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Theorem hdmap1neglem1N 32640
Description: Lemma for hdmapneg 32661. 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 2296 . . . . . 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 2296 . . . . . 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 } ) )
20 eldifi 3311 . . . . . . . . 9  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  Y  e.  V )
2117, 20syl 15 . . . . . . . 8  |-  ( ph  ->  Y  e.  V )
222, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 18, 19, 15, 21hdmap1cl 32617 . . . . . . 7  |-  ( ph  ->  ( I `  <. X ,  F ,  Y >. )  e.  D )
231, 22eqeltrrd 2371 . . . . . 6  |-  ( ph  ->  G  e.  D )
242, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 23, 19, 18hdmap1eq 32614 . . . . 5  |-  ( ph  ->  ( ( I `  <. X ,  F ,  Y >. )  =  G  <-> 
( ( M `  ( N `  { Y } ) )  =  ( L `  { G } )  /\  ( M `  ( N `  { ( X (
-g `  U ) Y ) } ) )  =  ( L `
 { ( F ( -g `  C
) G ) } ) ) ) )
251, 24mpbid 201 . . . 4  |-  ( ph  ->  ( ( M `  ( N `  { Y } ) )  =  ( L `  { G } )  /\  ( M `  ( N `  { ( X (
-g `  U ) Y ) } ) )  =  ( L `
 { ( F ( -g `  C
) G ) } ) ) )
2625simpld 445 . . 3  |-  ( ph  ->  ( M `  ( N `  { Y } ) )  =  ( L `  { G } ) )
272, 3, 14dvhlmod 31922 . . . . 5  |-  ( ph  ->  U  e.  LMod )
28 hdmap1neglem1.r . . . . . 6  |-  R  =  ( inv g `  U )
294, 28, 7lspsnneg 15779 . . . . 5  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( N `  { ( R `  Y ) } )  =  ( N `  { Y } ) )
3027, 21, 29syl2anc 642 . . . 4  |-  ( ph  ->  ( N `  {
( R `  Y
) } )  =  ( N `  { Y } ) )
3130fveq2d 5545 . . 3  |-  ( ph  ->  ( M `  ( N `  { ( R `  Y ) } ) )  =  ( M `  ( N `  { Y } ) ) )
322, 8, 14lcdlmod 32404 . . . 4  |-  ( ph  ->  C  e.  LMod )
33 hdmap1neglem1.s . . . . 5  |-  S  =  ( inv g `  C )
349, 33, 11lspsnneg 15779 . . . 4  |-  ( ( C  e.  LMod  /\  G  e.  D )  ->  ( L `  { ( S `  G ) } )  =  ( L `  { G } ) )
3532, 23, 34syl2anc 642 . . 3  |-  ( ph  ->  ( L `  {
( S `  G
) } )  =  ( L `  { G } ) )
3626, 31, 353eqtr4d 2338 . 2  |-  ( ph  ->  ( M `  ( N `  { ( R `  Y ) } ) )  =  ( L `  {
( S `  G
) } ) )
3725simprd 449 . . 3  |-  ( ph  ->  ( M `  ( N `  { ( X ( -g `  U
) Y ) } ) )  =  ( L `  { ( F ( -g `  C
) G ) } ) )
38 lmodabl 15688 . . . . . . . . 9  |-  ( U  e.  LMod  ->  U  e. 
Abel )
3927, 38syl 15 . . . . . . . 8  |-  ( ph  ->  U  e.  Abel )
40 eldifi 3311 . . . . . . . . 9  |-  ( X  e.  ( V  \  {  .0.  } )  ->  X  e.  V )
4115, 40syl 15 . . . . . . . 8  |-  ( ph  ->  X  e.  V )
424, 5, 28, 39, 41, 21ablsub2inv 15128 . . . . . . 7  |-  ( ph  ->  ( ( R `  X ) ( -g `  U ) ( R `
 Y ) )  =  ( Y (
-g `  U ) X ) )
4342sneqd 3666 . . . . . 6  |-  ( ph  ->  { ( ( R `
 X ) (
-g `  U )
( R `  Y
) ) }  =  { ( Y (
-g `  U ) X ) } )
4443fveq2d 5545 . . . . 5  |-  ( ph  ->  ( N `  {
( ( R `  X ) ( -g `  U ) ( R `
 Y ) ) } )  =  ( N `  { ( Y ( -g `  U
) X ) } ) )
454, 5, 7, 27, 21, 41lspsnsub 15780 . . . . 5  |-  ( ph  ->  ( N `  {
( Y ( -g `  U ) X ) } )  =  ( N `  { ( X ( -g `  U
) Y ) } ) )
4644, 45eqtrd 2328 . . . 4  |-  ( ph  ->  ( N `  {
( ( R `  X ) ( -g `  U ) ( R `
 Y ) ) } )  =  ( N `  { ( X ( -g `  U
) Y ) } ) )
4746fveq2d 5545 . . 3  |-  ( ph  ->  ( M `  ( N `  { (
( R `  X
) ( -g `  U
) ( R `  Y ) ) } ) )  =  ( M `  ( N `
 { ( X ( -g `  U
) Y ) } ) ) )
48 lmodabl 15688 . . . . . . . 8  |-  ( C  e.  LMod  ->  C  e. 
