Metamath Proof Explorer

Theorem mapdpglem30

Description: Lemma for mapdpg . Baer p. 45 line 18: "Hence we deduce (from mapdpglem28 , using lvecindp2 ) that v = 1 and v = u...". TODO: would it be shorter to have only the v = ( 1rA ) part and use mapdpglem28.u2 in mapdpglem31 ? (Contributed by NM, 22-Mar-2015)

Ref Expression
Hypotheses mapdpg.h 
|- H = ( LHyp ` K )
mapdpg.m 
|- M = ( ( mapd ` K ) ` W )
mapdpg.u 
|- U = ( ( DVecH ` K ) ` W )
mapdpg.v 
|- V = ( Base ` U )
mapdpg.s 
|- .- = ( -g ` U )
mapdpg.z 
|- .0. = ( 0g ` U )
mapdpg.n 
|- N = ( LSpan ` U )
mapdpg.c 
|- C = ( ( LCDual ` K ) ` W )
mapdpg.f 
|- F = ( Base ` C )
mapdpg.r 
|- R = ( -g ` C )
mapdpg.j 
|- J = ( LSpan ` C )
mapdpg.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
mapdpg.x 
|- ( ph -> X e. ( V \ { .0. } ) )
mapdpg.y 
|- ( ph -> Y e. ( V \ { .0. } ) )
mapdpg.g 
|- ( ph -> G e. F )
mapdpg.ne 
|- ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
mapdpg.e 
|- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { G } ) )
mapdpgem25.h1 
|- ( ph -> ( h e. F /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) ) )
mapdpgem25.i1 
|- ( ph -> ( i e. F /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) )
mapdpglem26.a 
|- A = ( Scalar ` U )
mapdpglem26.b 
|- B = ( Base ` A )
mapdpglem26.t 
|- .x. = ( .s ` C )
mapdpglem26.o 
|- O = ( 0g ` A )
mapdpglem28.ve 
|- ( ph -> v e. B )
mapdpglem28.u1 
|- ( ph -> h = ( u .x. i ) )
mapdpglem28.u2 
|- ( ph -> ( G R h ) = ( v .x. ( G R i ) ) )
mapdpglem28.ue 
|- ( ph -> u e. B )
Assertion mapdpglem30 
|- ( ph -> ( v = ( 1r ` A ) /\ v = u ) )

