Metamath Proof Explorer

Theorem lcfrlem31

Description: Lemma for lcfr . (Contributed by NM, 10-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 ) ) )
lcfrlem31.xi 
|- ( ph -> ( ( J ` X ) ` I ) =/= Q )
lcfrlem31.cn 
|- ( ph -> C = ( 0g ` D ) )
Assertion lcfrlem31 
|- ( ph -> ( N ` { X } ) = ( N ` { 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 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 lcfrlem31.xi 
 |-  ( ph -> ( ( J ` X ) ` I ) =/= Q )
27 lcfrlem31.cn 
 |-  ( ph -> C = ( 0g ` D ) )
28 25 27 eqtr3id 
 |-  ( ph -> ( ( J ` X ) .- ( ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) ( .s ` D ) ( J ` Y ) ) ) = ( 0g ` D ) )
29 1 3 9 dvhlmod 
 |-  ( ph -> U e. LMod )
30 21 29 lduallmod 
 |-  ( ph -> D e. LMod )
31 eqid 
 |-  ( LFnl ` U ) = ( LFnl ` U )
32 eqid 
 |-  ( Base ` D ) = ( Base ` D )
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 31 20 21 33 34 18 9 10 lcfrlem10 
 |-  ( ph -> ( J ` X ) e. ( LFnl ` U ) )
36 31 21 32 29 35 ldualelvbase 
 |-  ( ph -> ( J ` X ) e. ( Base ` D ) )
37 eqid 
 |-  ( .s ` D ) = ( .s ` D )
38 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 )
39 1 2 3 4 5 14 15 17 6 31 20 21 33 34 18 9 11 lcfrlem10 
 |-  ( ph -> ( J ` Y ) e. ( LFnl ` U ) )
40 31 15 17 21 37 29 38 39 ldualvscl 
 |-  ( ph -> ( ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) ( .s ` D ) ( J ` Y ) ) e. ( LFnl ` U ) )
41 31 21 32 29 40 ldualelvbase 
 |-  ( ph -> ( ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) ( .s ` D ) ( J ` Y ) ) e. ( Base ` D ) )
42 32 33 24 lmodsubeq0 
 |-  ( ( D e. LMod /\ ( J ` X ) e. ( Base ` D ) /\ ( ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) ( .s ` D ) ( J ` Y ) ) e. ( Base ` D ) ) -> ( ( ( J ` X ) .- ( ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) ( .s ` D ) ( J ` Y ) ) ) = ( 0g ` D ) <-> ( J ` X ) = ( ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) ( .s ` D ) ( J ` Y ) ) ) )
43 30 36 41 42 syl3anc 
 |-  ( ph -> ( ( ( J ` X ) .- ( ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) ( .s ` D ) ( J ` Y ) ) ) = ( 0g ` D ) <-> ( J ` X ) = ( ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) ( .s ` D ) ( J ` Y ) ) ) )
44 28 43 mpbid 
 |-  ( ph -> ( J ` X ) = ( ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) ( .s ` D ) ( J ` Y ) ) )
45 44 fveq2d 
 |-  ( ph -> ( L ` ( J ` X ) ) = ( L ` ( ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) ( .s ` D ) ( J ` Y ) ) ) )
46 1 3 9 dvhlvec 
 |-  ( ph -> U e. LVec )
47 15 lvecdrng 
 |-  ( U e. LVec -> S e. DivRing )
48 46 47 syl 
 |-  ( ph -> S e. DivRing )
49 1 2 3 4 5 6 7 8 9 10 11 12 13 lcfrlem22 
 |-  ( ph -> B e. A )
50 4 8 29 49 lsatssv 
 |-  ( ph -> B C_ V )
51 50 19 sseldd 
 |-  ( ph -> I e. V )
52 15 17 4 31 lflcl 
 |-  ( ( U e. LMod /\ ( J ` Y ) e. ( LFnl ` U ) /\ I e. V ) -> ( ( J ` Y ) ` I ) e. R )
53 29 39 51 52 syl3anc 
 |-  ( ph -> ( ( J ` Y ) ` I ) e. R )
54 17 16 23 drnginvrn0 
 |-  ( ( S e. DivRing /\ ( ( J ` Y ) ` I ) e. R /\ ( ( J ` Y ) ` I ) =/= Q ) -> ( F ` ( ( J ` Y ) ` I ) ) =/= Q )
55 48 53 22 54 syl3anc 
 |-  ( ph -> ( F ` ( ( J ` Y ) ` I ) ) =/= Q )
56 eqid 
 |-  ( .r ` S ) = ( .r ` S )
57 17 16 23 drnginvrcl 
 |-  ( ( S e. DivRing /\ ( ( J ` Y ) ` I ) e. R /\ ( ( J ` Y ) ` I ) =/= Q ) -> ( F ` ( ( J ` Y ) ` I ) ) e. R )
58 48 53 22 57 syl3anc 
 |-  ( ph -> ( F ` ( ( J ` Y ) ` I ) ) e. R )
59 15 17 4 31 lflcl 
 |-  ( ( U e. LMod /\ ( J ` X ) e. ( LFnl ` U ) /\ I e. V ) -> ( ( J ` X ) ` I ) e. R )
60 29 35 51 59 syl3anc 
 |-  ( ph -> ( ( J ` X ) ` I ) e. R )
61 17 16 56 48 58 60 drngmulne0 
 |-  ( ph -> ( ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) =/= Q <-> ( ( F ` ( ( J ` Y ) ` I ) ) =/= Q /\ ( ( J ` X ) ` I ) =/= Q ) ) )
62 55 26 61 mpbir2and 
 |-  ( ph -> ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) =/= Q )
63 15 17 16 31 20 21 37 46 39 38 62 ldualkrsc 
 |-  ( ph -> ( L ` ( ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) ( .s ` D ) ( J ` Y ) ) ) = ( L ` ( J ` Y ) ) )
64 45 63 eqtrd 
 |-  ( ph -> ( L ` ( J ` X ) ) = ( L ` ( J ` Y ) ) )
65 64 fveq2d 
 |-  ( ph -> ( ._|_ ` ( L ` ( J ` X ) ) ) = ( ._|_ ` ( L ` ( J ` Y ) ) ) )
66 1 2 3 4 5 14 15 17 6 31 20 21 33 34 18 9 10 7 lcfrlem14 
 |-  ( ph -> ( ._|_ ` ( L ` ( J ` X ) ) ) = ( N ` { X } ) )
67 1 2 3 4 5 14 15 17 6 31 20 21 33 34 18 9 11 7 lcfrlem14 
 |-  ( ph -> ( ._|_ ` ( L ` ( J ` Y ) ) ) = ( N ` { Y } ) )
68 65 66 67 3eqtr3d 
 |-  ( ph -> ( N ` { X } ) = ( N ` { Y } ) )