Abel )
4932, 48syl 15 . . . . . . 7  |-  ( ph  ->  C  e.  Abel )
509, 10, 33, 49, 16, 23ablsub2inv 15128 . . . . . 6  |-  ( ph  ->  ( ( S `  F ) ( -g `  C ) ( S `
 G ) )  =  ( G (
-g `  C ) F ) )
5150sneqd 3666 . . . . 5  |-  ( ph  ->  { ( ( S `
 F ) (
-g `  C )
( S `  G
) ) }  =  { ( G (
-g `  C ) F ) } )
5251fveq2d 5545 . . . 4  |-  ( ph  ->  ( L `  {
( ( S `  F ) ( -g `  C ) ( S `
 G ) ) } )  =  ( L `  { ( G ( -g `  C
) F ) } ) )
539, 10, 11, 32, 23, 16lspsnsub 15780 . . . 4  |-  ( ph  ->  ( L `  {
( G ( -g `  C ) F ) } )  =  ( L `  { ( F ( -g `  C
) G ) } ) )
5452, 53eqtrd 2328 . . 3  |-  ( ph  ->  ( L `  {
( ( S `  F ) ( -g `  C ) ( S `
 G ) ) } )  =  ( L `  { ( F ( -g `  C
) G ) } ) )
5537, 47, 543eqtr4d 2338 . 2  |-  ( ph  ->  ( M `  ( N `  { (
( R `  X
) ( -g `  U
) ( R `  Y ) ) } ) )  =  ( L `  { ( ( S `  F
) ( -g `  C
) ( S `  G ) ) } ) )
56 lmodgrp 15650 . . . . 5  |-  ( U  e.  LMod  ->  U  e. 
Grp )
5727, 56syl 15 . . . 4  |-  ( ph  ->  U  e.  Grp )
584, 6, 28grpinvnzcl 14556 . . . 4  |-  ( ( U  e.  Grp  /\  X  e.  ( V  \  {  .0.  } ) )  ->  ( R `  X )  e.  ( V  \  {  .0.  } ) )
5957, 15, 58syl2anc 642 . . 3  |-  ( ph  ->  ( R `  X
)  e.  ( V 
\  {  .0.  }
) )
609, 33lmodvnegcl 15681 . . . 4  |-  ( ( C  e.  LMod  /\  F  e.  D )  ->  ( S `  F )  e.  D )
6132, 16, 60syl2anc 642 . . 3  |-  ( ph  ->  ( S `  F
)  e.  D )
624, 6, 28grpinvnzcl 14556 . . . 4  |-  ( ( U  e.  Grp  /\  Y  e.  ( V  \  {  .0.  } ) )  ->  ( R `  Y )  e.  ( V  \  {  .0.  } ) )
6357, 17, 62syl2anc 642 . . 3  |-  ( ph  ->  ( R `  Y
)  e.  ( V 
\  {  .0.  }
) )
649, 33lmodvnegcl 15681 . . . 4  |-  ( ( C  e.  LMod  /\  G  e.  D )  ->  ( S `  G )  e.  D )
6532, 23, 64syl2anc 642 . . 3  |-  ( ph  ->  ( S `  G
)  e.  D )
664, 28, 7lspsnneg 15779 . . . . 5  |-  ( ( U  e.  LMod  /\  X  e.  V )  ->  ( N `  { ( R `  X ) } )  =  ( N `  { X } ) )
6727, 41, 66syl2anc 642 . . . 4  |-  ( ph  ->  ( N `  {
( R `  X
) } )  =  ( N `  { X } ) )
6819, 67, 303netr4d 2486 . . 3  |-  ( ph  ->  ( N `  {
( R `  X
) } )  =/=  ( N `  {
( R `  Y
) } ) )
6967fveq2d 5545 . . . 4  |-  ( ph  ->  ( M `  ( N `  { ( R `  X ) } ) )  =  ( M `  ( N `  { X } ) ) )
709, 33, 11lspsnneg 15779 . . . . 5  |-  ( ( C  e.  LMod  /\  F  e.  D )  ->  ( L `  { ( S `  F ) } )  =  ( L `  { F } ) )
7132, 16, 70syl2anc 642 . . . 4  |-  ( ph  ->  ( L `  {
( S `  F
) } )  =  ( L `  { F } ) )
7218, 69, 713eqtr4d 2338 . . 3  |-  ( ph  ->  ( M `  ( N `  { ( R `  X ) } ) )  =  ( L `  {
( S `  F
) } ) )
732, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 59, 61, 63, 65, 68, 72hdmap1eq 32614 . 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 ) ) } ) ) ) )
7436, 55, 73mpbir2and 888 1  |-  ( ph  ->  ( I `  <. ( R `  X ) ,  ( S `  F ) ,  ( R `  Y )
>. )  =  ( S `  G )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162   {csn 3653   <.cotp 3657   ` cfv 5271  (class class class)co 5874   Basecbs 13164   0gc0g 13416   Grpcgrp 14378   inv gcminusg 14379   -gcsg 14381   Abelcabel 15106   LModclmod 15643   LSpanclspn 15744   HLchlt 30162   LHypclh 30795   DVecHcdvh 31890  LCDualclcd 32398  mapdcmpd 32436  HDMap1chdma1 32604
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-ot 3663  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-tpos 6250  df-undef 6314  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-0g 13420  df-mre 13504  df-mrc 13505  df-acs 13507  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-cntz 14809  df-oppg 14835  df-lsm 14963  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-dvr 15481  df-drng 15530  df-lmod 15645  df-lss 15706  df-lsp 15745  df-lvec 15872  df-lsatoms 29788  df-lshyp 29789  df-lcv 29831  df-lfl 29870  df-lkr 29898  df-ldual 29936  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310  df-lvols 30311  df-lines 30312  df-psubsp 30314  df-pmap 30315  df-padd 30607  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916  df-trl 30970  df-tgrp 31554  df-tendo 31566  df-edring 31568  df-dveca 31814  df-disoa 31841  df-dvech 31891  df-dib 31951  df-dic 31985  df-dih 32041  df-doch 32160  df-djh 32207  df-lcdual 32399  df-mapd 32437  df-hdmap1 32606
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