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Theorem hdmaprnlem9N 37688
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 37467 and mapdcnv11N 37487. 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 37686 . . . . 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 37687 . . . . . 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 37684 . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { t } ) )  =  ( L `  {
s } ) )
2523, 24eleqtrd 2547 . . . . 5  |-  ( ph  ->  ( s ( -g `  C ) ( S `
 t ) )  e.  ( L `  { s } ) )
2622, 25elind 3684 . . . 4  |-  ( ph  ->  ( s ( -g `  C ) ( S `
 t ) )  e.  ( ( L `
 { ( ( S `  u ) 
.+b  s ) } )  i^i  ( L `
 { s } ) ) )
271, 5, 9lcdlvec 37419 . . . . 5  |-  ( ph  ->  C  e.  LVec )
281, 5, 9lcdlmod 37420 . . . . . 6  |-  ( ph  ->  C  e.  LMod )
291, 2, 3, 5, 15, 8, 9, 13hdmapcl 37661 . . . . . 6  |-  ( ph  ->  ( S `  u
)  e.  D )
3010eldifad 3483 . . . . . 6  |-  ( ph  ->  s  e.  D )
3115, 18lmodvacl 17652 . . . . . 6  |-  ( ( C  e.  LMod  /\  ( S `  u )  e.  D  /\  s  e.  D )  ->  (
( S `  u
)  .+b  s )  e.  D )
3228, 29, 30, 31syl3anc 1228 . . . . 5  |-  ( ph  ->  ( ( S `  u )  .+b  s
)  e.  D )
33 eqid 2457 . . . . . . . . . . . . . 14  |-  ( LSubSp `  C )  =  (
LSubSp `  C )
3415, 33, 6lspsncl 17749 . . . . . . . . . . . . 13  |-  ( ( C  e.  LMod  /\  s  e.  D )  ->  ( L `  { s } )  e.  (
LSubSp `  C ) )
3528, 30, 34syl2anc 661 . . . . . . . . . . . 12  |-  ( ph  ->  ( L `  {
s } )  e.  ( LSubSp `  C )
)
361, 7, 5, 33, 9mapdrn2 37479 . . . . . . . . . . . 12  |-  ( ph  ->  ran  M  =  (
LSubSp `  C ) )
3735, 36eleqtrrd 2548 . . . . . . . . . . 11  |-  ( ph  ->  ( L `  {
s } )  e. 
ran  M )
381, 7, 9, 37mapdcnvid2 37485 . . . . . . . . . 10  |-  ( ph  ->  ( M `  ( `' M `  ( L `
 { s } ) ) )  =  ( L `  {
s } ) )
3912, 38eqtr4d 2501 . . . . . . . . 9  |-  ( ph  ->  ( M `  ( N `  { v } ) )  =  ( M `  ( `' M `  ( L `
 { s } ) ) ) )
40 eqid 2457 . . . . . . . . . 10  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
411, 2, 9dvhlmod 36938 . . . . . . . . . . 11  |-  ( ph  ->  U  e.  LMod )
423, 40, 4lspsncl 17749 . . . . . . . . . . 11  |-  ( ( U  e.  LMod  /\  v  e.  V )  ->  ( N `  { v } )  e.  (
LSubSp `  U ) )
4341, 11, 42syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( N `  {
v } )  e.  ( LSubSp `  U )
)
441, 7, 2, 40, 9, 37mapdcnvcl 37480 . . . . . . . . . 10  |-  ( ph  ->  ( `' M `  ( L `  { s } ) )  e.  ( LSubSp `  U )
)
451, 2, 40, 7, 9, 43, 44mapd11 37467 . . . . . . . . 9  |-  ( ph  ->  ( ( M `  ( N `  { v } ) )  =  ( M `  ( `' M `  ( L `
 { s } ) ) )  <->  ( N `  { v } )  =  ( `' M `  ( L `  {
s } ) ) ) )
4639, 45mpbid 210 . . . . . . . 8  |-  ( ph  ->  ( N `  {
v } )  =  ( `' M `  ( L `  { s } ) ) )
471, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18hdmaprnlem3N 37681 . . . . . . . 8  |-  ( ph  ->  ( N `  {
v } )  =/=  ( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) ) )
4846, 47eqnetrrd 2751 . . . . . . 7  |-  ( ph  ->  ( `' M `  ( L `  { s } ) )  =/=  ( `' M `  ( L `  { ( ( S `  u
)  .+b  s ) } ) ) )
4915, 33, 6lspsncl 17749 . . . . . . . . . . 11  |-  ( ( C  e.  LMod  /\  (
( S `  u
)  .+b  s )  e.  D )  ->  ( L `  { (
( S `  u
)  .+b  s ) } )  e.  (
LSubSp `  C ) )
5028, 32, 49syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  e.  ( LSubSp `  C )
)
5150, 36eleqtrrd 2548 . . . . . . . . 9  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  e. 
