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Metamath Proof Explorer

Theorem pmapsubclN

Description: A projective map value is a closed projective subspace. (Contributed by NM, 24-Jan-2012) (New usage is discouraged.)

Ref Expression
Hypotheses pmapsubcl.b 
|- B = ( Base ` K )
pmapsubcl.m 
|- M = ( pmap ` K )
pmapsubcl.c 
|- C = ( PSubCl ` K )
Assertion pmapsubclN 
|- ( ( K e. HL /\ X e. B ) -> ( M ` X ) e. C )

Proof

Step Hyp Ref Expression
1 pmapsubcl.b 
 |-  B = ( Base ` K )
2 pmapsubcl.m 
 |-  M = ( pmap ` K )
3 pmapsubcl.c 
 |-  C = ( PSubCl ` K )
4 eqid 
 |-  ( Atoms ` K ) = ( Atoms ` K )
5 1 4 2 pmapssat 
 |-  ( ( K e. HL /\ X e. B ) -> ( M ` X ) C_ ( Atoms ` K ) )
6 eqid 
 |-  ( _|_P ` K ) = ( _|_P ` K )
7 1 2 6 2polpmapN 
 |-  ( ( K e. HL /\ X e. B ) -> ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` ( M ` X ) ) ) = ( M ` X ) )
8 4 6 3 ispsubclN 
 |-  ( K e. HL -> ( ( M ` X ) e. C <-> ( ( M ` X ) C_ ( Atoms ` K ) /\ ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` ( M ` X ) ) ) = ( M ` X ) ) ) )
9 8 adantr 
 |-  ( ( K e. HL /\ X e. B ) -> ( ( M ` X ) e. C <-> ( ( M ` X ) C_ ( Atoms ` K ) /\ ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` ( M ` X ) ) ) = ( M ` X ) ) ) )
10 5 7 9 mpbir2and 
 |-  ( ( K e. HL /\ X e. B ) -> ( M ` X ) e. C )