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Theorem hdmap1neglem1N 32465
Description: Lemma for hdmapneg 32486. TODO: Not used; delete. (Contributed by NM, 23-May-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
hdmap1neglem1.h  |-  H  =  ( LHyp `  K
)
hdmap1neglem1.u  |-  U  =  ( ( DVecH `  K
) `  W )
hdmap1neglem1.v  |-  V  =  ( Base `  U
)
hdmap1neglem1.r  |-  R  =  ( inv g `  U )
hdmap1neglem1.o  |-  .0.  =  ( 0g `  U )
hdmap1neglem1.n  |-  N  =  ( LSpan `  U )
hdmap1neglem1.c  |-  C  =  ( (LCDual `  K
) `  W )
hdmap1neglem1.d  |-  D  =  ( Base `  C
)
hdmap1neglem1.s  |-  S  =  ( inv g `  C )
hdmap1neglem1.l  |-  L  =  ( LSpan `  C )
hdmap1neglem1.m  |-  M  =  ( (mapd `  K
) `  W )
hdmap1neglem1.i  |-  I  =  ( (HDMap1 `  K
) `  W )
hdmap1neglem1.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
hdmap1neglem1.f  |-  ( ph  ->  F  e.  D )
hdmap1neglem1.mn  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( L `  { F } ) )
hdmap1neglem1.ne  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
hdmap1neglem1.x  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
hdmap1neglem1.y  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
hdmap1neglem1.e  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  G )
Assertion
Ref Expression
hdmap1neglem1N  |-  ( ph  ->  ( I `  <. ( R `  X ) ,  ( S `  F ) ,  ( R `  Y )
>. )  =  ( S `  G )
)

Proof of Theorem hdmap1neglem1N
StepHypRef Expression
1 hdmap1neglem1.e . . . . 5  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  =  G )
2 hdmap1neglem1.h . . . . . 6  |-  H  =  ( LHyp `  K
)
3 hdmap1neglem1.u . . . . . 6  |-  U  =  ( ( DVecH `  K
) `  W )
4 hdmap1neglem1.v . . . . . 6  |-  V  =  ( Base `  U
)
5 eqid 2435 . . . . . 6  |-  ( -g `  U )  =  (
-g `  U )
6 hdmap1neglem1.o . . . . . 6  |-  .0.  =  ( 0g `  U )
7 hdmap1neglem1.n . . . . . 6  |-  N  =  ( LSpan `  U )
8 hdmap1neglem1.c . . . . . 6  |-  C  =  ( (LCDual `  K
) `  W )
9 hdmap1neglem1.d . . . . . 6  |-  D  =  ( Base `  C
)
10 eqid 2435 . . . . . 6  |-  ( -g `  C )  =  (
-g `  C )
11 hdmap1neglem1.l . . . . . 6  |-  L  =  ( LSpan `  C )
12 hdmap1neglem1.m . . . . . 6  |-  M  =  ( (mapd `  K
) `  W )
13 hdmap1neglem1.i . . . . . 6  |-  I  =  ( (HDMap1 `  K
) `  W )
14 hdmap1neglem1.k . . . . . 6  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
15 hdmap1neglem1.x . . . . . 6  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
16 hdmap1neglem1.f . . . . . 6  |-  ( ph  ->  F  e.  D )
17 hdmap1neglem1.y . . . . . 6  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
18 hdmap1neglem1.mn . . . . . . . 8  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( L `  { F } ) )
19 hdmap1neglem1.ne . . . . . . . 8  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
2017eldifad 3324 . . . . . . . 8  |-  ( ph  ->  Y  e.  V )
212, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 18, 19, 15, 20hdmap1cl 32442 . . . . . . 7  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  e.  D )
221, 21eqeltrrd 2510 . . . . . 6  |-  ( ph  ->  G  e.  D )
232, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 22, 19, 18hdmap1eq 32439 . . . . 5  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Y >. )  =  G  <-> 
( ( M `  ( N `  { Y } ) )  =  ( L `  { G } )  /\  ( M `  ( N `  { ( X (
-g `  U ) Y ) } ) )  =  ( L `
