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Theorem hgmapvvlem1 31020
Description: Involution property of scalar sigma map. Line 10 in [Baer] p. 111, t sigma squared = t. Our  E,  C,  D,  Y,  X correspond to Baer's w, h, k, s, t. (Contributed by NM, 13-Jun-2015.)
Hypotheses
Ref Expression
hdmapglem6.h  |-  H  =  ( LHyp `  K
)
hdmapglem6.e  |-  E  = 
<. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  W ) ) >.
hdmapglem6.o  |-  O  =  ( ( ocH `  K
) `  W )
hdmapglem6.u  |-  U  =  ( ( DVecH `  K
) `  W )
hdmapglem6.v  |-  V  =  ( Base `  U
)
hdmapglem6.q  |-  .x.  =  ( .s `  U )
hdmapglem6.r  |-  R  =  (Scalar `  U )
hdmapglem6.b  |-  B  =  ( Base `  R
)
hdmapglem6.t  |-  .X.  =  ( .r `  R )
hdmapglem6.z  |-  .0.  =  ( 0g `  R )
hdmapglem6.i  |-  .1.  =  ( 1r `  R )
hdmapglem6.n  |-  N  =  ( invr `  R
)
hdmapglem6.s  |-  S  =  ( (HDMap `  K
) `  W )
hdmapglem6.g  |-  G  =  ( (HGMap `  K
) `  W )
hdmapglem6.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
hdmapglem6.x  |-  ( ph  ->  X  e.  ( B 
\  {  .0.  }
) )
hdmapglem6.c  |-  ( ph  ->  C  e.  ( O `
 { E }
) )
hdmapglem6.d  |-  ( ph  ->  D  e.  ( O `
 { E }
) )
hdmapglem6.cd  |-  ( ph  ->  ( ( S `  D ) `  C
)  =  .1.  )
hdmapglem6.y  |-  ( ph  ->  Y  e.  ( B 
\  {  .0.  }
) )
hdmapglem6.yx  |-  ( ph  ->  ( Y  .X.  ( G `  X )
)  =  .1.  )
Assertion
Ref Expression
hgmapvvlem1  |-  ( ph  ->  ( G `  ( G `  X )
)  =  X )

Proof of Theorem hgmapvvlem1
StepHypRef Expression
1 hdmapglem6.h . . . . . 6  |-  H  =  ( LHyp `  K
)
2 hdmapglem6.u . . . . . 6  |-  U  =  ( ( DVecH `  K
) `  W )
3 hdmapglem6.k . . . . . 6  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
41, 2, 3dvhlmod 30204 . . . . 5  |-  ( ph  ->  U  e.  LMod )
5 hdmapglem6.r . . . . . 6  |-  R  =  (Scalar `  U )
65lmodrng 15470 . . . . 5  |-  ( U  e.  LMod  ->  R  e. 
Ring )
74, 6syl 17 . . . 4  |-  ( ph  ->  R  e.  Ring )
8 hdmapglem6.b . . . . 5  |-  B  =  ( Base `  R
)
9 hdmapglem6.g . . . . 5  |-  G  =  ( (HGMap `  K
) `  W )
10 hdmapglem6.x . . . . . . 7  |-  ( ph  ->  X  e.  ( B 
\  {  .0.  }
) )
11 eldifi 3215 . . . . . . 7  |-  ( X  e.  ( B  \  {  .0.  } )  ->  X  e.  B )
1210, 11syl 17 . . . . . 6  |-  ( ph  ->  X  e.  B )
131, 2, 5, 8, 9, 3, 12hgmapcl 30986 . . . . 5  |-  ( ph  ->  ( G `  X
)  e.  B )
141, 2, 5, 8, 9, 3, 13hgmapcl 30986 . . . 4  |-  ( ph  ->  ( G `  ( G `  X )
)  e.  B )
15 hdmapglem6.y . . . . . 6  |-  ( ph  ->  Y  e.  ( B 
\  {  .0.  }
) )
16 eldifi 3215 . . . . . 6  |-  ( Y  e.  ( B  \  {  .0.  } )  ->  Y  e.  B )
1715, 16syl 17 . . . . 5  |-  ( ph  ->  Y  e.  B )
181, 2, 5, 8, 9, 3, 17hgmapcl 30986 . . . 4  |-  ( ph  ->  ( G `  Y
