8 9 syl3anc 
 |-  ( ( ( K e. V /\ X e. S ) /\ ( Y C_ X /\ X C_ ( U ` Y ) ) ) -> ( U ` Y ) C_ ( U ` X ) )
11 1 2 pclidN 
 |-  ( ( K e. V /\ X e. S ) -> ( U ` X ) = X )
12 11 adantr 
 |-  ( ( ( K e. V /\ X e. S ) /\ ( Y C_ X /\ X C_ ( U ` Y ) ) ) -> ( U ` X ) = X )
13 10 12 sseqtrd 
 |-  ( ( ( K e. V /\ X e. S ) /\ ( Y C_ X /\ X C_ ( U ` Y ) ) ) -> ( U ` Y ) C_ X )
14 3 13 eqssd 
 |-  ( ( ( K e. V /\ X e. S ) /\ ( Y C_ X /\ X C_ ( U ` Y ) ) ) -> X = ( U ` Y ) )