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Metamath Proof Explorer

Theorem dihglb

Description: Isomorphism H of a lattice glb. (Contributed by NM, 11-Apr-2014)

Ref Expression
Hypotheses dihglb.b 
|- B = ( Base ` K )
dihglb.g 
|- G = ( glb ` K )
dihglb.h 
|- H = ( LHyp ` K )
dihglb.i 
|- I = ( ( DIsoH ` K ) ` W )
Assertion dihglb 
|- ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) -> ( I ` ( G ` S ) ) = |^|_ x e. S ( I ` x ) )

Proof

Step Hyp Ref Expression
1 dihglb.b 
 |-  B = ( Base ` K )
2 dihglb.g 
 |-  G = ( glb ` K )
3 dihglb.h 
 |-  H = ( LHyp ` K )
4 dihglb.i 
 |-  I = ( ( DIsoH ` K ) ` W )
5 eqid 
 |-  ( le ` K ) = ( le ` K )
6 eqid 
 |-  ( meet ` K ) = ( meet ` K )
7 eqid 
 |-  ( Atoms ` K ) = ( Atoms ` K )
8 eqid 
 |-  ( ( DVecH ` K ) ` W ) = ( ( DVecH ` K ) ` W )
9 eqid 
 |-  ( LSubSp ` ( ( DVecH ` K ) ` W ) ) = ( LSubSp ` ( ( DVecH ` K ) ` W ) )
10 eqid 
 |-  ( LSAtoms ` ( ( DVecH ` K ) ` W ) ) = ( LSAtoms ` ( ( DVecH ` K ) ` W ) )
11 1 5 6 7 2 3 4 8 9 10 dihglblem6 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) ) -> ( I ` ( G ` S ) ) = |^|_ x e. S ( I ` x ) )