)  e.  B )
191, 2, 3dvhlvec 30203 . . . . . 6  |-  ( ph  ->  U  e.  LVec )
205lvecdrng 15693 . . . . . 6  |-  ( U  e.  LVec  ->  R  e.  DivRing )
2119, 20syl 17 . . . . 5  |-  ( ph  ->  R  e.  DivRing )
22 eldifsni 3654 . . . . . . 7  |-  ( Y  e.  ( B  \  {  .0.  } )  ->  Y  =/=  .0.  )
2315, 22syl 17 . . . . . 6  |-  ( ph  ->  Y  =/=  .0.  )
24 hdmapglem6.z . . . . . . . 8  |-  .0.  =  ( 0g `  R )
251, 2, 5, 8, 24, 9, 3, 17hgmapeq0 31001 . . . . . . 7  |-  ( ph  ->  ( ( G `  Y )  =  .0.  <->  Y  =  .0.  ) )
2625necon3bid 2447 . . . . . 6  |-  ( ph  ->  ( ( G `  Y )  =/=  .0.  <->  Y  =/=  .0.  ) )
2723, 26mpbird 225 . . . . 5  |-  ( ph  ->  ( G `  Y
)  =/=  .0.  )
28 hdmapglem6.n . . . . . 6  |-  N  =  ( invr `  R
)
298, 24, 28drnginvrcl 15364 . . . . 5  |-  ( ( R  e.  DivRing  /\  ( G `  Y )  e.  B  /\  ( G `  Y )  =/=  .0.  )  ->  ( N `  ( G `  Y ) )  e.  B )
3021, 18, 27, 29syl3anc 1187 . . . 4  |-  ( ph  ->  ( N `  ( G `  Y )
)  e.  B )
31 hdmapglem6.t . . . . 5  |-  .X.  =  ( .r `  R )
328, 31rngass 15192 . . . 4  |-  ( ( R  e.  Ring  /\  (
( G `  ( G `  X )
)  e.  B  /\  ( G `  Y )  e.  B  /\  ( N `  ( G `  Y ) )  e.  B ) )  -> 
( ( ( G `
 ( G `  X ) )  .X.  ( G `  Y ) )  .X.  ( N `  ( G `  Y
) ) )  =  ( ( G `  ( G `  X ) )  .X.  ( ( G `  Y )  .X.  ( N `  ( G `  Y )
) ) ) )
337, 14, 18, 30, 32syl13anc 1189 . . 3  |-  ( ph  ->  ( ( ( G `
 ( G `  X ) )  .X.  ( G `  Y ) )  .X.  ( N `  ( G `  Y
) ) )  =  ( ( G `  ( G `  X ) )  .X.  ( ( G `  Y )  .X.  ( N `  ( G `  Y )
) ) ) )
34 hdmapglem6.i . . . . . 6  |-  .1.  =  ( 1r `  R )
358, 24, 31, 34, 28drnginvrr 15367 . . . . 5  |-  ( ( R  e.  DivRing  /\  ( G `  Y )  e.  B  /\  ( G `  Y )  =/=  .0.  )  ->  (
( G `  Y
)  .X.  ( N `  ( G `  Y
) ) )  =  .1.  )
3621, 18, 27, 35syl3anc 1187 . . . 4  |-  ( ph  ->  ( ( G `  Y )  .X.  ( N `  ( G `  Y ) ) )  =  .1.  )
3736oveq2d 5726 . . 3  |-  ( ph  ->  ( ( G `  ( G `  X ) )  .X.  ( ( G `  Y )  .X.  ( N `  ( G `  Y )
) ) )  =  ( ( G `  ( G `  X ) )  .X.  .1.  )
)
388, 31, 34rngridm 15200 . . . 4  |-  ( ( R  e.  Ring  /\  ( G `  ( G `  X ) )  e.  B )  ->  (
( G `  ( G `  X )
)  .X.  .1.  )  =  ( G `  ( G `  X ) ) )
397, 14, 38syl2anc 645 . . 3  |-  ( ph  ->  ( ( G `  ( G `  X ) )  .X.  .1.  )  =  ( G `  ( G `  X ) ) )
4033, 37, 393eqtrrd 2290 . 2  |-  ( ph  ->  ( G `  ( G `  X )
)  =  ( ( ( G `  ( G `  X )
)  .X.  ( G `  Y ) )  .X.  ( N `  ( G `
 Y ) ) ) )
41 hdmapglem6.yx . . . . . . 7  |-  ( ph  ->  ( Y  .X.  ( G `  X )
)  =  .1.  )
4241fveq2d 5381 . . . . . 6  |-  ( ph  ->  ( G `  ( Y  .X.  ( G `  X ) ) )  =  ( G `  .1.  ) )
