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Theorem hdmap1neglem1N 37952
Description: Lemma for hdmapneg 37973. 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  =  ( invg `  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  =  ( invg `  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 2454 . . . . . 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 2454 . . . . . 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 3473 . . . . . . . 8  |-  ( ph  ->  Y  e.  V )
212, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 18, 19, 15, 20hdmap1cl 37929 . . . . . . 7  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  e.  D )
221, 21eqeltrrd 2543 . . . . . 6  |-  ( ph  ->  G  e.  D )
232, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 22, 19, 18hdmap1eq 37926 . . . . 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 210 . . . 4  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) )  =  ( L `  { G } )  /\  ( M `  ( N `  { ( X (
-g `  U ) Y ) } ) )  =  ( L `
 { ( F ( -g `  C
) G ) } ) ) )
2524simpld 457 . . 3  |-  ( ph  ->  ( M `  ( N `  { Y } ) )  =  ( L `  { G } ) )
262, 3, 14dvhlmod 37234 . . . . 5  |-  ( ph  ->  U  e.  LMod )
27 hdmap1neglem1.r . . . . . 6  |-  R  =  ( invg `  U )
284, 27, 7lspsnneg 17847 . . . . 5  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( N `  { ( R `  Y ) } )  =  ( N `  { Y } ) )
2926, 20, 28syl2anc 659 . . . 4  |-  ( ph  ->  ( N `  {
( R `  Y
) } )  =  ( N `  { Y } ) )
3029fveq2d 5852 . . 3  |-  ( ph  ->  ( M `  ( N `  { ( R `  Y ) } ) )  =  ( M `  ( N `  { Y } ) ) )
312, 8, 14lcdlmod 37716 . . . 4  |-  ( ph  ->  C  e.  LMod )
32 hdmap1neglem1.s . . . . 5  |-  S  =  ( invg `  C )
339, 32, 11lspsnneg 17847 . . . 4  |-  ( ( C  e.  LMod  /\  G  e.  D )  ->  ( L `  { ( S `  G ) } )  =  ( L `  { G } ) )
3431, 22, 33syl2anc 659 . . 3  |-  ( ph  ->  ( L `  {
( S `  G
) } )  =  ( L `  { G } ) )
3525, 30, 343eqtr4d 2505 . 2  |-  ( ph  ->  ( M `  ( N `  { ( R `  Y ) } ) )  =  ( L `  {
( S `  G
) } ) )
3624simprd 461 . . 3  |-  ( ph  ->  ( M `  ( N `  { ( X ( -g `  U
) Y ) } ) )  =  ( L `  { ( F ( -g `  C
) G ) } ) )
37 lmodabl 17752 . . . . . . . . 9  |-  ( U  e.  LMod  ->  U  e. 
Abel )
3826, 37syl 16 . . . . . . . 8  |-  ( ph  ->  U  e.  Abel )
3915eldifad 3473 . . . . . . . 8  |-  ( ph  ->  X  e.  V )
404, 5, 27, 38, 39, 20ablsub2inv 17020 . . . . . . 7  |-  ( ph  ->  ( ( R `  X ) ( -g `  U ) ( R `
 Y ) )  =  ( Y (
-g `  U ) X ) )
4140sneqd 4028 . . . . . 6  |-  ( ph  ->  { ( ( R `
 X ) (
-g `  U )
( R `  Y
) ) }  =  { ( Y (
-g `  U ) X ) } )
4241fveq2d 5852 . . . . 5  |-  ( ph  ->  ( N `  {
( ( R `  X ) ( -g `  U ) ( R `
 Y ) ) } )  =  ( N `  { ( Y ( -g `  U
) X ) } ) )
434, 5, 7, 26, 20, 39lspsnsub 17848 . . . . 5  |-  ( ph  ->  ( N `  {
( Y ( -g `  U ) X ) } )  =  ( N `  { ( X ( -g `  U
) Y ) } ) )
4442, 43eqtrd 2495 . . . 4  |-  ( ph  ->  ( N `  {
( ( R `  X ) ( -g `  U ) ( R `
 Y ) ) } )  =  ( N `  { ( X ( -g `  U
) Y ) } ) )
4544fveq2d 5852 . . 3  |-  ( ph  ->  ( M `  ( N `  { (
( R `  X
) ( -g `  U
) ( R `  Y ) ) } ) )  =  ( M `  ( N `
 { ( X ( -g `  U
) Y ) } ) ) )
46 lmodabl 17752 . . . . . . . 8  |-  ( C  e.  LMod  ->  C  e. 
Abel )
4731, 46syl 16 . . . . . . 7  |-  ( ph  ->  C  e.  Abel )
489, 10, 32, 47, 16, 22ablsub2inv 17020 . . . . . 6  |-  ( ph  ->  ( ( S `  F ) ( -g `  C ) ( S `
 G ) )  =  ( G (
-g `  C ) F ) )
4948sneqd 4028 . . . . 5  |-  ( ph  ->  { ( ( S `
 F ) (
-g `  C )
( S `  G
) ) }  =  { ( G (
-g `  C ) F ) } )
5049fveq2d 5852 . . . 4  |-  ( ph  ->  ( L `  {
( ( S `  F ) ( -g `  C ) ( S `
 G ) ) } )  =  ( L `  { ( G ( -g `  C
) F ) } ) )
519, 10, 11, 31, 22, 16lspsnsub 17848 . . . 4  |-  ( ph  ->  ( L `  {
( G ( -g `  C ) F ) } )  =  ( L `  { ( F ( -g `  C
) G ) } ) )
5250, 51eqtrd 2495 . . 3  |-  ( ph  ->  ( L `  {
( ( S `  F ) ( -g `  C ) ( S `
 G ) ) } )  =  ( L `  { ( F ( -g `  C
) G ) } ) )
5336, 45, 523eqtr4d 2505 . 2  |-  ( ph  ->  ( M `  ( N `  { (
( R `  X
) ( -g `  U
) ( R `  Y ) ) } ) )  =  ( L `  { ( ( S `  F
) ( -g `  C
) ( S `  G ) ) } ) )
54 lmodgrp 17714 . . . . 5  |-  ( U  e.  LMod  ->  U  e. 
