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Metamath Proof Explorer

Theorem dochnel2

Description: A nonzero member of a subspace doesn't belong to the orthocomplement of the subspace. (Contributed by NM, 28-Feb-2015)

Ref Expression
Hypotheses dochnoncon.h 
|- H = ( LHyp ` K )
dochnoncon.u 
|- U = ( ( DVecH ` K ) ` W )
dochnoncon.s 
|- S = ( LSubSp ` U )
dochnoncon.z 
|- .0. = ( 0g ` U )
dochnoncon.o 
|- ._|_ = ( ( ocH ` K ) ` W )
dochnel2.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
dochnel2.t 
|- ( ph -> T e. S )
dochnel2.x 
|- ( ph -> X e. ( T \ { .0. } ) )
Assertion dochnel2 
|- ( ph -> -. X e. ( ._|_ ` T ) )

Proof

Step Hyp Ref Expression
1 dochnoncon.h 
 |-  H = ( LHyp ` K )
2 dochnoncon.u 
 |-  U = ( ( DVecH ` K ) ` W )
3 dochnoncon.s 
 |-  S = ( LSubSp ` U )
4 dochnoncon.z 
 |-  .0. = ( 0g ` U )
5 dochnoncon.o 
 |-  ._|_ = ( ( ocH ` K ) ` W )
6 dochnel2.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
7 dochnel2.t 
 |-  ( ph -> T e. S )
8 dochnel2.x 
 |-  ( ph -> X e. ( T \ { .0. } ) )
9 8 eldifbd 
 |-  ( ph -> -. X e. { .0. } )
10 8 eldifad 
 |-  ( ph -> X e. T )
11 elin 
 |-  ( X e. ( T i^i ( ._|_ ` T ) ) <-> ( X e. T /\ X e. ( ._|_ ` T ) ) )
12 1 2 3 4 5 dochnoncon 
 |-  ( ( ( K e. HL /\ W e. H ) /\ T e. S ) -> ( T i^i ( ._|_ ` T ) ) = { .0. } )
13 6 7 12 syl2anc 
 |-  ( ph -> ( T i^i ( ._|_ ` T ) ) = { .0. } )
14 13 eleq2d 
 |-  ( ph -> ( X e. ( T i^i ( ._|_ ` T ) ) <-> X e. { .0. } ) )
15 11 14 bitr3id 
 |-  ( ph -> ( ( X e. T /\ X e. ( ._|_ ` T ) ) <-> X e. { .0. } ) )
16 15 biimpd 
 |-  ( ph -> ( ( X e. T /\ X e. ( ._|_ ` T ) ) -> X e. { .0. } ) )
17 10 16 mpand 
 |-  ( ph -> ( X e. ( ._|_ ` T ) -> X e. { .0. } ) )
18 9 17 mtod 
 |-  ( ph -> -. X e. ( ._|_ ` T ) )