Metamath Proof Explorer

Theorem lcfrlem35

Description: Lemma for lcfr . (Contributed by NM, 2-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 ) ) )
Assertion lcfrlem35 
|- ( ph -> ( ._|_ ` { ( X .+ Y ) } ) = ( L ` C ) )

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 eqid 
 |-  ( LSSum ` U ) = ( LSSum ` U )
27 1 2 3 4 5 6 7 8 9 10 11 12 13 26 lcfrlem23 
 |-  ( ph -> ( ( ._|_ ` { X , Y } ) ( LSSum ` U ) B ) = ( ._|_ ` { ( X .+ Y ) } ) )
28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 lcfrlem24 
 |-  ( ph -> ( ._|_ ` { X , Y } ) = ( ( L ` ( J ` X ) ) i^i ( L ` ( J ` Y ) ) ) )
29 eqid 
 |-  ( .r ` S ) = ( .r ` S )
30 eqid 
 |-  ( LFnl ` U ) = ( LFnl ` U )
31 eqid 
 |-  ( .s ` D ) = ( .s ` D )
32 1 3 9 dvhlvec 
 |-  ( ph -> U e. LVec )
33 eqid 
 |-  ( 0g ` D ) = ( 0g ` D )
34 eqid 
 |-  { f e. ( LFnl ` U ) | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) } = { f e. ( LFnl ` U ) | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) }
35 1 2 3 4 5 14 15 17 6 30 20 21 33 34 18 9 10 lcfrlem10 
 |-  ( ph -> ( J ` X ) e. ( LFnl ` U ) )
36 1 2 3 4 5 14 15 17 6 30 20 21 33 34 18 9 11 lcfrlem10 
 |-  ( ph -> ( J ` Y ) e. ( LFnl ` U ) )
37 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
38 1 3 9 dvhlmod 
 |-  ( ph -> U e. LMod )
39 1 2 3 4 5 6 7 8 9 10 11 12 13 lcfrlem22 
 |-  ( ph -> B e. A )
40 37 8 38 39 lsatlssel 
 |-  ( ph -> B e. ( LSubSp ` U ) )
41 4 37 lssel 
 |-  ( ( B e. ( LSubSp ` U ) /\ I e. B ) -> I e. V )
42 40 19 41 syl2anc 
 |-  ( ph -> I e. V )
43 4 15 29 16 23 30 21 31 24 32 35 36 42 22 25 20 lcfrlem2 
 |-  ( ph -> ( ( L ` ( J ` X ) ) i^i ( L ` ( J ` Y ) ) ) C_ ( L ` C ) )
44 28 43 eqsstrd 
 |-  ( ph -> ( ._|_ ` { X , Y } ) C_ ( L ` C ) )
45 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 lcfrlem28 
 |-  ( ph -> I =/= .0. )
46 6 7 8 32 39 19 45 lsatel 
 |-  ( ph -> B = ( N ` { I } ) )
47 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 lcfrlem30 
 |-  ( ph -> C e. ( LFnl ` U ) )
48 30 20 37 lkrlss 
 |-  ( ( U e. LMod /\ C e. ( LFnl ` U ) ) -> ( L ` C ) e. ( LSubSp ` U ) )
49 38 47 48 syl2anc 
 |-  ( ph -> ( L ` C ) e. ( LSubSp ` U ) )
50 4 15 29 16 23 30 21 31 24 32 35 36 42 22 25 20 lcfrlem3 
 |-  ( ph -> I e. ( L ` C ) )
51 37 7 38 49 50 lspsnel5a 
 |-  ( ph -> ( N ` { I } ) C_ ( L ` C ) )
52 46 51 eqsstrd 
 |-  ( ph -> B C_ ( L ` C ) )
53 37 lsssssubg 
 |-  ( U e. LMod -> ( LSubSp ` U ) C_ ( SubGrp ` U ) )
54 38 53 syl 
 |-  ( ph -> ( LSubSp ` U ) C_ ( SubGrp ` U ) )
55 10 eldifad 
 |-  ( ph -> X e. V )
56 11 eldifad 
 |-  ( ph -> Y e. V )
57 prssi 
 |-  ( ( X e. V /\ Y e. V ) -> { X , Y } C_ V )
58 55 56 57 syl2anc 
 |-  ( ph -> { X , Y } C_ V )
59 1 3 4 37 2 dochlss 
 |-  ( ( ( K e. HL /\ W e. H ) /\ { X , Y } C_ V ) -> ( ._|_ ` { X , Y } ) e. ( LSubSp ` U ) )
60 9 58 59 syl2anc 
 |-  ( ph -> ( ._|_ ` { X , Y } ) e. ( LSubSp ` U ) )
61 54 60 sseldd 
 |-  ( ph -> ( ._|_ ` { X , Y } ) e. ( SubGrp ` U ) )
62 54 40 sseldd 
 |-  ( ph -> B e. ( SubGrp ` U ) )
63 54 49 sseldd 
 |-  ( ph -> ( L ` C ) e. ( SubGrp ` U ) )
64 26 lsmlub 
 |-  ( ( ( ._|_ ` { X , Y } ) e. ( SubGrp ` U ) /\ B e. ( SubGrp ` U ) /\ ( L ` C ) e. ( SubGrp ` U ) ) -> ( ( ( ._|_ ` { X , Y } ) C_ ( L ` C ) /\ B C_ ( L ` C ) ) <-> ( ( ._|_ ` { X , Y } ) ( LSSum ` U ) B ) C_ ( L ` C ) ) )
65 61 62 63 64 syl3anc 
 |-  ( ph -> ( ( ( ._|_ ` { X , Y } ) C_ ( L ` C ) /\ B C_ ( L ` C ) ) <-> ( ( ._|_ ` { X , Y } ) ( LSSum ` U ) B ) C_ ( L ` C ) ) )
66 44 52 65 mpbi2and 
 |-  ( ph -> ( ( ._|_ ` { X , Y } ) ( LSSum ` U ) B ) C_ ( L ` C ) )
67 27 66 eqsstrrd 
 |-  ( ph -> ( ._|_ ` { ( X .+ Y ) } ) C_ ( L ` C ) )
68 eqid 
 |-  ( LSHyp ` U ) = ( LSHyp ` U )
69 1 2 3 4 5 6 7 8 9 10 11 12 lcfrlem17 
 |-  ( ph -> ( X .+ Y ) e. ( V \ { .0. } ) )
70 1 2 3 4 6 68 9 69 dochsnshp 
 |-  ( ph -> ( ._|_ ` { ( X .+ Y ) } ) e. ( LSHyp ` U ) )
71 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 lcfrlem34 
 |-  ( ph -> C =/= ( 0g ` D ) )
72 68 30 20 21 33 32 47 lduallkr3 
 |-  ( ph -> ( ( L ` C ) e. ( LSHyp ` U ) <-> C =/= ( 0g ` D ) ) )
73 71 72 mpbird 
 |-  ( ph -> ( L ` C ) e. ( LSHyp ` U ) )
74 68 32 70 73 lshpcmp 
 |-  ( ph -> ( ( ._|_ ` { ( X .+ Y ) } ) C_ ( L ` C ) <-> ( ._|_ ` { ( X .+ Y ) } ) = ( L ` C ) ) )
75 67 74 mpbid 
 |-  ( ph -> ( ._|_ ` { ( X .+ Y ) } ) = ( L ` C ) )