# 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 ) )`