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Theorem hdmaprnlem7N 31316
Description: Part of proof of part 12 in [Baer] p. 49 line 19, s-St  e. G(u'+s) = P*. (Contributed by NM, 27-May-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
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 )
hdmaprnlem1.t2  |-  ( ph  ->  t  e.  ( ( N `  { v } )  \  {  .0.  } ) )
hdmaprnlem1.p  |-  .+  =  ( +g  `  U )
hdmaprnlem1.pt  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  =  ( M `  ( N `  { (
u  .+  t ) } ) ) )
Assertion
Ref Expression
hdmaprnlem7N  |-  ( ph  ->  ( s ( -g `  C ) ( S `
 t ) )  e.  ( L `  { ( ( S `
 u )  .+b  s ) } ) )

Proof of Theorem hdmaprnlem7N
StepHypRef Expression
1 hdmaprnlem1.d . . 3  |-  D  =  ( Base `  C
)
2 hdmaprnlem1.a . . 3  |-  .+b  =  ( +g  `  C )
3 eqid 2285 . . 3  |-  ( -g `  C )  =  (
-g `  C )
4 hdmaprnlem1.h . . . . 5  |-  H  =  ( LHyp `  K
)
5 hdmaprnlem1.c . . . . 5  |-  C  =  ( (LCDual `  K
) `  W )
6 hdmaprnlem1.k . . . . 5  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
74, 5, 6lcdlmod 31050 . . . 4  |-  ( ph  ->  C  e.  LMod )
8 lmodabl 15667 . . . 4  |-  ( C  e.  LMod  ->  C  e. 
Abel )
97, 8syl 17 . . 3  |-  ( ph  ->  C  e.  Abel )
10 hdmaprnlem1.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
11 hdmaprnlem1.v . . . 4  |-  V  =  ( Base `  U
)
12 hdmaprnlem1.s . . . 4  |-  S  =  ( (HDMap `  K
) `  W )
13 hdmaprnlem1.ue . . . 4  |-  ( ph  ->  u  e.  V )
144, 10, 11, 5, 1, 12, 6, 13hdmapcl 31291 . . 3  |-  ( ph  ->  ( S `  u
)  e.  D )
15 hdmaprnlem1.se . . . 4  |-  ( ph  ->  s  e.  ( D 
\  { Q }
) )
16 eldifi 3300 . . . 4  |-  ( s  e.  ( D  \  { Q } )  -> 
s  e.  D )
1715, 16syl 17 . . 3  |-  ( ph  ->  s  e.  D )
18 hdmaprnlem1.n . . . . 5  |-  N  =  ( LSpan `  U )
19 hdmaprnlem1.l . . . . 5  |-  L  =  ( LSpan `  C )
20 hdmaprnlem1.m . . . . 5  |-  M  =  ( (mapd `  K
) `  W )
21 hdmaprnlem1.ve . . . . 5  |-  ( ph  ->  v  e.  V )
22 hdmaprnlem1.e . . . . 5  |-  ( ph  ->  ( M `  ( N `  { v } ) )  =  ( L `  {
s } ) )
23 hdmaprnlem1.un . . . . 5  |-  ( ph  ->  -.  u  e.  ( N `  { v } ) )
24 hdmaprnlem1.q . . . . 5  |-  Q  =  ( 0g `  C
)
25 hdmaprnlem1.o . . . . 5  |-  .0.  =  ( 0g `  U )
26 hdmaprnlem1.t2 . . . . 5  |-  ( ph  ->  t  e.  ( ( N `  { v } )  \  {  .0.  } ) )
274, 10, 11, 18, 5, 19, 20, 12, 6, 15, 21, 22, 13, 23, 1, 24, 25, 2, 26hdmaprnlem4tN 31313 . . . 4  |-  ( ph  ->  t  e.  V )
284, 10, 11, 5, 1, 12, 6, 27hdmapcl 31291 . . 3  |-  ( ph  ->  ( S `  t
)  e.  D )
291, 2, 3, 9, 14, 17, 28, 9, 14, 17, 28ablpnpcan 15116 . 2  |-  ( ph  ->  ( ( ( S `
 u )  .+b  s ) ( -g `  C ) ( ( S `  u ) 
.+b  ( S `  t ) ) )  =  ( s (
-g `  C )
( S `  t
) ) )
301, 2lmodvacl 15636 . . . . 5  |-  ( ( C  e.  LMod  /\  ( S `  u )  e.  D  /\  s  e.  D )  ->  (
( S `  u
)  .+b  s )  e.  D )
317, 14, 17, 30syl3anc 1184 . . . 4  |-  ( ph  ->  ( ( S `  u )  .+b  s
)  e.  D )
32 eqid 2285 . . . . 5  |-  ( LSubSp `  C )  =  (
LSubSp `  C )
331, 32, 19lspsncl 15729 . . . 4  |-  ( ( C  e.  LMod  /\  (
( S `  u
)  .+b  s )  e.  D )  ->  ( L `  { (
( S `  u
)  .+b  s ) } )  e.  (
LSubSp `  C ) )
347, 31, 33syl2anc 644 . . 3  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  e.  ( LSubSp `  C )
)
351, 19lspsnid 15745 . . . 4  |-  ( ( C  e.  LMod  /\  (
( S `  u
)  .+b  s )  e.  D )  ->  (
( S `  u
)  .+b  s )  e.  ( L `  {
( ( S `  u )  .+b  s
) } ) )
367, 31, 35syl2anc 644 . . 3  |-  ( ph  ->  ( ( S `  u )  .+b  s
)  e.  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )
371, 2lmodvacl 15636 . . . . . 6  |-  ( ( C  e.  LMod  /\  ( S `  u )  e.  D  /\  ( S `  t )  e.  D )  ->  (
( S `  u
)  .+b  ( S `  t ) )  e.  D )
387, 14, 28, 37syl3anc 1184 . . . . 5  |-  ( ph  ->  ( ( S `  u )  .+b  ( S `  t )
)  e.  D )
391, 19lspsnid 15745 . . . . 5  |-  ( ( C  e.  LMod  /\  (
( S `  u
