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Theorem hdmaprnlem3N 32552
Description: Part of proof of part 12 in [Baer] p. 49 line 15, T  =/= P. Our  ( `' M `  ( L `  {
( ( S `  u )  .+b  s
) } ) ) is Baer's P, where P* = G(u'+s). (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 )
Assertion
Ref Expression
hdmaprnlem3N  |-  ( ph  ->  ( N `  {
v } )  =/=  ( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) ) )

Proof of Theorem hdmaprnlem3N
StepHypRef Expression
1 hdmaprnlem1.d . . . . 5  |-  D  =  ( Base `  C
)
2 hdmaprnlem1.l . . . . 5  |-  L  =  ( LSpan `  C )
3 hdmaprnlem1.h . . . . . 6  |-  H  =  ( LHyp `  K
)
4 hdmaprnlem1.c . . . . . 6  |-  C  =  ( (LCDual `  K
) `  W )
5 hdmaprnlem1.k . . . . . 6  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
63, 4, 5lcdlmod 32291 . . . . 5  |-  ( ph  ->  C  e.  LMod )
7 hdmaprnlem1.u . . . . . . 7  |-  U  =  ( ( DVecH `  K
) `  W )
8 hdmaprnlem1.v . . . . . . 7  |-  V  =  ( Base `  U
)
9 hdmaprnlem1.s . . . . . . 7  |-  S  =  ( (HDMap `  K
) `  W )
10 hdmaprnlem1.ue . . . . . . 7  |-  ( ph  ->  u  e.  V )
113, 7, 8, 4, 1, 9, 5, 10hdmapcl 32532 . . . . . 6  |-  ( ph  ->  ( S `  u
)  e.  D )
12 hdmaprnlem1.se . . . . . . 7  |-  ( ph  ->  s  e.  ( D 
\  { Q }
) )
1312eldifad 3324 . . . . . 6  |-  ( ph  ->  s  e.  D )
14 hdmaprnlem1.a . . . . . . 7  |-  .+b  =  ( +g  `  C )
151, 14lmodvacl 15954 . . . . . 6  |-  ( ( C  e.  LMod  /\  ( S `  u )  e.  D  /\  s  e.  D )  ->  (
( S `  u
)  .+b  s )  e.  D )
166, 11, 13, 15syl3anc 1184 . . . . 5  |-  ( ph  ->  ( ( S `  u )  .+b  s
)  e.  D )
17 eqid 2435 . . . . . 6  |-  ( LSubSp `  C )  =  (
LSubSp `  C )
181, 17, 2lspsncl 16043 . . . . . . 7  |-  ( ( C  e.  LMod  /\  s  e.  D )  ->  ( L `  { s } )  e.  (
LSubSp `  C ) )
196, 13, 18syl2anc 643 . . . . . 6  |-  ( ph  ->  ( L `  {
s } )  e.  ( LSubSp `  C )
)
201, 2lspsnid 16059 . . . . . . 7  |-  ( ( C  e.  LMod  /\  s  e.  D )  ->  s  e.  ( L `  {
s } ) )
216, 13, 20syl2anc 643 . . . . . 6  |-  ( ph  ->  s  e.  ( L `
 { s } ) )
22 hdmaprnlem1.q . . . . . . 7  |-  Q  =  ( 0g `  C
)
233, 4, 5lcdlvec 32290 . . . . . . 7  |-  ( ph  ->  C  e.  LVec )
24 hdmaprnlem1.o . . . . . . . 8  |-  .0.  =  ( 0g `  U )
25 eqid 2435 . . . . . . . . 9  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
263, 7, 5dvhlmod 31809 . . . . . . . . 9  |-  ( ph  ->  U  e.  LMod )
27 hdmaprnlem1.ve . . . . . . . . . 10  |-  ( ph  ->  v  e.  V )
28 hdmaprnlem1.n . . . . . . . . . . 11  |-  N  =  ( LSpan `  U )
298, 25, 28lspsncl 16043 . . . . . . . . . 10  |-  ( ( U  e.  LMod  /\  v  e.  V )  ->  ( N `  { v } )  e.  (
LSubSp `  U ) )
3026, 27, 29syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( N `  {
v } )  e.  ( LSubSp `  U )
)
31 hdmaprnlem1.un . . . . . . . . 9  |-  ( ph  ->  -.  u  e.  ( N `  { v } ) )
328, 24, 25, 26, 30, 10, 31lssneln0 16018 . . . . . . . 8  |-  ( ph  ->  u  e.  ( V 
\  {  .0.  }
) )
333, 7, 8, 24, 4, 22, 1, 9, 5, 32hdmapnzcl 32547 . . . . . . 7  |-  ( ph  ->  ( S `  u
)  e.  ( D 
\  { Q }
) )
34 hdmaprnlem1.m . . . . . . . 8  |-  M  =  ( (mapd `  K
) `  W )
35 hdmaprnlem1.e . . . . . . . 8  |-  ( ph  ->  ( M `  ( N `  { v } ) )  =  ( L `  {
s } ) )
363, 7, 8, 28, 4, 2, 34, 9, 5, 12, 27, 35, 10, 31hdmaprnlem1N 32551 . . . . . . 7  |-  ( ph  ->  ( L `  {
( S `  u
) } )  =/=  ( L `  {
s } ) )
371, 22, 2, 23, 33, 13, 36lspsnne1 16179 . . . . . 6  |-  ( ph  ->  -.  ( S `  u )  e.  ( L `  { s } ) )
381, 14, 17, 6, 19, 21, 11, 37lssvancl2 16012 . . . . 5  |-  ( ph  ->  -.  ( ( S `
 u )  .+b  s )  e.  ( L `  { s } ) )
391, 2, 6, 16, 13, 38lspsnne2 16180 . . . 4  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  =/=  ( L `  {
s } ) )
4039necomd 2681 . . 3  |-  ( ph  ->  ( L `  {
s } )  =/=  ( L `  {
( ( S `  u )  .+b  s
) } ) )
411, 17, 2lspsncl 16043 . . . . . 6  |-  ( ( C  e.  LMod  /\  (
( S `  u
)  .+b  s )  e.  D )  ->  ( L `  { (
( S `  u
)  .+b  s ) } )  e.  (
LSubSp `  C ) )
426, 16, 41syl2anc 643 . . . . 5  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  e.  ( LSubSp `  C )
)
433, 34, 4, 17, 5mapdrn2 32350 . . . . 5  |-  ( ph  ->  ran  M  =  (
LSubSp `  C ) )
4442, 43eleqtrrd 2512 . . . 4  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  e. 
