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Metamath Proof Explorer

Theorem dihglbcN

Description: Isomorphism H of a lattice glb when the glb is not under the fiducial hyperplane W . (Contributed by NM, 26-Mar-2014) (New usage is discouraged.)

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

Proof

Step Hyp Ref Expression
1 dihglbc.b 
 |-  B = ( Base ` K )
2 dihglbc.g 
 |-  G = ( glb ` K )
3 dihglbc.h 
 |-  H = ( LHyp ` K )
4 dihglbc.i 
 |-  I = ( ( DIsoH ` K ) ` W )
5 dihglbc.l 
 |-  .<_ = ( le ` K )
6 eqid 
 |-  ( join ` K ) = ( join ` K )
7 eqid 
 |-  ( meet ` K ) = ( meet ` K )
8 eqid 
 |-  ( Atoms ` K ) = ( Atoms ` K )
9 eqid 
 |-  ( ( oc ` K ) ` W ) = ( ( oc ` K ) ` W )
10 eqid 
 |-  ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
11 eqid 
 |-  ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
12 eqid 
 |-  ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
13 eqid 
 |-  ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = q ) = ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = q )
14 1 2 3 4 5 6 7 8 9 10 11 12 13 dihglbcpreN 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( S C_ B /\ S =/= (/) ) /\ -. ( G ` S ) .<_ W ) -> ( I ` ( G ` S ) ) = |^|_ x e. S ( I ` x ) )