Proof

Step Hyp Ref Expression
1 mapdpg.h 
 |-  H = ( LHyp ` K )
2 mapdpg.m 
 |-  M = ( ( mapd ` K ) ` W )
3 mapdpg.u 
 |-  U = ( ( DVecH ` K ) ` W )
4 mapdpg.v 
 |-  V = ( Base ` U )
5 mapdpg.s 
 |-  .- = ( -g ` U )
6 mapdpg.z 
 |-  .0. = ( 0g ` U )
7 mapdpg.n 
 |-  N = ( LSpan ` U )
8 mapdpg.c 
 |-  C = ( ( LCDual ` K ) ` W )
9 mapdpg.f 
 |-  F = ( Base ` C )
10 mapdpg.r 
 |-  R = ( -g ` C )
11 mapdpg.j 
 |-  J = ( LSpan ` C )
12 mapdpg.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
13 mapdpg.x 
 |-  ( ph -> X e. ( V \ { .0. } ) )
14 mapdpg.y 
 |-  ( ph -> Y e. ( V \ { .0. } ) )
15 mapdpg.g 
 |-  ( ph -> G e. F )
16 mapdpg.ne 
 |-  ( ph -> ( N ` { X } ) =/= ( N ` { Y } ) )
17 mapdpg.e 
 |-  ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { G } ) )
18 mapdpgem25.h1 
 |-  ( ph -> ( h e. F /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R h ) } ) ) ) )
19 mapdpgem25.i1 
 |-  ( ph -> ( i e. F /\ ( ( M ` ( N ` { Y } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( X .- Y ) } ) ) = ( J ` { ( G R i ) } ) ) ) )
20 mapdpglem26.a 
 |-  A = ( Scalar ` U )
21 mapdpglem26.b 
 |-  B = ( Base ` A )
22 mapdpglem26.t 
 |-  .x. = ( .s ` C )
23 mapdpglem26.o 
 |-  O = ( 0g ` A )
24 mapdpglem28.ve 
 |-  ( ph -> v e. B )
25 mapdpglem28.u1 
 |-  ( ph -> h = ( u .x. i ) )
26 mapdpglem28.u2 
 |-  ( ph -> ( G R h ) = ( v .x. ( G R i ) ) )
27 mapdpglem28.ue 
 |-  ( ph -> u e. B )
28 eqid 
 |-  ( +g ` C ) = ( +g ` C )
29 eqid 
 |-  ( Scalar ` C ) = ( Scalar ` C )
30 eqid 
 |-  ( Base ` ( Scalar ` C ) ) = ( Base ` ( Scalar ` C ) )
31 eqid 
 |-  ( 0g ` C ) = ( 0g ` C )
32 1 8 12 lcdlvec 
 |-  ( ph -> C e. LVec )
33 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 mapdpglem30a 
 |-  ( ph -> G =/= ( 0g ` C ) )
34 eldifsn 
 |-  ( G e. ( F \ { ( 0g ` C ) } ) <-> ( G e. F /\ G =/= ( 0g ` C ) ) )
35 15 33 34 sylanbrc 
 |-  ( ph -> G e. ( F \ { ( 0g ` C ) } ) )
36 19 simpld 
 |-  ( ph -> i e. F )
37 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 mapdpglem30b 
 |-  ( ph -> i =/= ( 0g ` C ) )
38 eldifsn 
 |-  ( i e. ( F \ { ( 0g ` C ) } ) <-> ( i e. F /\ i =/= ( 0g ` C ) ) )
39 36 37 38 sylanbrc 
 |-  ( ph -> i e. ( F \ { ( 0g ` C ) } ) )
40 1 3 20 21 8 29 30 12 lcdsbase 
 |-  ( ph -> ( Base ` ( Scalar ` C ) ) = B )
41 24 40 eleqtrrd 
 |-  ( ph -> v e. ( Base ` ( Scalar ` C ) ) )
42 1 3 12 dvhlmod 
 |-  ( ph -> U e. LMod )
43 20 lmodring 
 |-  ( U e. LMod -> A e. Ring )
44 42 43 syl 
 |-  ( ph -> A e. Ring )
45 ringgrp 
 |-  ( A e. Ring -> A e. Grp )
46 44 45 syl 
 |-  ( ph -> A e. Grp )
47 eqid 
 |-  ( 1r ` A ) = ( 1r ` A )
48 21 47 ringidcl 
 |-  ( A e. Ring -> ( 1r ` A ) e. B )
49 44 48 syl 
 |-  ( ph -> ( 1r ` A ) e. B )
50 eqid 
 |-  ( invg ` A ) = ( invg ` A )
51 21 50 grpinvcl 
 |-  ( ( A e. Grp /\ ( 1r ` A ) e. B ) -> ( ( invg ` A ) ` ( 1r ` A ) ) e. B )
52 46 49 51 syl2anc 
 |-  ( ph -> ( ( invg ` A ) ` ( 1r ` A ) ) e. B )
53 eqid 
 |-  ( .r ` A ) = ( .r ` A )
54 21 53 ringcl 
 |-  ( ( A e. Ring /\ v e. B /\ ( ( invg ` A ) ` ( 1r ` A ) ) e. B ) -> ( v ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) e. B )
55 44 24 52 54 syl3anc 
 |-  ( ph -> ( v ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) e. B )
56 55 40 eleqtrrd 
 |-  ( ph -> ( v ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) e. ( Base ` ( Scalar ` C ) ) )
57 49 40 eleqtrrd 
 |-  ( ph -> ( 1r ` A ) e. ( Base ` ( Scalar ` C ) ) )
58 21 53 ringcl 
 |-  ( ( A e. Ring /\ u e. B /\ ( ( invg ` A ) ` ( 1r ` A ) ) e. B ) -> ( u ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) e. B )
59 44 27 52 58 syl3anc 
 |-  ( ph -> ( u ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) e. B )
60 59 40 eleqtrrd 
 |-  ( ph -> ( u ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) e. ( Base ` ( Scalar ` C ) ) )
61 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 26 mapdpglem29 
 |-  ( ph -> ( J ` { G } ) =/= ( J ` { i } ) )
