Metamath Proof Explorer

Theorem hdmapevec

Description: Value of map from vectors to functionals at the reference vector E . (Contributed by NM, 16-May-2015)

Ref Expression
Hypotheses hdmapevec.h 
|- H = ( LHyp ` K )
hdmapevec.e 
|- E = <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` W ) ) >.
hdmapevec.j 
|- J = ( ( HVMap ` K ) ` W )
hdmapevec.s 
|- S = ( ( HDMap ` K ) ` W )
hdmapevec.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
Assertion hdmapevec 
|- ( ph -> ( S ` E ) = ( J ` E ) )

Proof

Step Hyp Ref Expression
1 hdmapevec.h 
 |-  H = ( LHyp ` K )
2 hdmapevec.e 
 |-  E = <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` W ) ) >.
3 hdmapevec.j 
 |-  J = ( ( HVMap ` K ) ` W )
4 hdmapevec.s 
 |-  S = ( ( HDMap ` K ) ` W )
5 hdmapevec.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
6 eqid 
 |-  ( ( DVecH ` K ) ` W ) = ( ( DVecH ` K ) ` W )
7 eqid 
 |-  ( Base ` ( ( DVecH ` K ) ` W ) ) = ( Base ` ( ( DVecH ` K ) ` W ) )
8 eqid 
 |-  ( LSpan ` ( ( DVecH ` K ) ` W ) ) = ( LSpan ` ( ( DVecH ` K ) ` W ) )
9 eqid 
 |-  ( Base ` K ) = ( Base ` K )
10 eqid 
 |-  ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
11 eqid 
 |-  ( 0g ` ( ( DVecH ` K ) ` W ) ) = ( 0g ` ( ( DVecH ` K ) ` W ) )
12 1 9 10 6 7 11 2 5 dvheveccl 
 |-  ( ph -> E e. ( ( Base ` ( ( DVecH ` K ) ` W ) ) \ { ( 0g ` ( ( DVecH ` K ) ` W ) ) } ) )
13 12 eldifad 
 |-  ( ph -> E e. ( Base ` ( ( DVecH ` K ) ` W ) ) )
14 1 6 7 8 5 13 dvh2dim 
 |-  ( ph -> E. z e. ( Base ` ( ( DVecH ` K ) ` W ) ) -. z e. ( ( LSpan ` ( ( DVecH ` K ) ` W ) ) ` { E } ) )
15 5 3ad2ant1 
 |-  ( ( ph /\ z e. ( Base ` ( ( DVecH ` K ) ` W ) ) /\ -. z e. ( ( LSpan ` ( ( DVecH ` K ) ` W ) ) ` { E } ) ) -> ( K e. HL /\ W e. H ) )
16 eqid 
 |-  ( ( LCDual ` K ) ` W ) = ( ( LCDual ` K ) ` W )
17 eqid 
 |-  ( Base ` ( ( LCDual ` K ) ` W ) ) = ( Base ` ( ( LCDual ` K ) ` W ) )
18 eqid 
 |-  ( ( HDMap1 ` K ) ` W ) = ( ( HDMap1 ` K ) ` W )
19 simp2 
 |-  ( ( ph /\ z e. ( Base ` ( ( DVecH ` K ) ` W ) ) /\ -. z e. ( ( LSpan ` ( ( DVecH ` K ) ` W ) ) ` { E } ) ) -> z e. ( Base ` ( ( DVecH ` K ) ` W ) ) )
20 ssid 
 |-  ( ( LSpan ` ( ( DVecH ` K ) ` W ) ) ` { E } ) C_ ( ( LSpan ` ( ( DVecH ` K ) ` W ) ) ` { E } )
21 20 20 unssi 
 |-  ( ( ( LSpan ` ( ( DVecH ` K ) ` W ) ) ` { E } ) u. ( ( LSpan ` ( ( DVecH ` K ) ` W ) ) ` { E } ) ) C_ ( ( LSpan ` ( ( DVecH ` K ) ` W ) ) ` { E } )
22 21 sseli 
 |-  ( z e. ( ( ( LSpan ` ( ( DVecH ` K ) ` W ) ) ` { E } ) u. ( ( LSpan ` ( ( DVecH ` K ) ` W ) ) ` { E } ) ) -> z e. ( ( LSpan ` ( ( DVecH ` K ) ` W ) ) ` { E } ) )
23 22 con3i 
 |-  ( -. z e. ( ( LSpan ` ( ( DVecH ` K ) ` W ) ) ` { E } ) -> -. z e. ( ( ( LSpan ` ( ( DVecH ` K ) ` W ) ) ` { E } ) u. ( ( LSpan ` ( ( DVecH ` K ) ` W ) ) ` { E } ) ) )
24 23 3ad2ant3 
 |-  ( ( ph /\ z e. ( Base ` ( ( DVecH ` K ) ` W ) ) /\ -. z e. ( ( LSpan ` ( ( DVecH ` K ) ` W ) ) ` { E } ) ) -> -. z e. ( ( ( LSpan ` ( ( DVecH ` K ) ` W ) ) ` { E } ) u. ( ( LSpan ` ( ( DVecH ` K ) ` W ) ) ` { E } ) ) )
25 1 2 3 4 15 6 7 8 16 17 18 19 24 hdmapeveclem 
 |-  ( ( ph /\ z e. ( Base ` ( ( DVecH ` K ) ` W ) ) /\ -. z e. ( ( LSpan ` ( ( DVecH ` K ) ` W ) ) ` { E } ) ) -> ( S ` E ) = ( J ` E ) )
26 25 rexlimdv3a 
 |-  ( ph -> ( E. z e. ( Base ` ( ( DVecH ` K ) ` W ) ) -. z e. ( ( LSpan ` ( ( DVecH ` K ) ` W ) ) ` { E } ) -> ( S ` E ) = ( J ` E ) ) )
27 14 26 mpd 
 |-  ( ph -> ( S ` E ) = ( J ` E ) )