431, 2, 5, 8, 31, 9, 3, 17, 13hgmapmul 30992 . . . . . 6  |-  ( ph  ->  ( G `  ( Y  .X.  ( G `  X ) ) )  =  ( ( G `
 ( G `  X ) )  .X.  ( G `  Y ) ) )
4442, 43eqtr3d 2287 . . . . 5  |-  ( ph  ->  ( G `  .1.  )  =  ( ( G `  ( G `  X ) )  .X.  ( G `  Y ) ) )
45 hdmapglem6.cd . . . . . . 7  |-  ( ph  ->  ( ( S `  D ) `  C
)  =  .1.  )
4645fveq2d 5381 . . . . . 6  |-  ( ph  ->  ( G `  (
( S `  D
) `  C )
)  =  ( G `
 .1.  ) )
47 hdmapglem6.e . . . . . . 7  |-  E  = 
<. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  W ) ) >.
48 hdmapglem6.o . . . . . . 7  |-  O  =  ( ( ocH `  K
) `  W )
49 hdmapglem6.v . . . . . . 7  |-  V  =  ( Base `  U
)
50 eqid 2253 . . . . . . 7  |-  ( +g  `  U )  =  ( +g  `  U )
51 eqid 2253 . . . . . . 7  |-  ( -g `  U )  =  (
-g `  U )
52 hdmapglem6.q . . . . . . 7  |-  .x.  =  ( .s `  U )
53 hdmapglem6.s . . . . . . 7  |-  S  =  ( (HDMap `  K
) `  W )
54 hdmapglem6.c . . . . . . 7  |-  ( ph  ->  C  e.  ( O `
 { E }
) )
55 hdmapglem6.d . . . . . . 7  |-  ( ph  ->  D  e.  ( O `
 { E }
) )
561, 47, 48, 2, 49, 50, 51, 52, 5, 8, 31, 24, 53, 9, 3, 54, 55, 17, 12hdmapglem5 31019 . . . . . 6  |-  ( ph  ->  ( G `  (
( S `  D
) `  C )
)  =  ( ( S `  C ) `
 D ) )
5746, 56eqtr3d 2287 . . . . 5  |-  ( ph  ->  ( G `  .1.  )  =  ( ( S `  C ) `  D ) )
5844, 57eqtr3d 2287 . . . 4  |-  ( ph  ->  ( ( G `  ( G `  X ) )  .X.  ( G `  Y ) )  =  ( ( S `  C ) `  D
) )
5941, 45eqtr4d 2288 . . . . 5  |-  ( ph  ->  ( Y  .X.  ( G `  X )
)  =  ( ( S `  D ) `
 C ) )
601, 47, 48, 2, 49, 50, 51, 52, 5, 8, 31, 24, 53, 9, 3, 54, 55, 17, 12, 59hdmapinvlem4 31018 . . . 4  |-  ( ph  ->  ( X  .X.  ( G `  Y )
)  =  ( ( S `  C ) `
 D ) )
6158, 60eqtr4d 2288 . . 3  |-  ( ph  ->  ( ( G `  ( G `  X ) )  .X.  ( G `  Y ) )  =  ( X  .X.  ( G `  Y )
) )
6261oveq1d 5725 . 2  |-  ( ph  ->  ( ( ( G `
 ( G `  X ) )  .X.  ( G `  Y ) )  .X.  ( N `  ( G `  Y
) ) )  =  ( ( X  .X.  ( G `  Y ) )  .X.  ( N `  ( G `  Y
) ) ) )
638, 31rngass 15192 . . . 4  |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  ( G `  Y )  e.  B  /\  ( N `  ( G `  Y ) )  e.  B ) )  -> 
( ( X  .X.  ( G `  Y ) )  .X.  ( N `  ( G `  Y
) ) )  =  ( X  .X.  (
( G `  Y
)  .X.  ( N `  ( G `  Y
) ) ) ) )
647, 12, 18, 30, 63syl13anc 1189 . . 3  |-  ( ph  ->  ( ( X  .X.  ( G `  Y ) )  .X.  ( N `  ( G `  Y
) ) )  =  ( X  .X.  (
( G `  Y
)  .X.  ( N `  ( G `  Y
) ) ) ) )
6536oveq2d 5726 . . 3  |-  ( ph  ->  ( X  .X.  (
( G `  Y
)  .X.  ( N `  ( G `  Y
) ) ) )  =  ( X  .X.  .1.  ) )
668, 31, 34rngridm 15200 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( X  .X.  .1.  )  =  X )
677, 12, 66syl2anc 645 . . 3  |-  ( ph  ->  ( X  .X.  .1.  )  =  X )