 { ( F ( -g `  C
) G ) } ) ) ) )
241, 23mpbid 202 . . . 4  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) )  =  ( L `  { G } )  /\  ( M `  ( N `  { ( X (
-g `  U ) Y ) } ) )  =  ( L `
 { ( F ( -g `  C
) G ) } ) ) )
2524simpld 446 . . 3  |-  ( ph  ->  ( M `  ( N `  { Y } ) )  =  ( L `  { G } ) )
262, 3, 14dvhlmod 31747 . . . . 5  |-  ( ph  ->  U  e.  LMod )
27 hdmap1neglem1.r . . . . . 6  |-  R  =  ( inv g `  U )
284, 27, 7lspsnneg 16070 . . . . 5  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( N `  { ( R `  Y ) } )  =  ( N `  { Y } ) )
2926, 20, 28syl2anc 643 . . . 4  |-  ( ph  ->  ( N `  {
( R `  Y
) } )  =  ( N `  { Y } ) )
3029fveq2d 5723 . . 3  |-  ( ph  ->  ( M `  ( N `  { ( R `  Y ) } ) )  =  ( M `  ( N `  { Y } ) ) )
312, 8, 14lcdlmod 32229 . . . 4  |-  ( ph  ->  C  e.  LMod )
32 hdmap1neglem1.s . . . . 5  |-  S  =  ( inv g `  C )
339, 32, 11lspsnneg 16070 . . . 4  |-  ( ( C  e.  LMod  /\  G  e.  D )  ->  ( L `  { ( S `  G ) } )  =  ( L `  { G } ) )
3431, 22, 33syl2anc 643 . . 3  |-  ( ph  ->  ( L `  {
( S `  G
) } )  =  ( L `  { G } ) )
3525, 30, 343eqtr4d 2477 . 2  |-  ( ph  ->  ( M `  ( N `  { ( R `  Y ) } ) )  =  ( L `  {
( S `  G
) } ) )
3624simprd 450 . . 3  |-  ( ph  ->  ( M `  ( N `  { ( X ( -g `  U
) Y ) } ) )  =  ( L `  { ( F ( -g `  C
) G ) } ) )
37 lmodabl 15979 . . . . . . . . 9  |-  ( U  e.  LMod  ->  U  e. 
Abel )
3826, 37syl 16 . . . . . . . 8  |-  ( ph  ->  U  e.  Abel )
3915eldifad 3324 . . . . . . . 8  |-  ( ph  ->  X  e.  V )
404, 5, 27, 38, 39, 20ablsub2inv 15423 . . . . . . 7  |-  ( ph  ->  ( ( R `  X ) ( -g `  U ) ( R `
 Y ) )  =  ( Y (
-g `  U ) X ) )
4140sneqd 3819 . . . . . 6  |-  ( ph  ->  { ( ( R `
 X ) (
-g `  U )
( R `  Y
) ) }  =  { ( Y (
-g `  U ) X ) } )
4241fveq2d 5723 . . . . 5  |-  ( ph  ->  ( N `  {
( ( R `  X ) ( -g `  U ) ( R `
 Y ) ) } )  =  ( N `  { ( Y ( -g `  U
) X ) } ) )
434, 5, 7, 26, 20, 39lspsnsub 16071 . . . . 5  |-  ( ph  ->  ( N `  {
( Y ( -g `  U ) X ) } )  =  ( N `  { ( X ( -g `  U
) Y ) } ) )
4442, 43eqtrd 2467 . . . 4  |-  ( ph  ->  ( N `  {
( ( R `  X ) ( -g `  U ) ( R `
 Y ) ) } )  =  ( N `  { ( X ( -g `  U
) Y ) } ) )
4544fveq2d 5723 . . 3  |-  ( ph  ->  ( M `  ( N `  { (
( R `  X
) ( -g `  U
) ( R `  Y ) ) } ) )  =  ( M `  ( N `
 { ( X ( -g `  U
) Y ) } ) ) )
46 lmodabl 15979 . . . . . . . 8  |-  ( C  e.  LMod  ->  C  e. 
Abel )
4731, 46syl 16 . . . . . . 7  |-  ( ph  ->  C  e.  Abel )
489, 10, 32, 47, 16, 22ablsub2inv 15423 . . . . . 6  |-  ( ph  ->  ( ( S `  F ) ( -g `  C ) ( S `
 G ) )  =  ( G (
-g `  C ) F ) )
4948sneqd 3819 . . . . 5  |-  ( ph  ->  { ( ( S `
 F ) (
-g `  C )
( S `  G
) ) }  =  { ( G (
-g `  C ) F ) } )
5049fveq2d 5723 . . . 4  |-  ( ph  ->  ( L `  {
( ( S `  F ) ( -g `  C ) ( S `
 G ) ) } )  =  ( L `  { ( G ( -g `  C
) F ) } ) )
519, 10, 11, 31, 22, 16lspsnsub 16071 . . . 4  |-  ( ph  ->  ( L `  {
( G ( -g `  C ) F ) } )  =  ( L `  { ( F ( -g `  C
) G ) } ) )
5250, 51eqtrd 2467 . . 3  |-  ( ph  ->  ( L `  {
( ( S `  F ) ( -g `  C ) ( S `
 G ) ) } )  =  ( L `  { ( F ( -g `  C
) G ) } ) )
5336, 45, 523eqtr4d 2477 . 2  |-  ( ph  ->  ( M `  ( N `  { (
( R `  X
) ( -g `  U
) ( R `  Y ) ) } ) )  =  ( L `  { ( ( S `  F
) ( -g `  C
) ( S `  G ) ) } ) )
54 lmodgrp 15945 . . . . 5  |-  ( U  e.  LMod  ->  U  e. 
