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Metamath Proof Explorer

Theorem pclidN

Description: The projective subspace closure of a projective subspace is itself. (Contributed by NM, 8-Sep-2013) (New usage is discouraged.)

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

Proof

Step Hyp Ref Expression
1 pclid.s 
 |-  S = ( PSubSp ` K )
2 pclid.c 
 |-  U = ( PCl ` K )
3 eqid 
 |-  ( Atoms ` K ) = ( Atoms ` K )
4 3 1 psubssat 
 |-  ( ( K e. V /\ X e. S ) -> X C_ ( Atoms ` K ) )
5 3 1 2 pclvalN 
 |-  ( ( K e. V /\ X C_ ( Atoms ` K ) ) -> ( U ` X ) = |^| { y e. S | X C_ y } )
6 4 5 syldan 
 |-  ( ( K e. V /\ X e. S ) -> ( U ` X ) = |^| { y e. S | X C_ y } )
7 intmin 
 |-  ( X e. S -> |^| { y e. S | X C_ y } = X )
8 7 adantl 
 |-  ( ( K e. V /\ X e. S ) -> |^| { y e. S | X C_ y } = X )
9 6 8 eqtrd 
 |-  ( ( K e. V /\ X e. S ) -> ( U ` X ) = X )