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Theorem hdmaprnlem9N 32497
Description: Part of proof of part 12 in [Baer] p. 49 line 21, s=S(t). TODO: we seem to be going back and forth with mapd11 32276 and mapdcnv11N 32296. Use better hypotheses and/or theorems? (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
hdmaprnlem9N  |-  ( ph  ->  s  =  ( S `
 t ) )

Proof of Theorem hdmaprnlem9N
StepHypRef Expression
1 hdmaprnlem1.h . . . . . 6  |-  H  =  ( LHyp `  K
)
2 hdmaprnlem1.u . . . . . 6  |-  U  =  ( ( DVecH `  K
) `  W )
3 hdmaprnlem1.v . . . . . 6  |-  V  =  ( Base `  U
)
4 hdmaprnlem1.n . . . . . 6  |-  N  =  ( LSpan `  U )
5 hdmaprnlem1.c . . . . . 6  |-  C  =  ( (LCDual `  K
) `  W )
6 hdmaprnlem1.l . . . . . 6  |-  L  =  ( LSpan `  C )
7 hdmaprnlem1.m . . . . . 6  |-  M  =  ( (mapd `  K
) `  W )
8 hdmaprnlem1.s . . . . . 6  |-  S  =  ( (HDMap `  K
) `  W )
9 hdmaprnlem1.k . . . . . 6  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
10 hdmaprnlem1.se . . . . . 6  |-  ( ph  ->  s  e.  ( D 
\  { Q }
) )
11 hdmaprnlem1.ve . . . . . 6  |-  ( ph  ->  v  e.  V )
12 hdmaprnlem1.e . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { v } ) )  =  ( L `  {
s } ) )
13 hdmaprnlem1.ue . . . . . 6  |-  ( ph  ->  u  e.  V )
14 hdmaprnlem1.un . . . . . 6  |-  ( ph  ->  -.  u  e.  ( N `  { v } ) )
15 hdmaprnlem1.d . . . . . 6  |-  D  =  ( Base `  C
)
16 hdmaprnlem1.q . . . . . 6  |-  Q  =  ( 0g `  C
)
17 hdmaprnlem1.o . . . . . 6  |-  .0.  =  ( 0g `  U )
18 hdmaprnlem1.a . . . . . 6  |-  .+b  =  ( +g  `  C )
19 hdmaprnlem1.t2 . . . . . 6  |-  ( ph  ->  t  e.  ( ( N `  { v } )  \  {  .0.  } ) )
20 hdmaprnlem1.p . . . . . 6  |-  .+  =  ( +g  `  U )
21 hdmaprnlem1.pt . . . . . 6  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  =  ( M `  ( N `  { (
u  .+  t ) } ) ) )
221, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21hdmaprnlem7N 32495 . . . . 5  |-  ( ph  ->  ( s ( -g `  C ) ( S `
 t ) )  e.  ( L `  { ( ( S `
 u )  .+b  s ) } ) )
231, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21hdmaprnlem8N 32496 . . . . . 6  |-  ( ph  ->  ( s ( -g `  C ) ( S `
 t ) )  e.  ( M `  ( N `  { t } ) ) )
241, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19hdmaprnlem4N 32493 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { t } ) )  =  ( L `  {
s } ) )
2523, 24eleqtrd 2511 . . . . 5  |-  ( ph  ->  ( s ( -g `  C ) ( S `
 t ) )  e.  ( L `  { s } ) )
26 elin 3522 . . . . 5  |-  ( ( s ( -g `  C
) ( S `  t ) )  e.  ( ( L `  { ( ( S `
 u )  .+b  s ) } )  i^i  ( L `  { s } ) )  <->  ( ( s ( -g `  C
) ( S `  t ) )  e.  ( L `  {
( ( S `  u )  .+b  s
) } )  /\  ( s ( -g `  C ) ( S `
 t ) )  e.  ( L `  { s } ) ) )
2722, 25, 26sylanbrc 646 . . . 4  |-  ( ph  ->  ( s ( -g `  C ) ( S `
 t ) )  e.  ( ( L `
 { ( ( S `  u ) 
.+b  s ) } )  i^i  ( L `
 { s } ) ) )
281, 5, 9lcdlvec 32228 . . . . 5  |-  ( ph  ->  C  e.  LVec )
291, 5, 9lcdlmod 32229 . . . . . 6  |-  ( ph  ->  C  e.  LMod )
301, 2, 3, 5, 15, 8, 9, 13hdmapcl 32470 . . . . . 6  |-  ( ph  ->  ( S `  u
)  e.  D )
3110eldifad 3324 . . . . . 6  |-  ( ph  ->  s  e.  D )
3215, 18lmodvacl 15952 . . . . . 6  |-  ( ( C  e.  LMod  /\  ( S `  u )  e.  D  /\  s  e.  D )  ->  (
( S `  u
)  .+b  s )  e.  D )
3329, 30, 31, 32syl3anc 1184 . . . . 5  |-  ( ph  ->  ( ( S `  u )  .+b  s
)  e.  D )
34 eqid 2435 . . . . . . . . . . . . . 14  |-  ( LSubSp `  C )  =  (
LSubSp `  C )
3515, 34, 6lspsncl 16041 . . . . . . . . . . . . 13  |-  ( ( C  e.  LMod  /\  s  e.  D )  ->  ( L `  { s } )  e.  (
LSubSp `  C ) )
3629, 31, 35syl2anc 643 . . . . . . . . . . . 12  |-  ( ph  ->  ( L `  {
s } )  e.  ( LSubSp `  C )
)
371, 7, 5, 34, 9mapdrn2 32288 . . . . . . . . . . . 12  |-  ( ph  ->  ran  M  =  (
LSubSp `  C ) )
3836, 37eleqtrrd 2512 . . . . . . . . . . 11  |-  ( ph  ->  ( L `  {
s } )  e. 
