Metamath Proof Explorer

Theorem hdmaprnlem3eN

Description: Lemma for hdmaprnN . (Contributed by NM, 29-May-2015) (New usage is discouraged.)

Ref Expression
Hypotheses hdmaprnlem1.h 
|- H = ( LHyp ` K )
hdmaprnlem1.u 
|- U = ( ( DVecH ` K ) ` W )
hdmaprnlem1.v 
|- V = ( Base ` U )
hdmaprnlem1.n 
|- N = ( LSpan ` U )
hdmaprnlem1.c 
|- C = ( ( LCDual ` K ) ` W )
hdmaprnlem1.l 
|- L = ( LSpan ` C )
hdmaprnlem1.m 
|- M = ( ( mapd ` K ) ` W )
hdmaprnlem1.s 
|- S = ( ( HDMap ` K ) ` W )
hdmaprnlem1.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
hdmaprnlem1.se 
|- ( ph -> s e. ( D \ { Q } ) )
hdmaprnlem1.ve 
|- ( ph -> v e. V )
hdmaprnlem1.e 
|- ( ph -> ( M ` ( N ` { v } ) ) = ( L ` { s } ) )
hdmaprnlem1.ue 
|- ( ph -> u e. V )
hdmaprnlem1.un 
|- ( ph -> -. u e. ( N ` { v } ) )
hdmaprnlem1.d 
|- D = ( Base ` C )
hdmaprnlem1.q 
|- Q = ( 0g ` C )
hdmaprnlem1.o 
|- .0. = ( 0g ` U )
hdmaprnlem1.a 
|- .+b = ( +g ` C )
hdmaprnlem3e.p 
|- .+ = ( +g ` U )
Assertion hdmaprnlem3eN 
|- ( ph -> E. t e. ( ( N ` { v } ) \ { .0. } ) ( L ` { ( ( S ` u ) .+b s ) } ) = ( M ` ( N ` { ( u .+ t ) } ) ) )

Proof

Step Hyp Ref Expression
1 hdmaprnlem1.h 
 |-  H = ( LHyp ` K )
2 hdmaprnlem1.u 
 |-  U = ( ( DVecH ` K ) ` W )
3 hdmaprnlem1.v 
 |-  V = ( Base ` U )
4 hdmaprnlem1.n 
 |-  N = ( LSpan ` U )
5 hdmaprnlem1.c 
 |-  C = ( ( LCDual ` K ) ` W )
6 hdmaprnlem1.l 
 |-  L = ( LSpan ` C )
7 hdmaprnlem1.m 
 |-  M = ( ( mapd ` K ) ` W )
8 hdmaprnlem1.s 
 |-  S = ( ( HDMap ` K ) ` W )
9 hdmaprnlem1.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
10 hdmaprnlem1.se 
 |-  ( ph -> s e. ( D \ { Q } ) )
11 hdmaprnlem1.ve 
 |-  ( ph -> v e. V )
12 hdmaprnlem1.e 
 |-  ( ph -> ( M ` ( N ` { v } ) ) = ( L ` { s } ) )
13 hdmaprnlem1.ue 
 |-  ( ph -> u e. V )
14 hdmaprnlem1.un 
 |-  ( ph -> -. u e. ( N ` { v } ) )
15 hdmaprnlem1.d 
 |-  D = ( Base ` C )
16 hdmaprnlem1.q 
 |-  Q = ( 0g ` C )
17 hdmaprnlem1.o 
 |-  .0. = ( 0g ` U )
18 hdmaprnlem1.a 
 |-  .+b = ( +g ` C )
19 hdmaprnlem3e.p 
 |-  .+ = ( +g ` U )
20 eqid 
 |-  ( LSAtoms ` U ) = ( LSAtoms ` U )
21 1 2 9 dvhlvec 
 |-  ( ph -> U e. LVec )
22 eqid 
 |-  ( LSAtoms ` C ) = ( LSAtoms ` C )
23 1 5 9 lcdlmod 
 |-  ( ph -> C e. LMod )
24 1 2 3 5 15 8 9 13 hdmapcl 
 |-  ( ph -> ( S ` u ) e. D )
25 10 eldifad 
 |-  ( ph -> s e. D )
26 15 18 lmodvacl 
 |-  ( ( C e. LMod /\ ( S ` u ) e. D /\ s e. D ) -> ( ( S ` u ) .+b s ) e. D )
27 23 24 25 26 syl3anc 
 |-  ( ph -> ( ( S ` u ) .+b s ) e. D )
28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 hdmaprnlem1N 
 |-  ( ph -> ( L ` { ( S ` u ) } ) =/= ( L ` { s } ) )
