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Theorem hdmaprnlem7N 35503
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 2443 . . 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 35237 . . . 4  |-  ( ph  ->  C  e.  LMod )
8 lmodabl 16992 . . . 4  |-  ( C  e.  LMod  ->  C  e. 
Abel )
97, 8syl 16 . . 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 35478 . . 3  |-  ( ph  ->  ( S `  u
)  e.  D )
15 hdmaprnlem1.se . . . 4  |-  ( ph  ->  s  e.  ( D 
\  { Q }
) )
1615eldifad 3340 . . 3  |-  ( ph  ->  s  e.  D )
17 hdmaprnlem1.n . . . . 5  |-  N  =  ( LSpan `  U )
18 hdmaprnlem1.l . . . . 5  |-  L  =  ( LSpan `  C )
19 hdmaprnlem1.m . . . . 5  |-  M  =  ( (mapd `  K
) `  W )
20 hdmaprnlem1.ve . . . . 5  |-  ( ph  ->  v  e.  V )
21 hdmaprnlem1.e . . . . 5  |-  ( ph  ->  ( M `  ( N `  { v } ) )  =  ( L `  {
s } ) )
22 hdmaprnlem1.un . . . . 5  |-  ( ph  ->  -.  u  e.  ( N `  { v } ) )
23 hdmaprnlem1.q . . . . 5  |-  Q  =  ( 0g `  C
)
24 hdmaprnlem1.o . . . . 5  |-  .0.  =  ( 0g `  U )
25 hdmaprnlem1.t2 . . . . 5  |-  ( ph  ->  t  e.  ( ( N `  { v } )  \  {  .0.  } ) )
264, 10, 11, 17, 5, 18, 19, 12, 6, 15, 20, 21, 13, 22, 1, 23, 24, 2, 25hdmaprnlem4tN 35500 . . . 4  |-  ( ph  ->  t  e.  V )
274, 10, 11, 5, 1, 12, 6, 26hdmapcl 35478 . . 3  |-  ( ph  ->  ( S `  t
)  e.  D )
281, 2, 3, 9, 14, 16, 27, 9, 14, 16, 27ablpnpcan 16309 . 2  |-  ( ph  ->  ( ( ( S `
 u )  .+b  s ) ( -g `  C ) ( ( S `  u ) 
.+b  ( S `  t ) ) )  =  ( s (
-g `  C )
( S `  t
) ) )
291, 2lmodvacl 16962 . . . . 5  |-  ( ( C  e.  LMod  /\  ( S `  u )  e.  D  /\  s  e.  D )  ->  (
( S `  u
)  .+b  s )  e.  D )
307, 14, 16, 29syl3anc 1218 . . . 4  |-  ( ph  ->  ( ( S `  u )  .+b  s
)  e.  D )
31 eqid 2443 . . . . 5  |-  ( LSubSp `  C )  =  (
LSubSp `  C )
321, 31, 18lspsncl 17058 . . . 4  |-  ( ( C  e.  LMod  /\  (
( S `  u
)  .+b  s )  e.  D )  ->  ( L `  { (
( S `  u
)  .+b  s ) } )  e.  (
LSubSp `  C ) )
337, 30, 32syl2anc 661 . . 3  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  e.  ( LSubSp `  C )
)
341, 18lspsnid 17074 . . . 4  |-  ( ( C  e.  LMod  /\  (
( S `  u
)  .+b  s )  e.  D )  ->  (
( S `  u
)  .+b  s )  e.  ( L `  {
( ( S `  u )  .+b  s
) } ) )
357, 30, 34syl2anc 661 . . 3  |-  ( ph  ->  ( ( S `  u )  .+b  s
)  e.  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )
361, 2lmodvacl 16962 . . . . . 6  |-  ( ( C  e.  LMod  /\  ( S `  u )  e.  D  /\  ( S `  t )  e.  D )  ->  (
( S `  u
)  .+b  ( S `  t ) )  e.  D )
377, 14, 27, 36syl3anc 1218 . . . . 5  |-  ( ph  ->  ( ( S `  u )  .+b  ( S `  t )
)  e.  D )
381, 18lspsnid 17074 . . . . 5  |-  ( ( C  e.  LMod  /\  (
( S `  u
)  .+b  ( S `  t ) )  e.  D )  ->  (
( S `  u
)  .+b  ( S `  t ) )  e.  ( L `  {
( ( S `  u )  .+b  ( S `  t )
) } ) )
397, 37, 38syl2anc 661 . . . 4  |-  ( ph  ->  ( ( S `  u )  .+b  ( S `  t )
)  e.  ( L `
 { ( ( S `  u ) 
.+b  ( S `  t ) ) } ) )