ran  M )
391, 7, 9, 38mapdcnvid2 32294 . . . . . . . . . 10  |-  ( ph  ->  ( M `  ( `' M `  ( L `
 { s } ) ) )  =  ( L `  {
s } ) )
4012, 39eqtr4d 2470 . . . . . . . . 9  |-  ( ph  ->  ( M `  ( N `  { v } ) )  =  ( M `  ( `' M `  ( L `
 { s } ) ) ) )
41 eqid 2435 . . . . . . . . . 10  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
421, 2, 9dvhlmod 31747 . . . . . . . . . . 11  |-  ( ph  ->  U  e.  LMod )
433, 41, 4lspsncl 16041 . . . . . . . . . . 11  |-  ( ( U  e.  LMod  /\  v  e.  V )  ->  ( N `  { v } )  e.  (
LSubSp `  U ) )
4442, 11, 43syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  ( N `  {
v } )  e.  ( LSubSp `  U )
)
451, 7, 2, 41, 9, 38mapdcnvcl 32289 . . . . . . . . . 10  |-  ( ph  ->  ( `' M `  ( L `  { s } ) )  e.  ( LSubSp `  U )
)
461, 2, 41, 7, 9, 44, 45mapd11 32276 . . . . . . . . 9  |-  ( ph  ->  ( ( M `  ( N `  { v } ) )  =  ( M `  ( `' M `  ( L `
 { s } ) ) )  <->  ( N `  { v } )  =  ( `' M `  ( L `  {
s } ) ) ) )
4740, 46mpbid 202 . . . . . . . 8  |-  ( ph  ->  ( N `  {
v } )  =  ( `' M `  ( L `  { s } ) ) )
481, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18hdmaprnlem3N 32490 . . . . . . . 8  |-  ( ph  ->  ( N `  {
v } )  =/=  ( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) ) )
4947, 48eqnetrrd 2618 . . . . . . 7  |-  ( ph  ->  ( `' M `  ( L `  { s } ) )  =/=  ( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) ) )
5015, 34, 6lspsncl 16041 . . . . . . . . . . 11  |-  ( ( C  e.  LMod  /\  (
( S `  u
)  .+b  s )  e.  D )  ->  ( L `  { (
( S `  u
)  .+b  s ) } )  e.  (
LSubSp `  C ) )
5129, 33, 50syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  e.  ( LSubSp `  C )
)
5251, 37eleqtrrd 2512 . . . . . . . . 9  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  e. 
ran  M )
531, 7, 9, 38, 52mapdcnv11N 32296 . . . . . . . 8  |-  ( ph  ->  ( ( `' M `  ( L `  {
s } ) )  =  ( `' M `  ( L `  {
( ( S `  u )  .+b  s
) } ) )  <-> 
( L `  {
s } )  =  ( L `  {
( ( S `  u )  .+b  s
) } ) ) )
5453necon3bid 2633 . . . . . . 7  |-  ( ph  ->  ( ( `' M `  ( L `  {
s } ) )  =/=  ( `' M `  ( L `  {
( ( S `  u )  .+b  s
) } ) )  <-> 
( L `  {
s } )  =/=  ( L `  {
( ( S `  u )  .+b  s
) } ) ) )
5549, 54mpbid 202 . . . . . 6  |-  ( ph  ->  ( L `  {
s } )  =/=  ( L `  {
( ( S `  u )  .+b  s
) } ) )
5655necomd 2681 . . . . 5  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  =/=  ( L `  {
s } ) )
5715, 16, 6, 28, 33, 31, 56lspdisj2 16187 . . . 4  |-  ( ph  ->  ( ( L `  { ( ( S `
 u )  .+b  s ) } )  i^i  ( L `  { s } ) )  =  { Q } )
5827, 57eleqtrd 2511 . . 3  |-  ( ph  ->  ( s ( -g `  C ) ( S `
 t ) )  e.  { Q }
)
59 elsni 3830 . . 3  |-  ( ( s ( -g `  C
) ( S `  t ) )  e. 