)  .+b  ( S `  t ) )  e.  D )  ->  (
( S `  u
)  .+b  ( S `  t ) )  e.  ( L `  {
( ( S `  u )  .+b  ( S `  t )
) } ) )
407, 38, 39syl2anc 644 . . . 4  |-  ( ph  ->  ( ( S `  u )  .+b  ( S `  t )
)  e.  ( L `
 { ( ( S `  u ) 
.+b  ( S `  t ) ) } ) )
41 hdmaprnlem1.p . . . . 5  |-  .+  =  ( +g  `  U )
42 hdmaprnlem1.pt . . . . 5  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  =  ( M `  ( N `  { (
u  .+  t ) } ) ) )
434, 10, 11, 18, 5, 19, 20, 12, 6, 15, 21, 22, 13, 23, 1, 24, 25, 2, 26, 41, 42hdmaprnlem6N 31315 . . . 4  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  =  ( L `  {
( ( S `  u )  .+b  ( S `  t )
) } ) )
4440, 43eleqtrrd 2362 . . 3  |-  ( ph  ->  ( ( S `  u )  .+b  ( S `  t )
)  e.  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )
453, 32lssvsubcl 15696 . . 3  |-  ( ( ( C  e.  LMod  /\  ( L `  {
( ( S `  u )  .+b  s
) } )  e.  ( LSubSp `  C )
)  /\  ( (
( S `  u
)  .+b  s )  e.  ( L `  {
( ( S `  u )  .+b  s
) } )  /\  ( ( S `  u )  .+b  ( S `  t )
)  e.  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) ) )  -> 
( ( ( S `
 u )  .+b  s ) ( -g `  C ) ( ( S `  u ) 
.+b  ( S `  t ) ) )  e.  ( L `  { ( ( S `
 u )  .+b  s ) } ) )
467, 34, 36, 44, 45syl22anc 1185 . 2  |-  ( ph  ->  ( ( ( S `
 u )  .+b  s ) ( -g `  C ) ( ( S `  u ) 
.+b  ( S `  t ) ) )  e.  ( L `  { ( ( S `
 u )  .+b  s ) } ) )
4729, 46eqeltrrd 2360 1  |-  ( ph  ->  ( s ( -g `  C ) ( S `
 t ) )  e.  ( L `  { ( ( S `
 u )  .+b  s ) } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    = wceq 1624    e. wcel 1685    \ cdif 3151   {csn 3642   ` cfv 5222  (class class class)co 5820   Basecbs 13143   +g cplusg 13203   0gc0g 13395   -gcsg 14360   Abelcabel 15085   LModclmod 15622   LSubSpclss 15684   LSpanclspn 15723   HLchlt 28808   LHypclh 29441   DVecHcdvh 30536  LCDualclcd 31044  mapdcmpd 31082  HDMapchdma 31251
This theorem is referenced by:  hdmaprnlem9N  31318
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-fal 1313  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-ot 3652  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-of 6040  df-1st 6084  df-2nd 6085  df-tpos 6196  df-iota 6253  df-undef 6292  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-oadd 6479  df-er 6656  df-map 6770  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-nn 9743  df-2 9800  df-3 9801  df-4 9802  df-5 9803  df-6 9804  df-n0 9962  df-z 10021  df-uz 10227  df-fz 10778  df-struct 13145  df-ndx 13146  df-slot 13147  df-base 13148  df-sets 13149  df-ress 13150  df-plusg 13216  df-mulr 13217  df-sca 13219  df-vsca 13220  df-0g 13399  df-mre 13483  df-mrc 13484  df-acs 13486  df-poset 14075  df-plt 14087  df-lub 14103  df-glb 14104  df-join 14105  df-meet 14106  df-p0 14140  df-p1 14141  df-lat 14147  df-clat 14209  df-mnd 14362  df-submnd 14411  df-grp 14484  df-minusg 14485  df-sbg 14486  df-subg 14613  df-cntz 14788  df-oppg 14814  df-lsm 14942  df-cmn 15086  df-abl 15087  df-mgp 15321  df-rng 15335  df-ur 15337  df-oppr 15400  df-dvdsr 15418  df-unit 15419  df-invr 15449  df-dvr 15460  df-drng 15509  df-lmod 15624  df-lss 15685  df-lsp 15724  df-lvec 15851  df-lsatoms 28434  df-lshyp 28435  df-lcv 28477  df-lfl 28516  df-lkr 28544  df-ldual 28582  df-oposet 28634  df-ol 28636  df-oml 28637  df-covers 28724  df-ats 28725  df-atl 28756  df-cvlat 28780  df-hlat 28809  df-llines 28955  df-lplanes 28956  df-lvols 28957  df-lines 28958  df-psubsp 28960  df-pmap 28961  df-padd 29253  df-lhyp 29445  df-laut 29446  df-ldil 29561  df-ltrn 29562  df-trl 29616  df-tgrp 30200  df-tendo 30212  df-edring 30214  df-dveca 30460  df-disoa 30487  df-dvech 30537  df-dib 30597  df-dic 30631  df-dih 30687  df-doch 30806  df-djh 30853  df-lcdual 31045  df-mapd 31083  df-hvmap 31215  df-hdmap1 31252  df-hdmap 31253
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