Metamath Proof Explorer

Theorem lcfrlem33

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 ) ) )
lcfrlem33.xi 
|- ( ph -> ( ( J ` X ) ` I ) = Q )
Assertion lcfrlem33 
|- ( ph -> C =/= ( 0g ` D ) )

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 lcfrlem33.xi 
 |-  ( ph -> ( ( J ` X ) ` I ) = Q )
27 26 oveq2d 
 |-  ( ph -> ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) = ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) Q ) )
28 1 3 9 dvhlmod 
 |-  ( ph -> U e. LMod )
29 15 lmodring 
 |-  ( U e. LMod -> S e. Ring )
30 28 29 syl 
 |-  ( ph -> S e. Ring )
31 1 3 9 dvhlvec 
 |-  ( ph -> U e. LVec )
32 15 lvecdrng 
 |-  ( U e. LVec -> S e. DivRing )
33 31 32 syl 
 |-  ( ph -> S e. DivRing )
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 1 2 3 4 5 6 7 8 9 10 11 12 13 lcfrlem22 
 |-  ( ph -> B e. A )
39 4 8 28 38 lsatssv 
 |-  ( ph -> B C_ V )
40 39 19 sseldd 
 |-  ( ph -> I e. V )
41 15 17 4 34 lflcl 
 |-  ( ( U e. LMod /\ ( J ` Y ) e. ( LFnl ` U ) /\ I e. V ) -> ( ( J ` Y ) ` I ) e. R )
42 28 37 40 41 syl3anc 
 |-  ( ph -> ( ( J ` Y ) ` I ) e. R )
43 17 16 23 drnginvrcl 
 |-  ( ( S e. DivRing /\ ( ( J ` Y ) ` I ) e. R /\ ( ( J ` Y ) ` I ) =/= Q ) -> ( F ` ( ( J ` Y ) ` I ) ) e. R )
44 33 42 22 43 syl3anc 
 |-  ( ph -> ( F ` ( ( J ` Y ) ` I ) ) e. R )
45 eqid 
 |-  ( .r ` S ) = ( .r ` S )
46 17 45 16 ringrz 
 |-  ( ( S e. Ring /\ ( F ` ( ( J ` Y ) ` I ) ) e. R ) -> ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) Q ) = Q )
47 30 44 46 syl2anc 
 |-  ( ph -> ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) Q ) = Q )
48 27 47 eqtrd 
 |-  ( ph -> ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) = Q )
49 48 oveq1d 
 |-  ( ph -> ( ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) ( .s ` D ) ( J ` Y ) ) = ( Q ( .s ` D ) ( J ` Y ) ) )
50 eqid 
 |-  ( .s ` D ) = ( .s ` D )
51 34 15 16 21 50 35 28 37 ldual0vs 
 |-  ( ph -> ( Q ( .s ` D ) ( J ` Y ) ) = ( 0g ` D ) )
52 49 51 eqtrd 
 |-  ( ph -> ( ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) ( .s ` D ) ( J ` Y ) ) = ( 0g ` D ) )
53 52 oveq2d 
 |-  ( ph -> ( ( J ` X ) .- ( ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) ( .s ` D ) ( J ` Y ) ) ) = ( ( J ` X ) .- ( 0g ` D ) ) )
54 21 28 ldualgrp 
 |-  ( ph -> D e. Grp )
55 eqid 
 |-  ( Base ` D ) = ( Base ` D )
56 1 2 3 4 5 14 15 17 6 34 20 21 35 36 18 9 10 lcfrlem10 
 |-  ( ph -> ( J ` X ) e. ( LFnl ` U ) )
57 34 21 55 28 56 ldualelvbase 
 |-  ( ph -> ( J ` X ) e. ( Base ` D ) )
58 55 35 24 grpsubid1 
 |-  ( ( D e. Grp /\ ( J ` X ) e. ( Base ` D ) ) -> ( ( J ` X ) .- ( 0g ` D ) ) = ( J ` X ) )
59 54 57 58 syl2anc 
 |-  ( ph -> ( ( J ` X ) .- ( 0g ` D ) ) = ( J ` X ) )
60 53 59 eqtrd 
 |-  ( ph -> ( ( J ` X ) .- ( ( ( F ` ( ( J ` Y ) ` I ) ) ( .r ` S ) ( ( J ` X ) ` I ) ) ( .s ` D ) ( J ` Y ) ) ) = ( J ` X ) )
61 25 60 eqtrid 
 |-  ( ph -> C = ( J ` X ) )
62 1 2 3 4 5 14 15 17 6 34 20 21 35 36 18 9 10 lcfrlem13 
 |-  ( ph -> ( J ` X ) e. ( { f e. ( LFnl ` U ) | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) } \ { ( 0g ` D ) } ) )
63 eldifsni 
 |-  ( ( J ` X ) e. ( { f e. ( LFnl ` U ) | ( ._|_ ` ( ._|_ ` ( L ` f ) ) ) = ( L ` f ) } \ { ( 0g ` D ) } ) -> ( J ` X ) =/= ( 0g ` D ) )
64 62 63 syl 
 |-  ( ph -> ( J ` X ) =/= ( 0g ` D ) )
65 61 64 eqnetrd 
 |-  ( ph -> C =/= ( 0g ` D ) )