{ Q }  ->  ( s ( -g `  C
) ( S `  t ) )  =  Q )
6058, 59syl 16 . 2  |-  ( ph  ->  ( s ( -g `  C ) ( S `
 t ) )  =  Q )
611, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19hdmaprnlem4tN 32492 . . . 4  |-  ( ph  ->  t  e.  V )
621, 2, 3, 5, 15, 8, 9, 61hdmapcl 32470 . . 3  |-  ( ph  ->  ( S `  t
)  e.  D )
63 eqid 2435 . . . 4  |-  ( -g `  C )  =  (
-g `  C )
6415, 16, 63lmodsubeq0 15991 . . 3  |-  ( ( C  e.  LMod  /\  s  e.  D  /\  ( S `  t )  e.  D )  ->  (
( s ( -g `  C ) ( S `
 t ) )  =  Q  <->  s  =  ( S `  t ) ) )
6529, 31, 62, 64syl3anc 1184 . 2  |-  ( ph  ->  ( ( s (
-g `  C )
( S `  t
) )  =  Q  <-> 
s  =  ( S `
 t ) ) )
6660, 65mpbid 202 1  |-  ( ph  ->  s  =  ( S `
 t ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598    \ cdif 3309    i^i cin 3311   {csn 3806   `'ccnv 4868   ran crn 4870   ` cfv 5445  (class class class)co 6072   Basecbs 13457   +g cplusg 13517   0gc0g 13711   -gcsg 14676   LModclmod 15938   LSubSpclss 15996   LSpanclspn 16035   HLchlt 29987   LHypclh 30620   DVecHcdvh 31715  LCDualclcd 32223  mapdcmpd 32261  HDMapchdma 32430
This theorem is referenced by:  hdmaprnlem10N  32499
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 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056
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 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-of 6296  df-1st 6340  df-2nd 6341  df-tpos 6470  df-undef 6534  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-oadd 6719  df-er 6896  df-map 7011  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-nn 9990  df-2 10047  df-3 10048  df-4 10049  df-5 10050  df-6 10051  df-n0 10211  df-z 10272  df-uz 10478  df-fz 11033  df-struct 13459  df-ndx 13460  df-slot 13461  df-base 13462  df-sets 13463  df-ress 13464  df-plusg 13530  df-mulr 13531  df-sca 13533  df-vsca 13534  df-0g 13715  df-mre 13799  df-mrc 13800  df-acs 13802  df-poset 14391  df-plt 14403  df-lub 14419  df-glb 14420  df-join 14421  df-meet 14422  df-p0 14456  df-p1 14457  df-lat 14463  df-clat 14525  df-mnd 14678  df-submnd 14727  df-grp 14800  df-minusg 14801  df-sbg 14802  df-subg 14929  df-cntz 15104  df-oppg 15130  df-lsm 15258  df-cmn 15402  df-abl 15403  df-mgp 15637  df-rng 15651  df-ur 15653  df-oppr 15716  df-dvdsr 15734  df-unit 15735  df-invr 15765  df-dvr 15776  df-drng 15825  df-lmod 15940  df-lss 15997  df-lsp 16036  df-lvec 16163  df-lsatoms 29613  df-lshyp 29614  df-lcv 29656  df-lfl 29695  df-lkr 29723  df-ldual 29761  df-oposet 29813  df-ol 29815  df-oml 29816  df-covers 29903  df-ats 29904  df-atl 29935  df-cvlat 29959  df-hlat 29988  df-llines 30134  df-lplanes 30135  df-lvols 30136  df-lines 30137  df-psubsp 30139  df-pmap 30140  df-padd 30432  df-lhyp 30624  df-laut 30625  df-ldil 30740  df-ltrn 30741  df-trl 30795  df-tgrp 31379  df-tendo 31391  df-edring 31393  df-dveca 31639  df-disoa 31666  df-dvech 31716  df-dib 31776  df-dic 31810  df-dih 31866  df-doch 31985  df-djh 32032  df-lcdual 32224  df-mapd 32262  df-hvmap 32394  df-hdmap1 32431  df-hdmap 32432
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