Grp )
5526, 54syl 16 . . . 4  |-  ( ph  ->  U  e.  Grp )
564, 6, 27grpinvnzcl 16309 . . . 4  |-  ( ( U  e.  Grp  /\  X  e.  ( V  \  {  .0.  } ) )  ->  ( R `  X )  e.  ( V  \  {  .0.  } ) )
5755, 15, 56syl2anc 659 . . 3  |-  ( ph  ->  ( R `  X
)  e.  ( V 
\  {  .0.  }
) )
589, 32lmodvnegcl 17746 . . . 4  |-  ( ( C  e.  LMod  /\  F  e.  D )  ->  ( S `  F )  e.  D )
5931, 16, 58syl2anc 659 . . 3  |-  ( ph  ->  ( S `  F
)  e.  D )
604, 6, 27grpinvnzcl 16309 . . . 4  |-  ( ( U  e.  Grp  /\  Y  e.  ( V  \  {  .0.  } ) )  ->  ( R `  Y )  e.  ( V  \  {  .0.  } ) )
6155, 17, 60syl2anc 659 . . 3  |-  ( ph  ->  ( R `  Y
)  e.  ( V 
\  {  .0.  }
) )
629, 32lmodvnegcl 17746 . . . 4  |-  ( ( C  e.  LMod  /\  G  e.  D )  ->  ( S `  G )  e.  D )
6331, 22, 62syl2anc 659 . . 3  |-  ( ph  ->  ( S `  G
)  e.  D )
644, 27, 7lspsnneg 17847 . . . . 5  |-  ( ( U  e.  LMod  /\  X  e.  V )  ->  ( N `  { ( R `  X ) } )  =  ( N `  { X } ) )
6526, 39, 64syl2anc 659 . . . 4  |-  ( ph  ->  ( N `  {
( R `  X
) } )  =  ( N `  { X } ) )
6619, 65, 293netr4d 2759 . . 3  |-  ( ph  ->  ( N `  {
( R `  X
) } )  =/=  ( N `  {
( R `  Y
) } ) )
6765fveq2d 5852 . . . 4  |-  ( ph  ->  ( M `  ( N `  { ( R `  X ) } ) )  =  ( M `  ( N `  { X } ) ) )
689, 32, 11lspsnneg 17847 . . . . 5  |-  ( ( C  e.  LMod  /\  F  e.  D )  ->  ( L `  { ( S `  F ) } )  =  ( L `  { F } ) )
6931, 16, 68syl2anc 659 . . . 4  |-  ( ph  ->  ( L `  {
( S `  F
) } )  =  ( L `  { F } ) )
7018, 67, 693eqtr4d 2505 . . 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 37926 . 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 920 1  |-  ( ph  ->  ( I `  <. ( R `  X ) ,  ( S `  F ) ,  ( R `  Y )
>. )  =  ( S `  G )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649    \ cdif 3458   {csn 4016   <.cotp 4024   ` cfv 5570  (class class class)co 6270   Basecbs 14716   0gc0g 14929   Grpcgrp 16252   invgcminusg 16253   -gcsg 16254   Abelcabl 16998   LModclmod 17707   LSpanclspn 17812   HLchlt 35472   LHypclh 36105   DVecHcdvh 37202  LCDualclcd 37710  mapdcmpd 37748  HDMap1chdma1 37916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-riotaBAD 35081
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-ot 4025  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-tpos 6947  df-undef 6994  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-sca 14800  df-vsca 14801  df-0g 14931  df-mre 15075  df-mrc 15076  df-acs 15078  df-preset 15756  df-poset 15774  df-plt 15787  df-lub 15803  df-glb 15804  df-join 15805  df-meet 15806  df-p0 15868  df-p1 15869  df-lat 15875  df-clat 15937  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-grp 16256  df-minusg 16257  df-sbg 16258  df-subg 16397  df-cntz 16554  df-oppg 16580  df-lsm 16855  df-cmn 16999  df-abl 17000  df-mgp 17337  df-ur 17349  df-ring 17395  df-oppr 17467  df-dvdsr 17485  df-unit 17486  df-invr 17516  df-dvr 17527  df-drng 17593  df-lmod 17709  df-lss 17774  df-lsp 17813  df-lvec 17944  df-lsatoms 35098  df-lshyp 35099  df-lcv 35141  df-lfl 35180  df-lkr 35208  df-ldual 35246  df-oposet 35298  df-ol 35300  df-oml 35301  df-covers 35388  df-ats 35389  df-atl 35420  df-cvlat 35444  df-hlat 35473  df-llines 35619  df-lplanes 35620  df-lvols 35621  df-lines 35622  df-psubsp 35624  df-pmap 35625  df-padd 35917  df-lhyp 36109  df-laut 36110  df-ldil 36225  df-ltrn 36226  df-trl 36281  df-tgrp 36866  df-tendo 36878  df-edring 36880  df-dveca 37126  df-disoa 37153  df-dvech 37203  df-dib 37263  df-dic 37297  df-dih 37353  df-doch 37472  df-djh 37519  df-lcdual 37711  df-mapd 37749  df-hdmap1 37918
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator