Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hdmap1neglem1N Structured version   Unicode version

Theorem hdmap1neglem1N 35782
Description: Lemma for hdmapneg 35803. 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 2451 . . . . . 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 2451 . . . . . 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 3441 . . . . . . . 8  |-  ( ph  ->  Y  e.  V )
212, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 18, 19, 15, 20hdmap1cl 35759 . . . . . . 7  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  e.  D )
221, 21eqeltrrd 2540 . . . . . 6  |-  ( ph  ->  G  e.  D )
232, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 22, 19, 18hdmap1eq 35756 . . . . 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 459 . . 3  |-  ( ph  ->  ( M `  ( N `  { Y } ) )  =  ( L `  { G } ) )
262, 3, 14dvhlmod 35064 . . . . 5  |-  ( ph  ->  U  e.  LMod )
27 hdmap1neglem1.r . . . . . 6  |-  R  =  ( invg `  U )
284, 27, 7lspsnneg 17202 . . . . 5  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( N `  { ( R `  Y ) } )  =  ( N `  { Y } ) )
2926, 20, 28syl2anc 661 . . . 4  |-  ( ph  ->  ( N `  {
( R `  Y
) } )  =  ( N `  { Y } ) )
3029fveq2d 5796 . . 3  |-  ( ph  ->  ( M `  ( N `  { ( R `  Y ) } ) )  =  ( M `  ( N `  { Y } ) ) )
312, 8, 14lcdlmod 35546 . . . 4  |-  ( ph  ->  C  e.  LMod )
32 hdmap1neglem1.s . . . . 5  |-  S  =  ( invg `  C )
339, 32, 11lspsnneg 17202 . . . 4  |-  ( ( C  e.  LMod  /\  G  e.  D )  ->  ( L `  { ( S `  G ) } )  =  ( L `  { G } ) )
3431, 22, 33syl2anc 661 . . 3  |-  ( ph  ->  ( L `  {
( S `  G
) } )  =  ( L `  { G } ) )
3525, 30, 343eqtr4d 2502 . 2  |-  ( ph  ->  ( M `  ( N `  { ( R `  Y ) } ) )  =  ( L `  {
( S `  G
) } ) )
3624simprd 463 . . 3  |-  ( ph  ->  ( M `  ( N `  { ( X ( -g `  U
) Y ) } ) )  =  ( L `  { ( F ( -g `  C
) G ) } ) )
37 lmodabl 17107 . . . . . . . . 9  |-  ( U  e.  LMod  ->  U  e. 
Abel )
3826, 37syl 16 . . . . . . . 8  |-  ( ph  ->  U  e.  Abel )
3915eldifad 3441 . . . . . . . 8  |-  ( ph  ->  X  e.  V )
404, 5, 27, 38, 39, 20ablsub2inv 16413 . . . . . . 7  |-  ( ph  ->  ( ( R `  X ) ( -g `  U ) ( R `
 Y ) )  =  ( Y (
-g `  U ) X ) )
4140sneqd 3990 . . . . . 6  |-  ( ph  ->  { ( ( R `
 X ) (
-g `  U )
( R `  Y
) ) }  =  { ( Y (
-g `  U ) X ) } )
4241fveq2d 5796 . . . . 5  |-  ( ph  ->  ( N `  {
( ( R `  X ) ( -g `  U ) ( R `
 Y ) ) } )  =  ( N `  { ( Y ( -g `  U
) X ) } ) )
434, 5, 7, 26, 20, 39lspsnsub 17203 . . . . 5  |-  ( ph  ->  ( N `  {
( Y ( -g `  U ) X ) } )  =  ( N `  { ( X ( -g `  U
) Y ) } ) )
4442, 43eqtrd 2492 . . . 4  |-  ( ph  ->  ( N `  {
( ( R `  X ) ( -g `  U ) ( R `
 Y ) ) } )  =  ( N `  { ( X ( -g `  U
) Y ) } ) )
4544fveq2d 5796 . . 3  |-  ( ph  ->  ( M `  ( N `  { (
( R `  X
) ( -g `  U
) ( R `  Y ) ) } ) )  =  ( M `  ( N `
 { ( X ( -g `  U
) Y ) } ) ) )
46 lmodabl 17107 . . . . . . . 8  |-  ( C  e.  LMod  ->  C  e. 
Abel )
4731, 46syl 16 . . . . . . 7  |-  ( ph  ->  C  e.  Abel )
489, 10, 32, 47, 16, 22ablsub2inv 16413 . . . . . 6  |-  ( ph  ->  ( ( S `  F ) ( -g `  C ) ( S `
 G ) )  =  ( G (
-g `  C ) F ) )
4948sneqd 3990 . . . . 5  |-  ( ph  ->  { ( ( S `
 F ) (
-g `  C )
( S `  G
) ) }  =  { ( G (
-g `  C ) F ) } )
5049fveq2d 5796 . . . 4  |-  ( ph  ->  ( L `  {
( ( S `  F ) ( -g `  C ) ( S `
 G ) ) } )  =  ( L `  { ( G ( -g `  C
) F ) } ) )
519, 10, 11, 31, 22, 16lspsnsub 17203 . . . 4  |-  ( ph  ->  ( L `  {
( G ( -g `  C ) F ) } )  =  ( L `  { ( F ( -g `  C
) G ) } ) )
5250, 51eqtrd 2492 . . 3  |-  ( ph  ->  ( L `  {
( ( S `  F ) ( -g `  C ) ( S `
 G ) ) } )  =  ( L `  { ( F ( -g `  C
) G ) } ) )
5336, 45, 523eqtr4d 2502 . 2  |-  ( ph  ->  ( M `  ( N `  { (
( R `  X
) ( -g `  U
) ( R `  Y ) ) } ) )  =  ( L `  { ( ( S `  F
) ( -g `  C
) ( S `  G ) ) } ) )
54 lmodgrp 17070 . . . . 5  |-  ( U  e.  LMod  ->  U  e. 