29 15 18 16 6 23 24 25 28 lmodindp1 
 |-  ( ph -> ( ( S ` u ) .+b s ) =/= Q )
30 eldifsn 
 |-  ( ( ( S ` u ) .+b s ) e. ( D \ { Q } ) <-> ( ( ( S ` u ) .+b s ) e. D /\ ( ( S ` u ) .+b s ) =/= Q ) )
31 27 29 30 sylanbrc 
 |-  ( ph -> ( ( S ` u ) .+b s ) e. ( D \ { Q } ) )
32 15 6 16 22 23 31 lsatlspsn 
 |-  ( ph -> ( L ` { ( ( S ` u ) .+b s ) } ) e. ( LSAtoms ` C ) )
33 1 7 2 20 5 22 9 32 mapdcnvatN 
 |-  ( ph -> ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) e. ( LSAtoms ` U ) )
34 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 hdmaprnlem3uN 
 |-  ( ph -> ( N ` { u } ) =/= ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) )
35 34 necomd 
 |-  ( ph -> ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) =/= ( N ` { u } ) )
36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 hdmaprnlem3N 
 |-  ( ph -> ( N ` { v } ) =/= ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) )
37 36 necomd 
 |-  ( ph -> ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) =/= ( N ` { v } ) )
38 eqid 
 |-  ( LSubSp ` C ) = ( LSubSp ` C )
39 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
40 1 2 9 dvhlmod 
 |-  ( ph -> U e. LMod )
41 3 39 4 lspsncl 
 |-  ( ( U e. LMod /\ u e. V ) -> ( N ` { u } ) e. ( LSubSp ` U ) )
42 40 13 41 syl2anc 
 |-  ( ph -> ( N ` { u } ) e. ( LSubSp ` U ) )
43 1 7 2 39 5 38 9 42 mapdcl2 
 |-  ( ph -> ( M ` ( N ` { u } ) ) e. ( LSubSp ` C ) )
44 3 39 4 lspsncl 
 |-  ( ( U e. LMod /\ v e. V ) -> ( N ` { v } ) e. ( LSubSp ` U ) )
45 40 11 44 syl2anc 
 |-  ( ph -> ( N ` { v } ) e. ( LSubSp ` U ) )
46 1 7 2 39 5 38 9 45 mapdcl2 
 |-  ( ph -> ( M ` ( N ` { v } ) ) e. ( LSubSp ` C ) )
47 eqid 
 |-  ( LSSum ` C ) = ( LSSum ` C )
48 38 47 lsmcl 
 |-  ( ( C e. LMod /\ ( M ` ( N ` { u } ) ) e. ( LSubSp ` C ) /\ ( M ` ( N ` { v } ) ) e. ( LSubSp ` C ) ) -> ( ( M ` ( N ` { u } ) ) ( LSSum ` C ) ( M ` ( N ` { v } ) ) ) e. ( LSubSp ` C ) )
49 23 43 46 48 syl3anc 
 |-  ( ph -> ( ( M ` ( N ` { u } ) ) ( LSSum ` C ) ( M ` ( N ` { v } ) ) ) e. ( LSubSp ` C ) )
50 38 lsssssubg 
 |-  ( C e. LMod -> ( LSubSp ` C ) C_ ( SubGrp ` C ) )
51 23 50 syl 
 |-  ( ph -> ( LSubSp ` C ) C_ ( SubGrp ` C ) )
52 51 43 sseldd 
 |-  ( ph -> ( M ` ( N ` { u } ) ) e. ( SubGrp ` C ) )
53 51 46 sseldd 
 |-  ( ph -> ( M ` ( N ` { v } ) ) e. ( SubGrp ` C ) )
54 15 6 lspsnid 
 |-  ( ( C e. LMod /\ ( S ` u ) e. D ) -> ( S ` u ) e. ( L ` { ( S ` u ) } ) )
55 23 24 54 syl2anc 
 |-  ( ph -> ( S ` u ) e. ( L ` { ( S ` u ) } ) )
56 1 2 3 4 5 6 7 8 9 13 hdmap10 
 |-  ( ph -> ( M ` ( N ` { u } ) ) = ( L ` { ( S ` u ) } ) )
