Metamath Proof Explorer

Theorem lcfrlem26

Description: Lemma for lcfr . Special case of lcfrlem36 when ( ( JY )I ) is zero. (Contributed by NM, 11-Mar-2015)

Ref Expression
Hypotheses lcfrlem17.h 
|- H = ( LHyp ` K )
lcfrlem17.o 
|- ._|_ = ( ( ocH ` K ) ` W )
lcfrlem17.u 
|- U = ( ( DVecH ` K ) ` W )
lcfrlem17.v 
|- V = ( Base ` U )
lcfrlem17.p 
|- .+ = ( +g ` U )
lcfrlem17.z 
|- .0. = ( 0g ` U )
lcfrlem17.n 
|- N = ( LSpan ` U )
lcfrlem17.a 
|- A = ( LSAtoms ` U )
lcfrlem17.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
lcfrlem17.x 
|- ( ph -> X e. ( V \ { .0. } ) )
lcfrlem17.y 
|- ( ph -> Y e. ( V \ { .0. } ) )
lcfrlem17.ne 
|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
lcfrlem22.b 
|- B = ( ( N ` { X , Y } ) i^i ( ._|_ ` { ( X .+ Y ) } ) )
lcfrlem24.t 
|- .x. = ( .s ` U )
lcfrlem24.s 
|- S = ( Scalar ` U )
lcfrlem24.q 
|- Q = ( 0g ` S )
lcfrlem24.r 
|- R = ( Base ` S )
lcfrlem24.j 
|- J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )
lcfrlem24.ib 
|- ( ph -> I e. B )
lcfrlem24.l 
|- L = ( LKer ` U )
lcfrlem25.d 
|- D = ( LDual ` U )
lcfrlem25.jz 
|- ( ph -> ( ( J ` Y ) ` I ) = Q )
lcfrlem25.in 
|- ( ph -> I =/= .0. )
Assertion lcfrlem26 
|- ( ph -> ( X .+ Y ) e. ( ._|_ ` ( L ` ( J ` Y ) ) ) )

Proof

Step Hyp Ref Expression
1 lcfrlem17.h 
 |-  H = ( LHyp ` K )
2 lcfrlem17.o 
 |-  ._|_ = ( ( ocH ` K ) ` W )
3 lcfrlem17.u 
 |-  U = ( ( DVecH ` K ) ` W )
4 lcfrlem17.v 
 |-  V = ( Base ` U )
5 lcfrlem17.p 
 |-  .+ = ( +g ` U )
6 lcfrlem17.z 
 |-  .0. = ( 0g ` U )
7 lcfrlem17.n 
 |-  N = ( LSpan ` U )
8 lcfrlem17.a 
 |-  A = ( LSAtoms ` U )
9 lcfrlem17.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
10 lcfrlem17.x 
 |-  ( ph -> X e. ( V \ { .0. } ) )
11 lcfrlem17.y 
 |-  ( ph -> Y e. ( V \ { .0. } ) )
12 lcfrlem17.ne 
 |-  ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
13 lcfrlem22.b 
 |-  B = ( ( N ` { X , Y } ) i^i ( ._|_ ` { ( X .+ Y ) } ) )
14 lcfrlem24.t 
 |-  .x. = ( .s ` U )
15 lcfrlem24.s 
 |-  S = ( Scalar ` U )
16 lcfrlem24.q 
 |-  Q = ( 0g ` S )
17 lcfrlem24.r 
 |-  R = ( Base ` S )
18 lcfrlem24.j 
 |-  J = ( x e. ( V \ { .0. } ) |-> ( v e. V |-> ( iota_ k e. R E. w e. ( ._|_ ` { x } ) v = ( w .+ ( k .x. x ) ) ) ) )
19 lcfrlem24.ib 
 |-  ( ph -> I e. B )
20 lcfrlem24.l 
 |-  L = ( LKer ` U )
21 lcfrlem25.d 
 |-  D = ( LDual ` U )
22 lcfrlem25.jz 
 |-  ( ph -> ( ( J ` Y ) ` I ) = Q )
23 lcfrlem25.in 
 |-  ( ph -> I =/= .0. )
24 1 2 3 4 5 6 7 8 9 10 11 12 lcfrlem17 
 |-  ( ph -> ( X .+ Y ) e. ( V \ { .0. } ) )
25 24 eldifad 
 |-  ( ph -> ( X .+ Y ) e. V )
26 1 3 2 4 7 9 25 dochocsn 
 |-  ( ph -> ( ._|_ ` ( ._|_ ` { ( X .+ Y ) } ) ) = ( N ` { ( X .+ Y ) } ) )
27 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 lcfrlem25 
 |-  ( ph -> ( ._|_ ` { ( X .+ Y ) } ) = ( L ` ( J ` Y ) ) )
28 27 fveq2d 
 |-  ( ph -> ( ._|_ ` ( ._|_ ` { ( X .+ Y ) } ) ) = ( ._|_ ` ( L ` ( J ` Y ) ) ) )
29 26 28 eqtr3d 
 |-  ( ph -> ( N ` { ( X .+ Y ) } ) = ( ._|_ ` ( L ` ( J ` Y ) ) ) )
30 eqimss 
 |-  ( ( N ` { ( X .+ Y ) } ) = ( ._|_ ` ( L ` ( J ` Y ) ) ) -> ( N ` { ( X .+ Y ) } ) C_ ( ._|_ ` ( L ` ( J ` Y ) ) ) )
31 29 30 syl 
 |-  ( ph -> ( N ` { ( X .+ Y ) } ) C_ ( ._|_ ` ( L ` ( J ` Y ) ) ) )
32 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
33 1 3 9 dvhlmod 
 |-  ( ph -> U e. LMod )
34 eqid 
 |-  ( LFnl ` U ) = ( LFnl ` U )
35 eqid 
 |-  ( 0g ` D ) = ( 0g ` D )
36 eqid 
 |-  { f e. ( LFnl ` U ) | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) } = { f e. ( LFnl ` U ) | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }
37 1 2 3 4 5 14 15 17 6 34 20 21 35 36 18 9 11 lcfrlem10 
 |-  ( ph -> ( J ` Y ) e. ( LFnl ` U ) )
38 4 34 20 33 37 lkrssv 
 |-  ( ph -> ( L ` ( J ` Y ) ) C_ V )
39 1 3 4 32 2 dochlss 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( L ` ( J ` Y ) ) C_ V ) -> ( ._|_ ` ( L ` ( J ` Y ) ) ) e. ( LSubSp ` U ) )
40 9 38 39 syl2anc 
 |-  ( ph -> ( ._|_ ` ( L ` ( J ` Y ) ) ) e. ( LSubSp ` U ) )
41 4 32 7 33 40 25 lspsnel5 
 |-  ( ph -> ( ( X .+ Y ) e. ( ._|_ ` ( L ` ( J ` Y ) ) ) <-> ( N ` { ( X .+ Y ) } ) C_ ( ._|_ ` ( L ` ( J ` Y ) ) ) ) )
42 31 41 mpbird 
 |-  ( ph -> ( X .+ Y ) e. ( ._|_ ` ( L ` ( J ` Y ) ) ) )