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Metamath Proof Explorer

Theorem dih0cnv

Description: The isomorphism H converse value of the zero subspace is the lattice zero. (Contributed by NM, 19-Jun-2014)

Ref Expression
Hypotheses dih0cnv.h 
|- H = ( LHyp ` K )
dih0cnv.o 
|- .0. = ( 0. ` K )
dih0cnv.i 
|- I = ( ( DIsoH ` K ) ` W )
dih0cnv.u 
|- U = ( ( DVecH ` K ) ` W )
dih0cnv.z 
|- Z = ( 0g ` U )
Assertion dih0cnv 
|- ( ( K e. HL /\ W e. H ) -> ( `' I ` { Z } ) = .0. )

Proof

Step Hyp Ref Expression
1 dih0cnv.h 
 |-  H = ( LHyp ` K )
2 dih0cnv.o 
 |-  .0. = ( 0. ` K )
3 dih0cnv.i 
 |-  I = ( ( DIsoH ` K ) ` W )
4 dih0cnv.u 
 |-  U = ( ( DVecH ` K ) ` W )
5 dih0cnv.z 
 |-  Z = ( 0g ` U )
6 2 1 3 4 5 dih0 
 |-  ( ( K e. HL /\ W e. H ) -> ( I ` .0. ) = { Z } )
7 6 fveq2d 
 |-  ( ( K e. HL /\ W e. H ) -> ( `' I ` ( I ` .0. ) ) = ( `' I ` { Z } ) )
8 hlatl 
 |-  ( K e. HL -> K e. AtLat )
9 8 adantr 
 |-  ( ( K e. HL /\ W e. H ) -> K e. AtLat )
10 eqid 
 |-  ( Base ` K ) = ( Base ` K )
11 10 2 atl0cl 
 |-  ( K e. AtLat -> .0. e. ( Base ` K ) )
12 9 11 syl 
 |-  ( ( K e. HL /\ W e. H ) -> .0. e. ( Base ` K ) )
13 10 1 3 dihcnvid1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ .0. e. ( Base ` K ) ) -> ( `' I ` ( I ` .0. ) ) = .0. )
14 12 13 mpdan 
 |-  ( ( K e. HL /\ W e. H ) -> ( `' I ` ( I ` .0. ) ) = .0. )
15 7 14 eqtr3d 
 |-  ( ( K e. HL /\ W e. H ) -> ( `' I ` { Z } ) = .0. )