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Metamath Proof Explorer

Theorem pmapidclN

Description: Projective map of the LUB of a closed subspace. (Contributed by NM, 3-Feb-2012) (New usage is discouraged.)

Ref Expression
Hypotheses pmapidcl.u 
|- U = ( lub ` K )
pmapidcl.m 
|- M = ( pmap ` K )
pmapidcl.c 
|- C = ( PSubCl ` K )
Assertion pmapidclN 
|- ( ( K e. HL /\ X e. C ) -> ( M ` ( U ` X ) ) = X )

Proof

Step Hyp Ref Expression
1 pmapidcl.u 
 |-  U = ( lub ` K )
2 pmapidcl.m 
 |-  M = ( pmap ` K )
3 pmapidcl.c 
 |-  C = ( PSubCl ` K )
4 eqid 
 |-  ( Atoms ` K ) = ( Atoms ` K )
5 4 3 psubclssatN 
 |-  ( ( K e. HL /\ X e. C ) -> X C_ ( Atoms ` K ) )
6 eqid 
 |-  ( _|_P ` K ) = ( _|_P ` K )
7 1 4 2 6 2polvalN 
 |-  ( ( K e. HL /\ X C_ ( Atoms ` K ) ) -> ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` X ) ) = ( M ` ( U ` X ) ) )
8 5 7 syldan 
 |-  ( ( K e. HL /\ X e. C ) -> ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` X ) ) = ( M ` ( U ` X ) ) )
9 6 3 psubcli2N 
 |-  ( ( K e. HL /\ X e. C ) -> ( ( _|_P ` K ) ` ( ( _|_P ` K ) ` X ) ) = X )
10 8 9 eqtr3d 
 |-  ( ( K e. HL /\ X e. C ) -> ( M ` ( U ` X ) ) = X )