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 )