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Theorem hdmaprnlem7N 32341
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 2404 . . 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 32075 . . . 4  |-  ( ph  ->  C  e.  LMod )
8 lmodabl 15946 . . . 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 32316 . . 3  |-  ( ph  ->  ( S `  u
)  e.  D )
15 hdmaprnlem1.se . . . 4  |-  ( ph  ->  s  e.  ( D 
\  { Q }
) )
1615eldifad 3292 . . 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 32338 . . . 4  |-  ( ph  ->  t  e.  V )
274, 10, 11, 5, 1, 12, 6, 26hdmapcl 32316 . . 3  |-  ( ph  ->  ( S `  t
)  e.  D )
281, 2, 3, 9, 14, 16, 27, 9, 14, 16, 27ablpnpcan 15399 . 2  |-  ( ph  ->  ( ( ( S `
 u )  .+b  s ) ( -g `  C ) ( ( S `  u ) 
.+b  ( S `  t ) ) )  =  ( s (
-g `  C )
( S `  t
) ) )
291, 2lmodvacl 15919 . . . . 5  |-  ( ( C  e.  LMod  /\  ( S `  u )  e.  D  /\  s  e.  D )  ->  (
( S `  u
)  .+b  s )  e.  D )
307, 14, 16, 29syl3anc 1184 . . . 4  |-  ( ph  ->  ( ( S `  u )  .+b  s
)  e.  D )
31 eqid 2404 . . . . 5  |-  ( LSubSp `  C )  =  (
LSubSp `  C )
321, 31, 18lspsncl 16008 . . . 4  |-  ( ( C  e.  LMod  /\  (
( S `  u
)  .+b  s )  e.  D )  ->  ( L `  { (
( S `  u
)  .+b  s ) } )  e.  (
LSubSp `  C ) )
337, 30, 32syl2anc 643 . . 3  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  e.  ( LSubSp `  C )
)
341, 18lspsnid 16024 . . . 4  |-  ( ( C  e.  LMod  /\  (
( S `  u
)  .+b  s )  e.  D )  ->  (
( S `  u
)  .+b  s )  e.  ( L `  {
( ( S `  u )  .+b  s
) } ) )
357, 30, 34syl2anc 643 . . 3  |-  ( ph  ->  ( ( S `  u )  .+b  s
)  e.  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )
361, 2lmodvacl 15919 . . . . . 6  |-  ( ( C  e.  LMod  /\  ( S `  u )  e.  D  /\  ( S `  t )  e.  D )  ->  (
( S `  u
)  .+b  ( S `  t ) )  e.  D )
377, 14, 27, 36syl3anc 1184 . . . . 5  |-  ( ph  ->  ( ( S `  u )  .+b  ( S `  t )
)  e.  D )
381, 18lspsnid 16024 . . . . 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 643 . . . 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 32340 . . . 4  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  =  ( L `  {
( ( S `  u )  .+b  ( S `  t )
) } ) )
4339, 42eleqtrrd 2481 . . 3  |-  ( ph  ->  ( ( S `  u )  .+b  ( S `  t )
)  e.  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) )
443, 31lssvsubcl 15975 . . 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 1185 . 2  |-  ( ph  ->  ( ( ( S `
 u )  .+b  s ) ( -g `  C ) ( ( S `  u ) 
.+b  ( S `  t ) ) )  e.  ( L `  { ( ( S `
 u )  .+b  s ) } ) )
4628, 45eqeltrrd 2479 1  |-  ( ph  ->  ( s ( -g `  C ) ( S `
 t ) )  e.  ( L `  { ( ( S `
 u )  .+b  s ) } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    \ cdif 3277   {csn 3774   ` cfv 5413  (class class class)co 6040   Basecbs 13424   +g cplusg 13484   0gc0g 13678   -gcsg 14643   Abelcabel 15368   LModclmod 15905   LSubSpclss 15963   LSpanclspn 16002   HLchlt 29833   LHypclh 30466   DVecHcdvh 31561  LCDualclcd 32069  mapdcmpd 32107  HDMapchdma 32276
This theorem is referenced by:  hdmaprnlem9N  32343
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-fal 1326  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-ot 3784  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-tpos 6438  df-undef 6502  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-0g 13682  df-mre 13766  df-mrc 13767  df-acs 13769  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-mnd 14645  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-subg 14896  df-cntz 15071  df-oppg 15097  df-lsm 15225  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-dvr 15743  df-drng 15792  df-lmod 15907  df-lss 15964  df-lsp 16003  df-lvec 16130  df-lsatoms 29459  df-lshyp 29460  df-lcv 29502  df-lfl 29541  df-lkr 29569  df-ldual 29607  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641  df-tgrp 31225  df-tendo 31237  df-edring 31239  df-dveca 31485  df-disoa 31512  df-dvech 31562  df-dib 31622  df-dic 31656  df-dih 31712  df-doch 31831  df-djh 31878  df-lcdual 32070  df-mapd 32108  df-hvmap 32240  df-hdmap1 32277  df-hdmap 32278
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