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Metamath Proof Explorer

Theorem dih0rn

Description: The zero subspace belongs to the range of isomorphism H. (Contributed by NM, 27-Apr-2014)

Ref Expression
Hypotheses dih0rn.h 
|- H = ( LHyp ` K )
dih0rn.i 
|- I = ( ( DIsoH ` K ) ` W )
dih0rn.u 
|- U = ( ( DVecH ` K ) ` W )
dih0rn.o 
|- .0. = ( 0g ` U )
Assertion dih0rn 
|- ( ( K e. HL /\ W e. H ) -> { .0. } e. ran I )

Proof

Step Hyp Ref Expression
1 dih0rn.h 
 |-  H = ( LHyp ` K )
2 dih0rn.i 
 |-  I = ( ( DIsoH ` K ) ` W )
3 dih0rn.u 
 |-  U = ( ( DVecH ` K ) ` W )
4 dih0rn.o 
 |-  .0. = ( 0g ` U )
5 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
6 5 1 2 3 4 dih0 
 |-  ( ( K e. HL /\ W e. H ) -> ( I ` ( 0. ` K ) ) = { .0. } )
7 eqid 
 |-  ( Base ` K ) = ( Base ` K )
8 7 1 2 dihfn 
 |-  ( ( K e. HL /\ W e. H ) -> I Fn ( Base ` K ) )
9 hlop 
 |-  ( K e. HL -> K e. OP )
10 9 adantr 
 |-  ( ( K e. HL /\ W e. H ) -> K e. OP )
11 7 5 op0cl 
 |-  ( K e. OP -> ( 0. ` K ) e. ( Base ` K ) )
12 10 11 syl 
 |-  ( ( K e. HL /\ W e. H ) -> ( 0. ` K ) e. ( Base ` K ) )
13 fnfvelrn 
 |-  ( ( I Fn ( Base ` K ) /\ ( 0. ` K ) e. ( Base ` K ) ) -> ( I ` ( 0. ` K ) ) e. ran I )
14 8 12 13 syl2anc 
 |-  ( ( K e. HL /\ W e. H ) -> ( I ` ( 0. ` K ) ) e. ran I )
15 6 14 eqeltrrd 
 |-  ( ( K e. HL /\ W e. H ) -> { .0. } e. ran I )