Metamath Proof Explorer

Theorem lcd1

Description: The unit scalar of the closed kernel dual of a vector space. (Contributed by NM, 20-Mar-2015)

Ref Expression
Hypotheses lcd1.h 
|- H = ( LHyp ` K )
lcd1.u 
|- U = ( ( DVecH ` K ) ` W )
lcd1.f 
|- F = ( Scalar ` U )
lcd1.j 
|- .1. = ( 1r ` F )
lcd1.c 
|- C = ( ( LCDual ` K ) ` W )
lcd1.s 
|- S = ( Scalar ` C )
lcd1.i 
|- I = ( 1r ` S )
lcd1.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
Assertion lcd1 
|- ( ph -> I = .1. )

Proof

Step Hyp Ref Expression
1 lcd1.h 
 |-  H = ( LHyp ` K )
2 lcd1.u 
 |-  U = ( ( DVecH ` K ) ` W )
3 lcd1.f 
 |-  F = ( Scalar ` U )
4 lcd1.j 
 |-  .1. = ( 1r ` F )
5 lcd1.c 
 |-  C = ( ( LCDual ` K ) ` W )
6 lcd1.s 
 |-  S = ( Scalar ` C )
7 lcd1.i 
 |-  I = ( 1r ` S )
8 lcd1.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
9 eqid 
 |-  ( oppR ` F ) = ( oppR ` F )
10 1 2 3 9 5 6 8 lcdsca 
 |-  ( ph -> S = ( oppR ` F ) )
11 10 fveq2d 
 |-  ( ph -> ( 1r ` S ) = ( 1r ` ( oppR ` F ) ) )
12 9 4 oppr1 
 |-  .1. = ( 1r ` ( oppR ` F ) )
13 11 7 12 3eqtr4g 
 |-  ( ph -> I = .1. )