Metamath Proof Explorer

Theorem dihvalb

Description: Value of isomorphism H for a lattice K when X .<_ W . (Contributed by NM, 4-Mar-2014)

Ref Expression
Hypotheses dihvalb.b 
|- B = ( Base ` K )
dihvalb.l 
|- .<_ = ( le ` K )
dihvalb.h 
|- H = ( LHyp ` K )
dihvalb.i 
|- I = ( ( DIsoH ` K ) ` W )
dihvalb.d 
|- D = ( ( DIsoB ` K ) ` W )
Assertion dihvalb 
|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) = ( D ` X ) )

Proof

Step Hyp Ref Expression
1 dihvalb.b 
 |-  B = ( Base ` K )
2 dihvalb.l 
 |-  .<_ = ( le ` K )
3 dihvalb.h 
 |-  H = ( LHyp ` K )
4 dihvalb.i 
 |-  I = ( ( DIsoH ` K ) ` W )
5 dihvalb.d 
 |-  D = ( ( DIsoB ` K ) ` W )
6 eqid 
 |-  ( join ` K ) = ( join ` K )
7 eqid 
 |-  ( meet ` K ) = ( meet ` K )
8 eqid 
 |-  ( Atoms ` K ) = ( Atoms ` K )
9 eqid 
 |-  ( ( DIsoC ` K ) ` W ) = ( ( DIsoC ` K ) ` W )
10 eqid 
 |-  ( ( DVecH ` K ) ` W ) = ( ( DVecH ` K ) ` W )
11 eqid 
 |-  ( LSubSp ` ( ( DVecH ` K ) ` W ) ) = ( LSubSp ` ( ( DVecH ` K ) ` W ) )
12 eqid 
 |-  ( LSSum ` ( ( DVecH ` K ) ` W ) ) = ( LSSum ` ( ( DVecH ` K ) ` W ) )
13 1 2 6 7 8 3 4 5 9 10 11 12 dihval 
 |-  ( ( ( K e. V /\ W e. H ) /\ X e. B ) -> ( I ` X ) = if ( X .<_ W , ( D ` X ) , ( iota_ u e. ( LSubSp ` ( ( DVecH ` K ) ` W ) ) A. q e. ( Atoms ` K ) ( ( -. q .<_ W /\ ( q ( join ` K ) ( X ( meet ` K ) W ) ) = X ) -> u = ( ( ( ( DIsoC ` K ) ` W ) ` q ) ( LSSum ` ( ( DVecH ` K ) ` W ) ) ( D ` ( X ( meet ` K ) W ) ) ) ) ) ) )
14 iftrue 
 |-  ( X .<_ W -> if ( X .<_ W , ( D ` X ) , ( iota_ u e. ( LSubSp ` ( ( DVecH ` K ) ` W ) ) A. q e. ( Atoms ` K ) ( ( -. q .<_ W /\ ( q ( join ` K ) ( X ( meet ` K ) W ) ) = X ) -> u = ( ( ( ( DIsoC ` K ) ` W ) ` q ) ( LSSum ` ( ( DVecH ` K ) ` W ) ) ( D ` ( X ( meet ` K ) W ) ) ) ) ) ) = ( D ` X ) )
15 13 14 sylan9eq 
 |-  ( ( ( ( K e. V /\ W e. H ) /\ X e. B ) /\ X .<_ W ) -> ( I ` X ) = ( D ` X ) )
16 15 anasss 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) = ( D ` X ) )