6864, 65, 673eqtrd 2289 . 2  |-  ( ph  ->  ( ( X  .X.  ( G `  Y ) )  .X.  ( N `  ( G `  Y
) ) )  =  X )
6940, 62, 683eqtrd 2289 1  |-  ( ph  ->  ( G `  ( G `  X )
)  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2412    \ cdif 3075   {csn 3544   <.cop 3547    _I cid 4197    |` cres 4582   ` cfv 4592  (class class class)co 5710   Basecbs 13022   +g cplusg 13082   .rcmulr 13083  Scalarcsca 13085   .scvsca 13086   0gc0g 13274   -gcsg 14200   Ringcrg 15172   1rcur 15174   invrcinvr 15288   DivRingcdr 15347   LModclmod 15462   LVecclvec 15690   HLchlt 28444   LHypclh 29077   LTrncltrn 29194   DVecHcdvh 30172   ocHcoch 30441  HDMapchdma 30887  HGMapchg 30980
This theorem is referenced by:  hgmapvvlem2  31021
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-fal 1316  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-ot 3554  df-uni 3728  df-int 3761  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-of 5930  df-1st 5974  df-2nd 5975  df-tpos 6086  df-iota 6143  df-undef 6182  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-er 6546  df-map 6660  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-n 9627  df-2 9684  df-3 9685  df-4 9686  df-5 9687  df-6 9688  df-n0 9845  df-z 9904  df-uz 10110  df-fz 10661  df-struct 13024  df-ndx 13025  df-slot 13026  df-base 13027  df-sets 13028  df-ress 13029  df-plusg 13095  df-mulr 13096  df-sca 13098  df-vsca 13099  df-0g 13278  df-mre 13361  df-mrc 13362  df-acs 13363  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-mnd 14202  df-submnd 14251  df-grp 14324  df-minusg 14325  df-sbg 14326  df-subg 14453  df-cntz 14628  df-oppg 14654  df-lsm 14782  df-cmn 14926  df-abl 14927  df-mgp 15161  df-ring 15175  df-ur 15177  df-oppr 15240  df-dvdsr 15258  df-unit 15259  df-invr 15289  df-dvr 15300  df-drng 15349  df-lmod 15464  df-lss 15525  df-lsp 15564  df-lvec 15691  df-lsatoms 28070  df-lshyp 28071  df-lcv 28113  df-lfl 28152  df-lkr 28180  df-ldual 28218  df-oposet 28270  df-ol 28272  df-oml 28273  df-covers 28360  df-ats 28361  df-atl 28392  df-cvlat 28416  df-hlat 28445  df-llines 28591  df-lplanes 28592  df-lvols 28593  df-lines 28594  df-psubsp 28596  df-pmap 28597  df-padd 28889  df-lhyp 29081  df-laut 29082  df-ldil 29197  df-ltrn 29198  df-trl 29252  df-tgrp 29836  df-tendo 29848  df-edring 29850  df-dveca 30096  df-disoa 30123  df-dvech 30173  df-dib 30233  df-dic 30267  df-dih 30323  df-doch 30442  df-djh 30489  df-lcdual 30681  df-mapd 30719  df-hvmap 30851  df-hdmap1 30888  df-hdmap 30889  df-hgmap 30981
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