Grp )
5526, 54syl 16 . . . 4  |-  ( ph  ->  U  e.  Grp )
564, 6, 27grpinvnzcl 14851 . . . 4  |-  ( ( U  e.  Grp  /\  X  e.  ( V  \  {  .0.  } ) )  ->  ( R `  X )  e.  ( V  \  {  .0.  } ) )
5755, 15, 56syl2anc 643 . . 3  |-  ( ph  ->  ( R `  X
)  e.  ( V 
\  {  .0.  }
) )
589, 32lmodvnegcl 15973 . . . 4  |-  ( ( C  e.  LMod  /\  F  e.  D )  ->  ( S `  F )  e.  D )
5931, 16, 58syl2anc 643 . . 3  |-  ( ph  ->  ( S `  F
)  e.  D )
604, 6, 27grpinvnzcl 14851 . . . 4  |-  ( ( U  e.  Grp  /\  Y  e.  ( V  \  {  .0.  } ) )  ->  ( R `  Y )  e.  ( V  \  {  .0.  } ) )
6155, 17, 60syl2anc 643 . . 3  |-  ( ph  ->  ( R `  Y
)  e.  ( V 
\  {  .0.  }
) )
629, 32lmodvnegcl 15973 . . . 4  |-  ( ( C  e.  LMod  /\  G  e.  D )  ->  ( S `  G )  e.  D )
6331, 22, 62syl2anc 643 . . 3  |-  ( ph  ->  ( S `  G
)  e.  D )
644, 27, 7lspsnneg 16070 . . . . 5  |-  ( ( U  e.  LMod  /\  X  e.  V )  ->  ( N `  { ( R `  X ) } )  =  ( N `  { X } ) )
6526, 39, 64syl2anc 643 . . . 4  |-  ( ph  ->  ( N `  {
( R `  X
) } )  =  ( N `  { X } ) )
6619, 65, 293netr4d 2625 . . 3  |-  ( ph  ->  ( N `  {
( R `  X
) } )  =/=  ( N `  {
( R `  Y
) } ) )
6765fveq2d 5723 . . . 4  |-  ( ph  ->  ( M `  ( N `  { ( R `  X ) } ) )  =  ( M `  ( N `  { X } ) ) )
689, 32, 11lspsnneg 16070 . . . . 5  |-  ( ( C  e.  LMod  /\  F  e.  D )  ->  ( L `  { ( S `  F ) } )  =  ( L `  { F } ) )
6931, 16, 68syl2anc 643 . . . 4  |-  ( ph  ->  ( L `  {
( S `  F
) } )  =  ( L `  { F } ) )
7018, 67, 693eqtr4d 2477 . . 3  |-  ( ph  ->  ( M `  ( N `  { ( R `  X ) } ) )  =  ( L `  {
( S `  F
) } ) )
712, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 57, 59, 61, 63, 66, 70hdmap1eq 32439 . 2  |-  ( ph  ->  ( ( I `  <. ( R `  X
) ,  ( S `
 F ) ,  ( R `  Y
) >. )  =  ( S `  G )  <-> 
( ( M `  ( N `  { ( R `  Y ) } ) )  =  ( L `  {
( S `  G
) } )  /\  ( M `  ( N `
 { ( ( R `  X ) ( -g `  U
) ( R `  Y ) ) } ) )  =  ( L `  { ( ( S `  F
) ( -g `  C
) ( S `  G ) ) } ) ) ) )
7235, 53, 71mpbir2and 889 1  |-  ( ph  ->  ( I `  <. ( R `  X ) ,  ( S `  F ) ,  ( R `  Y )
>. )  =  ( S `  G )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598    \ cdif 3309   {csn 3806   <.cotp 3810   ` cfv 5445  (class class class)co 6072   Basecbs 13457   0gc0g 13711   Grpcgrp 14673   inv gcminusg 14674   -gcsg 14676   Abelcabel 15401   LModclmod 15938   LSpanclspn 16035   HLchlt 29987   LHypclh 30620   DVecHcdvh 31715  LCDualclcd 32223  mapdcmpd 32261  HDMap1chdma1 32429
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-hdmap1 32431
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