Metamath Proof Explorer

Theorem hdmapinvlem1

Description: Line 27 in Baer p. 110. We use C for Baer's u. Our unit vector E has the required properties for his w by hdmapevec2 . Our ( ( SE )C ) means the inner product <. C , E >. i.e. his f(u,w) (note argument reversal). (Contributed by NM, 11-Jun-2015)

Ref Expression
Hypotheses hdmapinvlem1.h 
|- H = ( LHyp ` K )
hdmapinvlem1.e 
|- E = <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` W ) ) >.
hdmapinvlem1.o 
|- O = ( ( ocH ` K ) ` W )
hdmapinvlem1.u 
|- U = ( ( DVecH ` K ) ` W )
hdmapinvlem1.v 
|- V = ( Base ` U )
hdmapinvlem1.r 
|- R = ( Scalar ` U )
hdmapinvlem1.b 
|- B = ( Base ` R )
hdmapinvlem1.t 
|- .x. = ( .r ` R )
hdmapinvlem1.z 
|- .0. = ( 0g ` R )
hdmapinvlem1.s 
|- S = ( ( HDMap ` K ) ` W )
hdmapinvlem1.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
hdmapinvlem1.c 
|- ( ph -> C e. ( O ` { E } ) )
Assertion hdmapinvlem1 
|- ( ph -> ( ( S ` E ) ` C ) = .0. )

Proof

Step Hyp Ref Expression
1 hdmapinvlem1.h 
 |-  H = ( LHyp ` K )
2 hdmapinvlem1.e 
 |-  E = <. ( _I |` ( Base ` K ) ) , ( _I |` ( ( LTrn ` K ) ` W ) ) >.
3 hdmapinvlem1.o 
 |-  O = ( ( ocH ` K ) ` W )
4 hdmapinvlem1.u 
 |-  U = ( ( DVecH ` K ) ` W )
5 hdmapinvlem1.v 
 |-  V = ( Base ` U )
6 hdmapinvlem1.r 
 |-  R = ( Scalar ` U )
7 hdmapinvlem1.b 
 |-  B = ( Base ` R )
8 hdmapinvlem1.t 
 |-  .x. = ( .r ` R )
9 hdmapinvlem1.z 
 |-  .0. = ( 0g ` R )
10 hdmapinvlem1.s 
 |-  S = ( ( HDMap ` K ) ` W )
11 hdmapinvlem1.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
12 hdmapinvlem1.c 
 |-  ( ph -> C e. ( O ` { E } ) )
13 eqid 
 |-  ( LFnl ` U ) = ( LFnl ` U )
14 eqid 
 |-  ( LKer ` U ) = ( LKer ` U )
15 eqid 
 |-  ( Base ` K ) = ( Base ` K )
16 eqid 
 |-  ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
17 eqid 
 |-  ( 0g ` U ) = ( 0g ` U )
18 1 15 16 4 5 17 2 11 dvheveccl 
 |-  ( ph -> E e. ( V \ { ( 0g ` U ) } ) )
19 18 eldifad 
 |-  ( ph -> E e. V )
20 1 3 4 5 13 14 10 11 19 hdmaplkr 
 |-  ( ph -> ( ( LKer ` U ) ` ( S ` E ) ) = ( O ` { E } ) )
21 12 20 eleqtrrd 
 |-  ( ph -> C e. ( ( LKer ` U ) ` ( S ` E ) ) )
22 1 4 11 dvhlmod 
 |-  ( ph -> U e. LMod )
23 eqid 
 |-  ( ( LCDual ` K ) ` W ) = ( ( LCDual ` K ) ` W )
24 eqid 
 |-  ( Base ` ( ( LCDual ` K ) ` W ) ) = ( Base ` ( ( LCDual ` K ) ` W ) )
25 1 4 5 23 24 10 11 19 hdmapcl 
 |-  ( ph -> ( S ` E ) e. ( Base ` ( ( LCDual ` K ) ` W ) ) )
26 1 23 24 4 13 11 25 lcdvbaselfl 
 |-  ( ph -> ( S ` E ) e. ( LFnl ` U ) )
27 19 snssd 
 |-  ( ph -> { E } C_ V )
28 1 4 5 3 dochssv 
 |-  ( ( ( K e. HL /\ W e. H ) /\ { E } C_ V ) -> ( O ` { E } ) C_ V )
29 11 27 28 syl2anc 
 |-  ( ph -> ( O ` { E } ) C_ V )
30 29 12 sseldd 
 |-  ( ph -> C e. V )
31 5 6 9 13 14 22 26 30 ellkr2 
 |-  ( ph -> ( C e. ( ( LKer ` U ) ` ( S ` E ) ) <-> ( ( S ` E ) ` C ) = .0. ) )
32 21 31 mpbid 
 |-  ( ph -> ( ( S ` E ) ` C ) = .0. )