Grp )
5526, 54syl 16 . . . 4  |-  ( ph  ->  U  e.  Grp )
564, 6, 27grpinvnzcl 15709 . . . 4  |-  ( ( U  e.  Grp  /\  X  e.  ( V  \  {  .0.  } ) )  ->  ( R `  X )  e.  ( V  \  {  .0.  } ) )
5755, 15, 56syl2anc 661 . . 3  |-  ( ph  ->  ( R `  X
)  e.  ( V 
\  {  .0.  }
) )
589, 32lmodvnegcl 17101 . . . 4  |-  ( ( C  e.  LMod  /\  F  e.  D )  ->  ( S `  F )  e.  D )
5931, 16, 58syl2anc 661 . . 3  |-  ( ph  ->  ( S `  F
)  e.  D )
604, 6, 27grpinvnzcl 15709 . . . 4  |-  ( ( U  e.  Grp  /\  Y  e.  ( V  \  {  .0.  } ) )  ->  ( R `  Y )  e.  ( V  \  {  .0.  } ) )
6155, 17, 60syl2anc 661 . . 3  |-  ( ph  ->  ( R `  Y
)  e.  ( V 
\  {  .0.  }
) )
629, 32lmodvnegcl 17101 . . . 4  |-  ( ( C  e.  LMod  /\  G  e.  D )  ->  ( S `  G )  e.  D )
6331, 22, 62syl2anc 661 . . 3  |-  ( ph  ->  ( S `  G
)  e.  D )
644, 27, 7lspsnneg 17202 . . . . 5  |-  ( ( U  e.  LMod  /\  X  e.  V )  ->  ( N `  { ( R `  X ) } )  =  ( N `  { X } ) )
6526, 39, 64syl2anc 661 . . . 4  |-  ( ph  ->  ( N `  {
( R `  X
) } )  =  ( N `  { X } ) )
6619, 65, 293netr4d 2753 . . 3  |-  ( ph  ->  ( N `  {
( R `  X
) } )  =/=  ( N `  {
( R `  Y
) } ) )
6765fveq2d 5796 . . . 4  |-  ( ph  ->  ( M `  ( N `  { ( R `  X ) } ) )  =  ( M `  ( N `  { X } ) ) )
689, 32, 11lspsnneg 17202 . . . . 5  |-  ( ( C  e.  LMod  /\  F  e.  D )  ->  ( L `  { ( S `  F ) } )  =  ( L `  { F } ) )
6931, 16, 68syl2anc 661 . . . 4  |-  ( ph  ->  ( L `  {
( S `  F
) } )  =  ( L `  { F } ) )
7018, 67, 693eqtr4d 2502 . . 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 35756 . 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 913 1  |-  ( ph  ->  ( I `  <. ( R `  X ) ,  ( S `  F ) ,  ( R `  Y )
>. )  =  ( S `  G )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644    \ cdif 3426   {csn 3978   <.cotp 3986   ` cfv 5519  (class class class)co 6193   Basecbs 14285   0gc0g 14489   Grpcgrp 15521   invgcminusg 15522   -gcsg 15524   Abelcabel 16391   LModclmod 17063   LSpanclspn 17167   HLchlt 33304   LHypclh 33937   DVecHcdvh 35032  LCDualclcd 35540  mapdcmpd 35578  HDMap1chdma1 35746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-riotaBAD 32913
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-ot 3987  df-uni 4193  df-int 4230  df-iun 4274  df-iin 4275  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-of 6423  df-om 6580  df-1st 6680  df-2nd 6681  df-tpos 6848  df-undef 6895  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-map 7319  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-n0 10684  df-z 10751  df-uz 10966  df-fz 11548  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-mulr 14363  df-sca 14365  df-vsca 14366  df-0g 14491  df-mre 14635  df-mrc 14636  df-acs 14638  df-poset 15227  df-plt 15239  df-lub 15255  df-glb 15256  df-join 15257  df-meet 15258  df-p0 15320  df-p1 15321  df-lat 15327  df-clat 15389  df-mnd 15526  df-submnd 15576  df-grp 15656  df-minusg 15657  df-sbg 15658  df-subg 15789  df-cntz 15946  df-oppg 15972  df-lsm 16248  df-cmn 16392  df-abl 16393  df-mgp 16706  df-ur 16718  df-rng 16762  df-oppr 16830  df-dvdsr 16848  df-unit 16849  df-invr 16879  df-dvr 16890  df-drng 16949  df-lmod 17065  df-lss 17129  df-lsp 17168  df-lvec 17299  df-lsatoms 32930  df-lshyp 32931  df-lcv 32973  df-lfl 33012  df-lkr 33040  df-ldual 33078  df-oposet 33130  df-ol 33132  df-oml 33133  df-covers 33220  df-ats 33221  df-atl 33252  df-cvlat 33276  df-hlat 33305  df-llines 33451  df-lplanes 33452  df-lvols 33453  df-lines 33454  df-psubsp 33456  df-pmap 33457  df-padd 33749  df-lhyp 33941  df-laut 33942  df-ldil 34057  df-ltrn 34058  df-trl 34112  df-tgrp 34696  df-tendo 34708  df-edring 34710  df-dveca 34956  df-disoa 34983  df-dvech 35033  df-dib 35093  df-dic 35127  df-dih 35183  df-doch 35302  df-djh 35349  df-lcdual 35541  df-mapd 35579  df-hdmap1 35748
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator