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Metamath Proof Explorer

Theorem dihmeetlem4N

Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014) (New usage is discouraged.)

Ref Expression
Hypotheses dihmeetlem4.b 
|- B = ( Base ` K )
dihmeetlem4.l 
|- .<_ = ( le ` K )
dihmeetlem4.m 
|- ./\ = ( meet ` K )
dihmeetlem4.a 
|- A = ( Atoms ` K )
dihmeetlem4.h 
|- H = ( LHyp ` K )
dihmeetlem4.i 
|- I = ( ( DIsoH ` K ) ` W )
dihmeetlem4.u 
|- U = ( ( DVecH ` K ) ` W )
dihmeetlem4.z 
|- .0. = ( 0g ` U )
Assertion dihmeetlem4N 
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( I ` Q ) i^i ( I ` ( X ./\ W ) ) ) = { .0. } )

Proof

Step Hyp Ref Expression
1 dihmeetlem4.b 
 |-  B = ( Base ` K )
2 dihmeetlem4.l 
 |-  .<_ = ( le ` K )
3 dihmeetlem4.m 
 |-  ./\ = ( meet ` K )
4 dihmeetlem4.a 
 |-  A = ( Atoms ` K )
5 dihmeetlem4.h 
 |-  H = ( LHyp ` K )
6 dihmeetlem4.i 
 |-  I = ( ( DIsoH ` K ) ` W )
7 dihmeetlem4.u 
 |-  U = ( ( DVecH ` K ) ` W )
8 dihmeetlem4.z 
 |-  .0. = ( 0g ` U )
9 eqid 
 |-  ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q ) = ( iota_ g e. ( ( LTrn ` K ) ` W ) ( g ` ( ( oc ` K ) ` W ) ) = Q )
10 eqid 
 |-  ( ( oc ` K ) ` W ) = ( ( oc ` K ) ` W )
11 eqid 
 |-  ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
12 eqid 
 |-  ( ( trL ` K ) ` W ) = ( ( trL ` K ) ` W )
13 eqid 
 |-  ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W )
14 eqid 
 |-  ( h e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) = ( h e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) )
15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 dihmeetlem4preN 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ -. X .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( ( I ` Q ) i^i ( I ` ( X ./\ W ) ) ) = { .0. } )