This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem dih1rn

Description: The full vector space belongs to the range of isomorphism H. (Contributed by NM, 19-Jun-2014)

Ref Expression
Hypotheses dih1rn.h 
|- H = ( LHyp ` K )
dih1rn.i 
|- I = ( ( DIsoH ` K ) ` W )
dih1rn.u 
|- U = ( ( DVecH ` K ) ` W )
dih1rn.v 
|- V = ( Base ` U )
Assertion dih1rn 
|- ( ( K e. HL /\ W e. H ) -> V e. ran I )

Proof

Step Hyp Ref Expression
1 dih1rn.h 
 |-  H = ( LHyp ` K )
2 dih1rn.i 
 |-  I = ( ( DIsoH ` K ) ` W )
3 dih1rn.u 
 |-  U = ( ( DVecH ` K ) ` W )
4 dih1rn.v 
 |-  V = ( Base ` U )
5 eqid 
 |-  ( 1. ` K ) = ( 1. ` K )
6 5 1 2 3 4 dih1 
 |-  ( ( K e. HL /\ W e. H ) -> ( I ` ( 1. ` K ) ) = V )
7 hlop 
 |-  ( K e. HL -> K e. OP )
8 7 adantr 
 |-  ( ( K e. HL /\ W e. H ) -> K e. OP )
9 eqid 
 |-  ( Base ` K ) = ( Base ` K )
10 9 5 op1cl 
 |-  ( K e. OP -> ( 1. ` K ) e. ( Base ` K ) )
11 8 10 syl 
 |-  ( ( K e. HL /\ W e. H ) -> ( 1. ` K ) e. ( Base ` K ) )
12 9 1 2 dihcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( 1. ` K ) e. ( Base ` K ) ) -> ( I ` ( 1. ` K ) ) e. ran I )
13 11 12 mpdan 
 |-  ( ( K e. HL /\ W e. H ) -> ( I ` ( 1. ` K ) ) e. ran I )
14 6 13 eqeltrrd 
 |-  ( ( K e. HL /\ W e. H ) -> V e. ran I )