ran  M )
453, 34, 5, 44mapdcnvid2 32356 . . 3  |-  ( ph  ->  ( M `  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) ) )  =  ( L `  {
( ( S `  u )  .+b  s
) } ) )
4640, 35, 453netr4d 2625 . 2  |-  ( ph  ->  ( M `  ( N `  { v } ) )  =/=  ( M `  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) ) ) )
473, 34, 7, 25, 5, 44mapdcnvcl 32351 . . . 4  |-  ( ph  ->  ( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) )  e.  ( LSubSp `  U )
)
483, 7, 25, 34, 5, 30, 47mapd11 32338 . . 3  |-  ( ph  ->  ( ( M `  ( N `  { v } ) )  =  ( M `  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) ) )  <->  ( N `  { v } )  =  ( `' M `  ( L `  {
( ( S `  u )  .+b  s
) } ) ) ) )
4948necon3bid 2633 . 2  |-  ( ph  ->  ( ( M `  ( N `  { v } ) )  =/=  ( M `  ( `' M `  ( L `
 { ( ( S `  u ) 
.+b  s ) } ) ) )  <->  ( N `  { v } )  =/=  ( `' M `  ( L `  {
( ( S `  u )  .+b  s
) } ) ) ) )
5046, 49mpbid 202 1  |-  ( ph  ->  ( N `  {
v } )  =/=  ( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598    \ cdif 3309   {csn 3806   `'ccnv 4869   ran crn 4871   ` cfv 5446  (class class class)co 6073   Basecbs 13459   +g cplusg 13519   0gc0g 13713   LModclmod 15940   LSubSpclss 15998   LSpanclspn 16037   HLchlt 30049   LHypclh 30682   DVecHcdvh 31777  LCDualclcd 32285  mapdcmpd 32323  HDMapchdma 32492
This theorem is referenced by:  hdmaprnlem9N  32559  hdmaprnlem3eN  32560
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-fal 1329  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-ot 3816  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-tpos 6471  df-undef 6535  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-nn 9991  df-2 10048  df-3 10049  df-4 10050  df-5 10051  df-6 10052  df-n0 10212  df-z 10273  df-uz 10479  df-fz 11034  df-struct 13461  df-ndx 13462  df-slot 13463  df-base 13464  df-sets 13465  df-ress 13466  df-plusg 13532  df-mulr 13533  df-sca 13535  df-vsca 13536  df-0g 13717  df-mre 13801  df-mrc 13802  df-acs 13804  df-poset 14393  df-plt 14405  df-lub 14421  df-glb 14422  df-join 14423  df-meet 14424  df-p0 14458  df-p1 14459  df-lat 14465  df-clat 14527  df-mnd 14680  df-submnd 14729  df-grp 14802  df-minusg 14803  df-sbg 14804  df-subg 14931  df-cntz 15106  df-oppg 15132  df-lsm 15260  df-cmn 15404  df-abl 15405  df-mgp 15639  df-rng 15653  df-ur 15655  df-oppr 15718  df-dvdsr 15736  df-unit 15737  df-invr 15767  df-dvr 15778  df-drng 15827  df-lmod 15942  df-lss 15999  df-lsp 16038  df-lvec 16165  df-lsatoms 29675  df-lshyp 29676  df-lcv 29718  df-lfl 29757  df-lkr 29785  df-ldual 29823  df-oposet 29875  df-ol 29877  df-oml 29878  df-covers 29965  df-ats 29966  df-atl 29997  df-cvlat 30021  df-hlat 30050  df-llines 30196  df-lplanes 30197  df-lvols 30198  df-lines 30199  df-psubsp 30201  df-pmap 30202  df-padd 30494  df-lhyp 30686  df-laut 30687  df-ldil 30802  df-ltrn 30803  df-trl 30857  df-tgrp 31441  df-tendo 31453  df-edring 31455  df-dveca 31701  df-disoa 31728  df-dvech 31778  df-dib 31838  df-dic 31872  df-dih 31928  df-doch 32047  df-djh 32094  df-lcdual 32286  df-mapd 32324  df-hvmap 32456  df-hdmap1 32493  df-hdmap 32494
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