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Metamath Proof Explorer

Theorem pmaplubN

Description: The LUB of a projective map is the projective map's argument. (Contributed by NM, 13-Mar-2012) (New usage is discouraged.)

Ref Expression
Hypotheses pmaplub.b 
|- B = ( Base ` K )
pmaplub.u 
|- U = ( lub ` K )
pmaplub.m 
|- M = ( pmap ` K )
Assertion pmaplubN 
|- ( ( K e. HL /\ X e. B ) -> ( U ` ( M ` X ) ) = X )

Proof

Step Hyp Ref Expression
1 pmaplub.b 
 |-  B = ( Base ` K )
2 pmaplub.u 
 |-  U = ( lub ` K )
3 pmaplub.m 
 |-  M = ( pmap ` K )
4 eqid 
 |-  ( le ` K ) = ( le ` K )
5 eqid 
 |-  ( Atoms ` K ) = ( Atoms ` K )
6 1 4 5 3 pmapval 
 |-  ( ( K e. HL /\ X e. B ) -> ( M ` X ) = { p e. ( Atoms ` K ) | p ( le ` K ) X } )
7 6 fveq2d 
 |-  ( ( K e. HL /\ X e. B ) -> ( U ` ( M ` X ) ) = ( U ` { p e. ( Atoms ` K ) | p ( le ` K ) X } ) )
8 hlomcmat 
 |-  ( K e. HL -> ( K e. OML /\ K e. CLat /\ K e. AtLat ) )
9 1 4 2 5 atlatmstc 
 |-  ( ( ( K e. OML /\ K e. CLat /\ K e. AtLat ) /\ X e. B ) -> ( U ` { p e. ( Atoms ` K ) | p ( le ` K ) X } ) = X )
10 8 9 sylan 
 |-  ( ( K e. HL /\ X e. B ) -> ( U ` { p e. ( Atoms ` K ) | p ( le ` K ) X } ) = X )
11 7 10 eqtrd 
 |-  ( ( K e. HL /\ X e. B ) -> ( U ` ( M ` X ) ) = X )