Metamath Proof Explorer


Theorem dochsnkr2

Description: Kernel of the explicit functional G determined by a nonzero vector X . Compare the more general lshpkr . (Contributed by NM, 27-Oct-2014)

Ref Expression
Hypotheses dochsnkr2.h
|- H = ( LHyp ` K )
dochsnkr2.o
|- ._|_ = ( ( ocH ` K ) ` W )
dochsnkr2.u
|- U = ( ( DVecH ` K ) ` W )
dochsnkr2.v
|- V = ( Base ` U )
dochsnkr2.z
|- .0. = ( 0g ` U )
dochsnkr2.a
|- .+ = ( +g ` U )
dochsnkr2.t
|- .x. = ( .s ` U )
dochsnkr2.l
|- L = ( LKer ` U )
dochsnkr2.d
|- D = ( Scalar ` U )
dochsnkr2.r
|- R = ( Base ` D )
dochsnkr2.g
|- G = ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { X } ) v = ( w .+ ( k .x. X ) ) ) )
dochsnkr2.k
|- ( ph -> ( K e. HL /\ W e. H ) )
dochsnkr2.x
|- ( ph -> X e. ( V \ { .0. } ) )
Assertion dochsnkr2
|- ( ph -> ( L ` G ) = ( ._|_ ` { X } ) )

Proof

Step Hyp Ref Expression
1 dochsnkr2.h
 |-  H = ( LHyp ` K )
2 dochsnkr2.o
 |-  ._|_ = ( ( ocH ` K ) ` W )
3 dochsnkr2.u
 |-  U = ( ( DVecH ` K ) ` W )
4 dochsnkr2.v
 |-  V = ( Base ` U )
5 dochsnkr2.z
 |-  .0. = ( 0g ` U )
6 dochsnkr2.a
 |-  .+ = ( +g ` U )
7 dochsnkr2.t
 |-  .x. = ( .s ` U )
8 dochsnkr2.l
 |-  L = ( LKer ` U )
9 dochsnkr2.d
 |-  D = ( Scalar ` U )
10 dochsnkr2.r
 |-  R = ( Base ` D )
11 dochsnkr2.g
 |-  G = ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { X } ) v = ( w .+ ( k .x. X ) ) ) )
12 dochsnkr2.k
 |-  ( ph -> ( K e. HL /\ W e. H ) )
13 dochsnkr2.x
 |-  ( ph -> X e. ( V \ { .0. } ) )
14 eqid
 |-  ( LSpan ` U ) = ( LSpan ` U )
15 eqid
 |-  ( LSSum ` U ) = ( LSSum ` U )
16 eqid
 |-  ( LSHyp ` U ) = ( LSHyp ` U )
17 1 3 12 dvhlvec
 |-  ( ph -> U e. LVec )
18 1 2 3 4 5 16 12 13 dochsnshp
 |-  ( ph -> ( ._|_ ` { X } ) e. ( LSHyp ` U ) )
19 13 eldifad
 |-  ( ph -> X e. V )
20 1 2 3 4 5 14 15 12 13 dochexmidat
 |-  ( ph -> ( ( ._|_ ` { X } ) ( LSSum ` U ) ( ( LSpan ` U ) ` { X } ) ) = V )
21 4 6 14 15 16 17 18 19 20 9 10 7 11 8 lshpkr
 |-  ( ph -> ( L ` G ) = ( ._|_ ` { X } ) )