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Metamath Proof Explorer

Theorem dochnel

Description: A nonzero vector doesn't belong to the orthocomplement of its singleton. (Contributed by NM, 27-Oct-2014)

Ref Expression
Hypotheses dochnel.h 
|- H = ( LHyp ` K )
dochnel.o 
|- ._|_ = ( ( ocH ` K ) ` W )
dochnel.u 
|- U = ( ( DVecH ` K ) ` W )
dochnel.v 
|- V = ( Base ` U )
dochnel.z 
|- .0. = ( 0g ` U )
dochnel.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
dochnel.x 
|- ( ph -> X e. ( V \ { .0. } ) )
Assertion dochnel 
|- ( ph -> -. X e. ( ._|_ ` { X } ) )

Proof

Step Hyp Ref Expression
1 dochnel.h 
 |-  H = ( LHyp ` K )
2 dochnel.o 
 |-  ._|_ = ( ( ocH ` K ) ` W )
3 dochnel.u 
 |-  U = ( ( DVecH ` K ) ` W )
4 dochnel.v 
 |-  V = ( Base ` U )
5 dochnel.z 
 |-  .0. = ( 0g ` U )
6 dochnel.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
7 dochnel.x 
 |-  ( ph -> X e. ( V \ { .0. } ) )
8 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
9 1 3 6 dvhlmod 
 |-  ( ph -> U e. LMod )
10 7 eldifad 
 |-  ( ph -> X e. V )
11 eqid 
 |-  ( LSpan ` U ) = ( LSpan ` U )
12 4 8 11 lspsncl 
 |-  ( ( U e. LMod /\ X e. V ) -> ( ( LSpan ` U ) ` { X } ) e. ( LSubSp ` U ) )
13 9 10 12 syl2anc 
 |-  ( ph -> ( ( LSpan ` U ) ` { X } ) e. ( LSubSp ` U ) )
14 4 11 lspsnid 
 |-  ( ( U e. LMod /\ X e. V ) -> X e. ( ( LSpan ` U ) ` { X } ) )
15 9 10 14 syl2anc 
 |-  ( ph -> X e. ( ( LSpan ` U ) ` { X } ) )
16 eldifsni 
 |-  ( X e. ( V \ { .0. } ) -> X =/= .0. )
17 7 16 syl 
 |-  ( ph -> X =/= .0. )
18 eldifsn 
 |-  ( X e. ( ( ( LSpan ` U ) ` { X } ) \ { .0. } ) <-> ( X e. ( ( LSpan ` U ) ` { X } ) /\ X =/= .0. ) )
19 15 17 18 sylanbrc 
 |-  ( ph -> X e. ( ( ( LSpan ` U ) ` { X } ) \ { .0. } ) )
20 1 3 8 5 2 6 13 19 dochnel2 
 |-  ( ph -> -. X e. ( ._|_ ` ( ( LSpan ` U ) ` { X } ) ) )
21 10 snssd 
 |-  ( ph -> { X } C_ V )
22 1 3 2 4 11 6 21 dochocsp 
 |-  ( ph -> ( ._|_ ` ( ( LSpan ` U ) ` { X } ) ) = ( ._|_ ` { X } ) )
23 20 22 neleqtrd 
 |-  ( ph -> -. X e. ( ._|_ ` { X } ) )