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Metamath Proof Explorer

Theorem dih1cnv

Description: The isomorphism H converse value of the full vector space is the lattice one. (Contributed by NM, 19-Jun-2014)

Ref Expression
Hypotheses dih1cnv.h 
|- H = ( LHyp ` K )
dih1cnv.m 
|- .1. = ( 1. ` K )
dih1cnv.i 
|- I = ( ( DIsoH ` K ) ` W )
dih1cnv.u 
|- U = ( ( DVecH ` K ) ` W )
dih1cnv.v 
|- V = ( Base ` U )
Assertion dih1cnv 
|- ( ( K e. HL /\ W e. H ) -> ( `' I ` V ) = .1. )

Proof

Step Hyp Ref Expression
1 dih1cnv.h 
 |-  H = ( LHyp ` K )
2 dih1cnv.m 
 |-  .1. = ( 1. ` K )
3 dih1cnv.i 
 |-  I = ( ( DIsoH ` K ) ` W )
4 dih1cnv.u 
 |-  U = ( ( DVecH ` K ) ` W )
5 dih1cnv.v 
 |-  V = ( Base ` U )
6 2 1 3 4 5 dih1 
 |-  ( ( K e. HL /\ W e. H ) -> ( I ` .1. ) = V )
7 6 fveq2d 
 |-  ( ( K e. HL /\ W e. H ) -> ( `' I ` ( I ` .1. ) ) = ( `' I ` V ) )
8 hlop 
 |-  ( K e. HL -> K e. OP )
9 8 adantr 
 |-  ( ( K e. HL /\ W e. H ) -> K e. OP )
10 eqid 
 |-  ( Base ` K ) = ( Base ` K )
11 10 2 op1cl 
 |-  ( K e. OP -> .1. e. ( Base ` K ) )
12 9 11 syl 
 |-  ( ( K e. HL /\ W e. H ) -> .1. e. ( Base ` K ) )
13 10 1 3 dihcnvid1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ .1. e. ( Base ` K ) ) -> ( `' I ` ( I ` .1. ) ) = .1. )
14 12 13 mpdan 
 |-  ( ( K e. HL /\ W e. H ) -> ( `' I ` ( I ` .1. ) ) = .1. )
15 7 14 eqtr3d 
 |-  ( ( K e. HL /\ W e. H ) -> ( `' I ` V ) = .1. )