40 hdmaprnlem1.p . . . . 5  |-  .+  =  ( +g  `  U )
41 hdmaprnlem1.pt . . . . 5  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  =  ( M `  ( N `  { (
u  .+  t ) } ) ) )
424, 10, 11, 17, 5, 18, 19, 12, 6, 15, 20, 21, 13, 22, 1, 23, 24, 2, 25, 40, 41hdmaprnlem6N 35502 . . . 4  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  =  ( L `  {
( ( S `  u )  .+b  ( S `  t )
) } ) )
4339, 42eleqtrrd 2520 . . 3  |-  ( ph  ->  ( ( S `  u )  .+b  ( S `  t )
)  e.  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )
443, 31lssvsubcl 17025 . . 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 ) } ) )
457, 33, 35, 43, 44syl22anc 1219 . 2  |-  ( ph  ->  ( ( ( S `
 u )  .+b  s ) ( -g `  C ) ( ( S `  u ) 
.+b  ( S `  t ) ) )  e.  ( L `  { ( ( S `
 u )  .+b  s ) } ) )
4628, 45eqeltrrd 2518 1  |-  ( ph  ->  ( s ( -g `  C ) ( S `
 t ) )  e.  ( L `  { ( ( S `
 u )  .+b  s ) } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    \ cdif 3325   {csn 3877   ` cfv 5418  (class class class)co 6091   Basecbs 14174   +g cplusg 14238   0gc0g 14378   -gcsg 15413   Abelcabel 16278   LModclmod 16948   LSubSpclss 17013   LSpanclspn 17052   HLchlt 32995   LHypclh 33628   DVecHcdvh 34723  LCDualclcd 35231  mapdcmpd 35269  HDMapchdma 35438
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-riotaBAD 32604
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-ot 3886  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-tpos 6745  df-undef 6792  df-recs 6832  df-rdg 6866  df-1o 6920  df-oadd 6924  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-sca 14254  df-vsca 14255  df-0g 14380  df-mre 14524  df-mrc 14525  df-acs 14527  df-poset 15116  df-plt 15128  df-lub 15144  df-glb 15145  df-join 15146  df-meet 15147  df-p0 15209  df-p1 15210  df-lat 15216  df-clat 15278  df-mnd 15415  df-submnd 15465  df-grp 15545  df-minusg 15546  df-sbg 15547  df-subg 15678  df-cntz 15835  df-oppg 15861  df-lsm 16135  df-cmn 16279  df-abl 16280  df-mgp 16592  df-ur 16604  df-rng 16647  df-oppr 16715  df-dvdsr 16733  df-unit 16734  df-invr 16764  df-dvr 16775  df-drng 16834  df-lmod 16950  df-lss 17014  df-lsp 17053  df-lvec 17184  df-lsatoms 32621  df-lshyp 32622  df-lcv 32664  df-lfl 32703  df-lkr 32731  df-ldual 32769  df-oposet 32821  df-ol 32823  df-oml 32824  df-covers 32911  df-ats 32912  df-atl 32943  df-cvlat 32967  df-hlat 32996  df-llines 33142  df-lplanes 33143  df-lvols 33144  df-lines 33145  df-psubsp 33147  df-pmap 33148  df-padd 33440  df-lhyp 33632  df-laut 33633  df-ldil 33748  df-ltrn 33749  df-trl 33803  df-tgrp 34387  df-tendo 34399  df-edring 34401  df-dveca 34647  df-disoa 34674  df-dvech 34724  df-dib 34784  df-dic 34818  df-dih 34874  df-doch 34993  df-djh 35040  df-lcdual 35232  df-mapd 35270  df-hvmap 35402  df-hdmap1 35439  df-hdmap 35440
This theorem is referenced by:  hdmaprnlem9N  35505
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