Metamath Proof Explorer

Theorem lcfrlem14

Description: Lemma for lcfr . (Contributed by NM, 10-Mar-2015)

Ref Expression
Hypotheses lcf1o.h 
|- H = ( LHyp ` K )
lcf1o.o 
|- ._|_ = ( ( ocH ` K ) ` W )
lcf1o.u 
|- U = ( ( DVecH ` K ) ` W )
lcf1o.v 
|- V = ( Base ` U )
lcf1o.a 
|- .+ = ( +g ` U )
lcf1o.t 
|- .x. = ( .s ` U )
lcf1o.s 
|- S = ( Scalar ` U )
lcf1o.r 
|- R = ( Base ` S )
lcf1o.z 
|- .0. = ( 0g ` U )
lcf1o.f 
|- F = ( LFnl ` U )
lcf1o.l 
|- L = ( LKer ` U )
lcf1o.d 
|- D = ( LDual ` U )
lcf1o.q 
|- Q = ( 0g ` D )
lcf1o.c 
|- C = { f e. F | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }
lcf1o.j 
|- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )
lcflo.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
lcfrlem10.x 
|- ( ph -> X e. ( V \ { .0. } ) )
lcfrlem14.n 
|- N = ( LSpan ` U )
Assertion lcfrlem14 
|- ( ph -> ( ._|_ ` ( L ` ( J ` X ) ) ) = ( N ` { X } ) )

Proof

Step Hyp Ref Expression
1 lcf1o.h 
 |-  H = ( LHyp ` K )
2 lcf1o.o 
 |-  ._|_ = ( ( ocH ` K ) ` W )
3 lcf1o.u 
 |-  U = ( ( DVecH ` K ) ` W )
4 lcf1o.v 
 |-  V = ( Base ` U )
5 lcf1o.a 
 |-  .+ = ( +g ` U )
6 lcf1o.t 
 |-  .x. = ( .s ` U )
7 lcf1o.s 
 |-  S = ( Scalar ` U )
8 lcf1o.r 
 |-  R = ( Base ` S )
9 lcf1o.z 
 |-  .0. = ( 0g ` U )
10 lcf1o.f 
 |-  F = ( LFnl ` U )
11 lcf1o.l 
 |-  L = ( LKer ` U )
12 lcf1o.d 
 |-  D = ( LDual ` U )
13 lcf1o.q 
 |-  Q = ( 0g ` D )
14 lcf1o.c 
 |-  C = { f e. F | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }
15 lcf1o.j 
 |-  J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )
16 lcflo.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
17 lcfrlem10.x 
 |-  ( ph -> X e. ( V \ { .0. } ) )
18 lcfrlem14.n 
 |-  N = ( LSpan ` U )
19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 lcfrlem11 
 |-  ( ph -> ( L ` ( J ` X ) ) = ( ._|_ ` { X } ) )
20 17 eldifad 
 |-  ( ph -> X e. V )
21 20 snssd 
 |-  ( ph -> { X } C_ V )
22 1 3 2 4 18 16 21 dochocsp 
 |-  ( ph -> ( ._|_ ` ( N ` { X } ) ) = ( ._|_ ` { X } ) )
23 19 22 eqtr4d 
 |-  ( ph -> ( L ` ( J ` X ) ) = ( ._|_ ` ( N ` { X } ) ) )
24 23 fveq2d 
 |-  ( ph -> ( ._|_ ` ( L ` ( J ` X ) ) ) = ( ._|_ ` ( ._|_ ` ( N ` { X } ) ) ) )
25 eqid 
 |-  ( ( DIsoH ` K ) ` W ) = ( ( DIsoH ` K ) ` W )
26 1 3 4 18 25 dihlsprn 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. V ) -> ( N ` { X } ) e. ran ( ( DIsoH ` K ) ` W ) )
27 16 20 26 syl2anc 
 |-  ( ph -> ( N ` { X } ) e. ran ( ( DIsoH ` K ) ` W ) )
28 1 25 2 dochoc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( N ` { X } ) e. ran ( ( DIsoH ` K ) ` W ) ) -> ( ._|_ ` ( ._|_ ` ( N ` { X } ) ) ) = ( N ` { X } ) )
29 16 27 28 syl2anc 
 |-  ( ph -> ( ._|_ ` ( ._|_ ` ( N ` { X } ) ) ) = ( N ` { X } ) )
30 24 29 eqtrd 
 |-  ( ph -> ( ._|_ ` ( L ` ( J ` X ) ) ) = ( N ` { X } ) )