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 } ) )