Metamath Proof Explorer

Theorem lcfrlem37

Description: Lemma for lcfr . (Contributed by NM, 8-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 )
lcfrlem28.jn 
|- ( ph -> ( ( J ` Y ) ` I ) =/= Q )
lcfrlem29.i 
|- F = ( invr ` S )
lcfrlem30.m 
|- .- = ( -g ` D )
lcfrlem30.c 
|- C = ( ( J ` X ) .- ( ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) ( .s ` D ) ( J ` Y ) ) )
lcfrlem37.g 
|- ( ph -> G e. ( LSubSp ` D ) )
lcfrlem37.gs 
|- ( ph -> G C_ { f e. ( LFnl ` U ) | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) } )
lcfrlem37.e 
|- E = U_ g e. G ( ._|_ ` ( L ` g ) )
lcfrlem37.xe 
|- ( ph -> X e. E )
lcfrlem37.ye 
|- ( ph -> Y e. E )
Assertion lcfrlem37 
|- ( ph -> ( X .+ Y ) e. E )

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 lcfrlem28.jn 
 |-  ( ph -> ( ( J ` Y ) ` I ) =/= Q )
23 lcfrlem29.i 
 |-  F = ( invr ` S )
24 lcfrlem30.m 
 |-  .- = ( -g ` D )
25 lcfrlem30.c 
 |-  C = ( ( J ` X ) .- ( ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) ( .s ` D ) ( J ` Y ) ) )
26 lcfrlem37.g 
 |-  ( ph -> G e. ( LSubSp ` D ) )
27 lcfrlem37.gs 
 |-  ( ph -> G C_ { f e. ( LFnl ` U ) | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) } )
28 lcfrlem37.e 
 |-  E = U_ g e. G ( ._|_ ` ( L ` g ) )
29 lcfrlem37.xe 
 |-  ( ph -> X e. E )
30 lcfrlem37.ye 
 |-  ( ph -> Y e. E )
31 eqid 
 |-  ( LSubSp ` D ) = ( LSubSp ` D )
32 1 3 9 dvhlmod 
 |-  ( ph -> U e. LMod )
33 eqid 
 |-  ( LFnl ` U ) = ( LFnl ` U )
34 eqid 
 |-  ( 0g ` D ) = ( 0g ` D )
35 eqid 
 |-  { f e. ( LFnl ` U ) | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) } = { f e. ( LFnl ` U ) | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }
36 eldifsni 
 |-  ( X e. ( V \ { .0. } ) -> X =/= .0. )
37 10 36 syl 
 |-  ( ph -> X =/= .0. )
38 eldifsn 
 |-  ( X e. ( E \ { .0. } ) <-> ( X e. E /\ X =/= .0. ) )
39 29 37 38 sylanbrc 
 |-  ( ph -> X e. ( E \ { .0. } ) )
40 1 2 3 4 5 14 15 17 6 33 20 21 34 35 18 9 31 26 27 28 39 lcfrlem16 
 |-  ( ph -> ( J ` X ) e. G )
41 eqid 
 |-  ( .s ` D ) = ( .s ` D )
42 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 lcfrlem29 
 |-  ( ph -> ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) e. R )
43 eldifsni 
 |-  ( Y e. ( V \ { .0. } ) -> Y =/= .0. )
44 11 43 syl 
 |-  ( ph -> Y =/= .0. )
45 eldifsn 
 |-  ( Y e. ( E \ { .0. } ) <-> ( Y e. E /\ Y =/= .0. ) )
46 30 44 45 sylanbrc 
 |-  ( ph -> Y e. ( E \ { .0. } ) )
47 1 2 3 4 5 14 15 17 6 33 20 21 34 35 18 9 31 26 27 28 46 lcfrlem16 
 |-  ( ph -> ( J ` Y ) e. G )
48 15 17 21 41 31 32 26 42 47 ldualssvscl 
 |-  ( ph -> ( ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) ( .s ` D ) ( J ` Y ) ) e. G )
49 21 24 31 32 26 40 48 ldualssvsubcl 
 |-  ( ph -> ( ( J ` X ) .- ( ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) ( .s ` D ) ( J ` Y ) ) ) e. G )
50 25 49 eqeltrid 
 |-  ( ph -> C e. G )
51 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 lcfrlem36 
 |-  ( ph -> ( X .+ Y ) e. ( ._|_ ` ( L ` C ) ) )
52 2fveq3 
 |-  ( g = C -> ( ._|_ ` ( L ` g ) ) = ( ._|_ ` ( L ` C ) ) )
53 52 eleq2d 
 |-  ( g = C -> ( ( X .+ Y ) e. ( ._|_ ` ( L ` g ) ) <-> ( X .+ Y ) e. ( ._|_ ` ( L ` C ) ) ) )
54 53 rspcev 
 |-  ( ( C e. G /\ ( X .+ Y ) e. ( ._|_ ` ( L ` C ) ) ) -> E. g e. G ( X .+ Y ) e. ( ._|_ ` ( L ` g ) ) )
55 50 51 54 syl2anc 
 |-  ( ph -> E. g e. G ( X .+ Y ) e. ( ._|_ ` ( L ` g ) ) )
56 eliun 
 |-  ( ( X .+ Y ) e. U_ g e. G ( ._|_ ` ( L ` g ) ) <-> E. g e. G ( X .+ Y ) e. ( ._|_ ` ( L ` g ) ) )
57 55 56 sylibr 
 |-  ( ph -> ( X .+ Y ) e. U_ g e. G ( ._|_ ` ( L ` g ) ) )
58 57 28 eleqtrrdi 
 |-  ( ph -> ( X .+ Y ) e. E )