57 55 56 eleqtrrd 
 |-  ( ph -> ( S ` u ) e. ( M ` ( N ` { u } ) ) )
58 eqimss2 
 |-  ( ( M ` ( N ` { v } ) ) = ( L ` { s } ) -> ( L ` { s } ) C_ ( M ` ( N ` { v } ) ) )
59 12 58 syl 
 |-  ( ph -> ( L ` { s } ) C_ ( M ` ( N ` { v } ) ) )
60 15 38 6 23 46 25 ellspsn5b 
 |-  ( ph -> ( s e. ( M ` ( N ` { v } ) ) <-> ( L ` { s } ) C_ ( M ` ( N ` { v } ) ) ) )
61 59 60 mpbird 
 |-  ( ph -> s e. ( M ` ( N ` { v } ) ) )
62 18 47 lsmelvali 
 |-  ( ( ( ( M ` ( N ` { u } ) ) e. ( SubGrp ` C ) /\ ( M ` ( N ` { v } ) ) e. ( SubGrp ` C ) ) /\ ( ( S ` u ) e. ( M ` ( N ` { u } ) ) /\ s e. ( M ` ( N ` { v } ) ) ) ) -> ( ( S ` u ) .+b s ) e. ( ( M ` ( N ` { u } ) ) ( LSSum ` C ) ( M ` ( N ` { v } ) ) ) )
63 52 53 57 61 62 syl22anc 
 |-  ( ph -> ( ( S ` u ) .+b s ) e. ( ( M ` ( N ` { u } ) ) ( LSSum ` C ) ( M ` ( N ` { v } ) ) ) )
64 38 6 23 49 63 ellspsn5 
 |-  ( ph -> ( L ` { ( ( S ` u ) .+b s ) } ) C_ ( ( M ` ( N ` { u } ) ) ( LSSum ` C ) ( M ` ( N ` { v } ) ) ) )
65 eqid 
 |-  ( LSSum ` U ) = ( LSSum ` U )
66 1 7 2 39 65 5 47 9 42 45 mapdlsm 
 |-  ( ph -> ( M ` ( ( N ` { u } ) ( LSSum ` U ) ( N ` { v } ) ) ) = ( ( M ` ( N ` { u } ) ) ( LSSum ` C ) ( M ` ( N ` { v } ) ) ) )
67 64 66 sseqtrrd 
 |-  ( ph -> ( L ` { ( ( S ` u ) .+b s ) } ) C_ ( M ` ( ( N ` { u } ) ( LSSum ` U ) ( N ` { v } ) ) ) )
68 15 38 6 lspsncl 
 |-  ( ( C e. LMod /\ ( ( S ` u ) .+b s ) e. D ) -> ( L ` { ( ( S ` u ) .+b s ) } ) e. ( LSubSp ` C ) )
69 23 27 68 syl2anc 
 |-  ( ph -> ( L ` { ( ( S ` u ) .+b s ) } ) e. ( LSubSp ` C ) )
70 1 7 5 38 9 mapdrn2 
 |-  ( ph -> ran M = ( LSubSp ` C ) )
71 69 70 eleqtrrd 
 |-  ( ph -> ( L ` { ( ( S ` u ) .+b s ) } ) e. ran M )
72 39 65 lsmcl 
 |-  ( ( U e. LMod /\ ( N ` { u } ) e. ( LSubSp ` U ) /\ ( N ` { v } ) e. ( LSubSp ` U ) ) -> ( ( N ` { u } ) ( LSSum ` U ) ( N ` { v } ) ) e. ( LSubSp ` U ) )
73 40 42 45 72 syl3anc 
 |-  ( ph -> ( ( N ` { u } ) ( LSSum ` U ) ( N ` { v } ) ) e. ( LSubSp ` U ) )
74 1 7 2 39 9 73 mapdcl 
 |-  ( ph -> ( M ` ( ( N ` { u } ) ( LSSum ` U ) ( N ` { v } ) ) ) e. ran M )
75 1 7 9 71 74 mapdcnvordN 
 |-  ( ph -> ( ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) C_ ( `' M ` ( M ` ( ( N ` { u } ) ( LSSum ` U ) ( N ` { v } ) ) ) ) <-> ( L ` { ( ( S ` u ) .+b s ) } ) C_ ( M ` ( ( N ` { u } ) ( LSSum ` U ) ( N ` { v } ) ) ) ) )
76 67 75 mpbird 
 |-  ( ph -> ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) C_ ( `' M ` ( M ` ( ( N ` { u } ) ( LSSum ` U ) ( N ` { v } ) ) ) ) )
77 3 4 65 40 13 11 lsmpr 
 |-  ( ph -> ( N ` { u , v } ) = ( ( N ` { u } ) ( LSSum ` U ) ( N ` { v } ) ) )