ran  M )
521, 7, 9, 37, 51mapdcnv11N 37487 . . . . . . . 8  |-  ( ph  ->  ( ( `' M `  ( L `  {
s } ) )  =  ( `' M `  ( L `  {
( ( S `  u )  .+b  s
) } ) )  <-> 
( L `  {
s } )  =  ( L `  {
( ( S `  u )  .+b  s
) } ) ) )
5352necon3bid 2715 . . . . . . 7  |-  ( ph  ->  ( ( `' M `  ( L `  {
s } ) )  =/=  ( `' M `  ( L `  {
( ( S `  u )  .+b  s
) } ) )  <-> 
( L `  {
s } )  =/=  ( L `  {
( ( S `  u )  .+b  s
) } ) ) )
5448, 53mpbid 210 . . . . . 6  |-  ( ph  ->  ( L `  {
s } )  =/=  ( L `  {
( ( S `  u )  .+b  s
) } ) )
5554necomd 2728 . . . . 5  |-  ( ph  ->  ( L `  {
( ( S `  u )  .+b  s
) } )  =/=  ( L `  {
s } ) )
5615, 16, 6, 27, 32, 30, 55lspdisj2 17899 . . . 4  |-  ( ph  ->  ( ( L `  { ( ( S `
 u )  .+b  s ) } )  i^i  ( L `  { s } ) )  =  { Q } )
5726, 56eleqtrd 2547 . . 3  |-  ( ph  ->  ( s ( -g `  C ) ( S `
 t ) )  e.  { Q }
)
58 elsni 4057 . . 3  |-  ( ( s ( -g `  C
) ( S `  t ) )  e. 
{ Q }  ->  ( s ( -g `  C
) ( S `  t ) )  =  Q )
5957, 58syl 16 . 2  |-  ( ph  ->  ( s ( -g `  C ) ( S `
 t ) )  =  Q )
601, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19hdmaprnlem4tN 37683 . . . 4  |-  ( ph  ->  t  e.  V )
611, 2, 3, 5, 15, 8, 9, 60hdmapcl 37661 . . 3  |-  ( ph  ->  ( S `  t
)  e.  D )
62 eqid 2457 . . . 4  |-  ( -g `  C )  =  (
-g `  C )
6315, 16, 62lmodsubeq0 17695 . . 3  |-  ( ( C  e.  LMod  /\  s  e.  D  /\  ( S `  t )  e.  D )  ->  (
( s ( -g `  C ) ( S `
 t ) )  =  Q  <->  s  =  ( S `  t ) ) )
6428, 30, 61, 63syl3anc 1228 . 2  |-  ( ph  ->  ( ( s (
-g `  C )
( S `  t
) )  =  Q  <-> 
s  =  ( S `
 t ) ) )
6559, 64mpbid 210 1  |-  ( ph  ->  s  =  ( S `
 t ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652    \ cdif 3468    i^i cin 3470   {csn 4032   `'ccnv 5007   ran crn 5009   ` cfv 5594  (class class class)co 6296   Basecbs 14643   +g cplusg 14711   0gc0g 14856   -gcsg 16181   LModclmod 17638   LSubSpclss 17704   LSpanclspn 17743   HLchlt 35176   LHypclh 35809   DVecHcdvh 36906  LCDualclcd 37414  mapdcmpd 37452  HDMapchdma 37621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-riotaBAD 34785
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-ot 4041  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-tpos 6973  df-undef 7020  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-mulr 14725  df-sca 14727  df-vsca 14728  df-0g 14858  df-mre 15002  df-mrc 15003  df-acs 15005  df-preset 15683  df-poset 15701  df-plt 15714  df-lub 15730  df-glb 15731  df-join 15732  df-meet 15733  df-p0 15795  df-p1 15796  df-lat 15802  df-clat 15864  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-submnd 16093  df-grp 16183  df-minusg 16184  df-sbg 16185  df-subg 16324  df-cntz 16481  df-oppg 16507  df-lsm 16782  df-cmn 16926  df-abl 16927  df-mgp 17268  df-ur 17280  df-ring 17326  df-oppr 17398  df-dvdsr 17416  df-unit 17417  df-invr 17447  df-dvr 17458  df-drng 17524  df-lmod 17640  df-lss 17705  df-lsp 17744  df-lvec 17875  df-lsatoms 34802  df-lshyp 34803  df-lcv 34845  df-lfl 34884  df-lkr 34912  df-ldual 34950  df-oposet 35002  df-ol 35004  df-oml 35005  df-covers 35092  df-ats 35093  df-atl 35124  df-cvlat 35148  df-hlat 35177  df-llines 35323  df-lplanes 35324  df-lvols 35325  df-lines 35326  df-psubsp 35328  df-pmap 35329  df-padd 35621  df-lhyp 35813  df-laut 35814  df-ldil 35929  df-ltrn 35930  df-trl 35985  df-tgrp 36570  df-tendo 36582  df-edring 36584  df-dveca 36830  df-disoa 36857  df-dvech 36907  df-dib 36967  df-dic 37001  df-dih 37057  df-doch 37176  df-djh 37223  df-lcdual 37415  df-mapd 37453  df-hvmap 37585  df-hdmap1 37622  df-hdmap 37623
This theorem is referenced by:  hdmaprnlem10N  37690
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