62 1 3 20 21 53 8 9 22 12 52 27 36 lcdvsass 
 |-  ( ph -> ( ( u ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) .x. i ) = ( ( ( invg ` A ) ` ( 1r ` A ) ) .x. ( u .x. i ) ) )
63 62 oveq2d 
 |-  ( ph -> ( ( ( 1r ` A ) .x. G ) ( +g ` C ) ( ( u ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) .x. i ) ) = ( ( ( 1r ` A ) .x. G ) ( +g ` C ) ( ( ( invg ` A ) ` ( 1r ` A ) ) .x. ( u .x. i ) ) ) )
64 1 3 20 21 8 9 22 12 49 15 lcdvscl 
 |-  ( ph -> ( ( 1r ` A ) .x. G ) e. F )
65 1 3 20 21 8 9 22 12 27 36 lcdvscl 
 |-  ( ph -> ( u .x. i ) e. F )
66 1 3 20 50 47 8 9 28 22 10 12 64 65 lcdvsub 
 |-  ( ph -> ( ( ( 1r ` A ) .x. G ) R ( u .x. i ) ) = ( ( ( 1r ` A ) .x. G ) ( +g ` C ) ( ( ( invg ` A ) ` ( 1r ` A ) ) .x. ( u .x. i ) ) ) )
67 1 3 20 21 53 8 9 22 12 52 24 36 lcdvsass 
 |-  ( ph -> ( ( v ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) .x. i ) = ( ( ( invg ` A ) ` ( 1r ` A ) ) .x. ( v .x. i ) ) )
68 67 oveq2d 
 |-  ( ph -> ( ( v .x. G ) ( +g ` C ) ( ( v ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) .x. i ) ) = ( ( v .x. G ) ( +g ` C ) ( ( ( invg ` A ) ` ( 1r ` A ) ) .x. ( v .x. i ) ) ) )
69 1 3 20 21 8 9 22 12 24 15 lcdvscl 
 |-  ( ph -> ( v .x. G ) e. F )
70 1 3 20 21 8 9 22 12 24 36 lcdvscl 
 |-  ( ph -> ( v .x. i ) e. F )
71 1 3 20 50 47 8 9 28 22 10 12 69 70 lcdvsub 
 |-  ( ph -> ( ( v .x. G ) R ( v .x. i ) ) = ( ( v .x. G ) ( +g ` C ) ( ( ( invg ` A ) ` ( 1r ` A ) ) .x. ( v .x. i ) ) ) )
72 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 26 mapdpglem28 
 |-  ( ph -> ( ( v .x. G ) R ( v .x. i ) ) = ( G R ( u .x. i ) ) )
73 eqid 
 |-  ( 1r ` ( Scalar ` C ) ) = ( 1r ` ( Scalar ` C ) )
74 1 3 20 47 8 29 73 12 lcd1 
 |-  ( ph -> ( 1r ` ( Scalar ` C ) ) = ( 1r ` A ) )
75 74 oveq1d 
 |-  ( ph -> ( ( 1r ` ( Scalar ` C ) ) .x. G ) = ( ( 1r ` A ) .x. G ) )
76 1 8 12 lcdlmod 
 |-  ( ph -> C e. LMod )
77 9 29 22 73 lmodvs1 
 |-  ( ( C e. LMod /\ G e. F ) -> ( ( 1r ` ( Scalar ` C ) ) .x. G ) = G )
78 76 15 77 syl2anc 
 |-  ( ph -> ( ( 1r ` ( Scalar ` C ) ) .x. G ) = G )
79 75 78 eqtr3d 
 |-  ( ph -> ( ( 1r ` A ) .x. G ) = G )
80 79 oveq1d 
 |-  ( ph -> ( ( ( 1r ` A ) .x. G ) R ( u .x. i ) ) = ( G R ( u .x. i ) ) )
81 72 80 eqtr4d 
 |-  ( ph -> ( ( v .x. G ) R ( v .x. i ) ) = ( ( ( 1r ` A ) .x. G ) R ( u .x. i ) ) )
82 68 71 81 3eqtr2rd 
 |-  ( ph -> ( ( ( 1r ` A ) .x. G ) R ( u .x. i ) ) = ( ( v .x. G ) ( +g ` C ) ( ( v ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) .x. i ) ) )
83 63 66 82 3eqtr2rd 
 |-  ( ph -> ( ( v .x. G ) ( +g ` C ) ( ( v ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) .x. i ) ) = ( ( ( 1r ` A ) .x. G ) ( +g ` C ) ( ( u ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) .x. i ) ) )
84 9 28 29 30 22 31 11 32 35 39 41 56 57 60 61 83 lvecindp2 
 |-  ( ph -> ( v = ( 1r ` A ) /\ ( v ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) = ( u ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) ) )
85 21 53 47 50 44 24 ringnegr 
 |-  ( ph -> ( v ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) = ( ( invg ` A ) ` v ) )
86 21 53 47 50 44 27 ringnegr 
 |-  ( ph -> ( u ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) = ( ( invg ` A ) ` u ) )
87 85 86 eqeq12d 
 |-  ( ph -> ( ( v ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) = ( u ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) <-> ( ( invg ` A ) ` v ) = ( ( invg ` A ) ` u ) ) )
88 21 50 46 24 27 grpinv11 
 |-  ( ph -> ( ( ( invg ` A ) ` v ) = ( ( invg ` A ) ` u ) <-> v = u ) )
89 87 88 bitrd 
 |-  ( ph -> ( ( v ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) = ( u ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) <-> v = u ) )
90 89 anbi2d 
 |-  ( ph -> ( ( v = ( 1r ` A ) /\ ( v ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) = ( u ( .r ` A ) ( ( invg ` A ) ` ( 1r ` A ) ) ) ) <-> ( v = ( 1r ` A ) /\ v = u ) ) )
91 84 90 mpbid 
 |-  ( ph -> ( v = ( 1r ` A ) /\ v = u ) )