78 1 7 2 39 9 73 mapdcnvid1N 
 |-  ( ph -> ( `' M ` ( M ` ( ( N ` { u } ) ( LSSum ` U ) ( N ` { v } ) ) ) ) = ( ( N ` { u } ) ( LSSum ` U ) ( N ` { v } ) ) )
79 77 78 eqtr4d 
 |-  ( ph -> ( N ` { u , v } ) = ( `' M ` ( M ` ( ( N ` { u } ) ( LSSum ` U ) ( N ` { v } ) ) ) ) )
80 76 79 sseqtrrd 
 |-  ( ph -> ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) C_ ( N ` { u , v } ) )
81 3 19 17 4 20 21 33 13 11 35 37 80 lsatfixedN 
 |-  ( ph -> E. t e. ( ( N ` { v } ) \ { .0. } ) ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) = ( N ` { ( u .+ t ) } ) )
82 simpr 
 |-  ( ( ( ph /\ t e. ( ( N ` { v } ) \ { .0. } ) ) /\ ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) = ( N ` { ( u .+ t ) } ) ) -> ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) = ( N ` { ( u .+ t ) } ) )
83 9 ad2antrr 
 |-  ( ( ( ph /\ t e. ( ( N ` { v } ) \ { .0. } ) ) /\ ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) = ( N ` { ( u .+ t ) } ) ) -> ( K e. HL /\ W e. H ) )
84 40 ad2antrr 
 |-  ( ( ( ph /\ t e. ( ( N ` { v } ) \ { .0. } ) ) /\ ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) = ( N ` { ( u .+ t ) } ) ) -> U e. LMod )
85 13 ad2antrr 
 |-  ( ( ( ph /\ t e. ( ( N ` { v } ) \ { .0. } ) ) /\ ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) = ( N ` { ( u .+ t ) } ) ) -> u e. V )
86 10 ad2antrr 
 |-  ( ( ( ph /\ t e. ( ( N ` { v } ) \ { .0. } ) ) /\ ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) = ( N ` { ( u .+ t ) } ) ) -> s e. ( D \ { Q } ) )
87 11 ad2antrr 
 |-  ( ( ( ph /\ t e. ( ( N ` { v } ) \ { .0. } ) ) /\ ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) = ( N ` { ( u .+ t ) } ) ) -> v e. V )
88 12 ad2antrr 
 |-  ( ( ( ph /\ t e. ( ( N ` { v } ) \ { .0. } ) ) /\ ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) = ( N ` { ( u .+ t ) } ) ) -> ( M ` ( N ` { v } ) ) = ( L ` { s } ) )
89 14 ad2antrr 
 |-  ( ( ( ph /\ t e. ( ( N ` { v } ) \ { .0. } ) ) /\ ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) = ( N ` { ( u .+ t ) } ) ) -> -. u e. ( N ` { v } ) )
90 simplr 
 |-  ( ( ( ph /\ t e. ( ( N ` { v } ) \ { .0. } ) ) /\ ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) = ( N ` { ( u .+ t ) } ) ) -> t e. ( ( N ` { v } ) \ { .0. } ) )
91 1 2 3 4 5 6 7 8 83 86 87 88 85 89 15 16 17 18 90 hdmaprnlem4tN 
 |-  ( ( ( ph /\ t e. ( ( N ` { v } ) \ { .0. } ) ) /\ ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) = ( N ` { ( u .+ t ) } ) ) -> t e. V )
92 3 19 lmodvacl 
 |-  ( ( U e. LMod /\ u e. V /\ t e. V ) -> ( u .+ t ) e. V )
93 84 85 91 92 syl3anc 
 |-  ( ( ( ph /\ t e. ( ( N ` { v } ) \ { .0. } ) ) /\ ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) = ( N ` { ( u .+ t ) } ) ) -> ( u .+ t ) e. V )
94 3 39 4 lspsncl 
 |-  ( ( U e. LMod /\ ( u .+ t ) e. V ) -> ( N ` { ( u .+ t ) } ) e. ( LSubSp ` U ) )
95 84 93 94 syl2anc 
 |-  ( ( ( ph /\ t e. ( ( N ` { v } ) \ { .0. } ) ) /\ ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) = ( N ` { ( u .+ t ) } ) ) -> ( N ` { ( u .+ t ) } ) e. ( LSubSp ` U ) )
96 1 7 2 39 83 95 mapdcnvid1N 
 |-  ( ( ( ph /\ t e. ( ( N ` { v } ) \ { .0. } ) ) /\ ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) = ( N ` { ( u .+ t ) } ) ) -> ( `' M ` ( M ` ( N ` { ( u .+ t ) } ) ) ) = ( N ` { ( u .+ t ) } ) )
97 82 96 eqtr4d 
 |-  ( ( ( ph /\ t e. ( ( N ` { v } ) \ { .0. } ) ) /\ ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) = ( N ` { ( u .+ t ) } ) ) -> ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) = ( `' M ` ( M ` ( N ` { ( u .+ t ) } ) ) ) )
98 71 ad2antrr 
 |-  ( ( ( ph /\ t e. ( ( N ` { v } ) \ { .0. } ) ) /\ ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) = ( N ` { ( u .+ t ) } ) ) -> ( L ` { ( ( S ` u ) .+b s ) } ) e. ran M )
99 1 7 2 39 83 95 mapdcl 
 |-  ( ( ( ph /\ t e. ( ( N ` { v } ) \ { .0. } ) ) /\ ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) = ( N ` { ( u .+ t ) } ) ) -> ( M ` ( N ` { ( u .+ t ) } ) ) e. ran M )
100 1 7 83 98 99 mapdcnv11N 
 |-  ( ( ( ph /\ t e. ( ( N ` { v } ) \ { .0. } ) ) /\ ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) = ( N ` { ( u .+ t ) } ) ) -> ( ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) = ( `' M ` ( M ` ( N ` { ( u .+ t ) } ) ) ) <-> ( L ` { ( ( S ` u ) .+b s ) } ) = ( M ` ( N ` { ( u .+ t ) } ) ) ) )
101 97 100 mpbid 
 |-  ( ( ( ph /\ t e. ( ( N ` { v } ) \ { .0. } ) ) /\ ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) = ( N ` { ( u .+ t ) } ) ) -> ( L ` { ( ( S ` u ) .+b s ) } ) = ( M ` ( N ` { ( u .+ t ) } ) ) )
102 101 ex 
 |-  ( ( ph /\ t e. ( ( N ` { v } ) \ { .0. } ) ) -> ( ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) = ( N ` { ( u .+ t ) } ) -> ( L ` { ( ( S ` u ) .+b s ) } ) = ( M ` ( N ` { ( u .+ t ) } ) ) ) )
103 102 reximdva 
 |-  ( ph -> ( E. t e. ( ( N ` { v } ) \ { .0. } ) ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) = ( N ` { ( u .+ t ) } ) -> E. t e. ( ( N ` { v } ) \ { .0. } ) ( L ` { ( ( S ` u ) .+b s ) } ) = ( M ` ( N ` { ( u .+ t ) } ) ) ) )
104 81 103 mpd 
 |-  ( ph -> E. t e. ( ( N ` { v } ) \ { .0. } ) ( L ` { ( ( S ` u ) .+b s ) } ) = ( M ` ( N ` { ( u .+ t ) } ) ) )