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Theorem hgmapvvlem1 31384
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 30568 . . . . 5  |-  ( ph  ->  U  e.  LMod )
5 hdmapglem6.r . . . . . 6  |-  R  =  (Scalar `  U )
65lmodrng 15630 . . . . 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 3300 . . . . . . 7  |-  ( X  e.  ( B  \  {  .0.  } )  ->  X  e.  B )
1210, 11syl 17 . . . . . 6  |-  ( ph  ->  X  e.  B )
131, 2, 5, 8, 9, 3, 12hgmapcl 31350 . . . . 5  |-  ( ph  ->  ( G `  X
)  e.  B )
141, 2, 5, 8, 9, 3, 13hgmapcl 31350 . . . 4  |-  ( ph  ->  ( G `  ( G `  X )
)  e.  B )
15 hdmapglem6.y . . . . . 6  |-  ( ph  ->  Y  e.  ( B 
\  {  .0.  }
) )
16 eldifi 3300 . . . . . 6  |-  ( Y  e.  ( B  \  {  .0.  } )  ->  Y  e.  B )
1715, 16syl 17 . . . . 5  |-  ( ph  ->  Y  e.  B )
181, 2, 5, 8, 9, 3, 17hgmapcl 31350 . . . 4  |-  ( ph  ->  ( G `  Y
)  e.  B )
191, 2, 3dvhlvec 30567 . . . . . 6  |-  ( ph  ->  U  e.  LVec )
205lvecdrng 15853 . . . . . 6  |-  ( U  e.  LVec  ->  R  e.  DivRing )
2119, 20syl 17 . . . . 5  |-  ( ph  ->  R  e.  DivRing )
22 eldifsni 3752 . . . . . . 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 31365 . . . . . . 7  |-  ( ph  ->  ( ( G `  Y )  =  .0.  <->  Y  =  .0.  ) )
2625necon3bid 2483 . . . . . 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 15524 . . . . 5  |-  ( ( R  e.  DivRing  /\  ( G `  Y )  e.  B  /\  ( G `  Y )  =/=  .0.  )  ->  ( N `  ( G `  Y ) )  e.  B )
3021, 18, 27, 29syl3anc 1184 . . . 4  |-  ( ph  ->  ( N `  ( G `  Y )
)  e.  B )
31 hdmapglem6.t . . . . 5  |-  .X.  =  ( .r `  R )
328, 31rngass 15352 . . . 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 1186 . . 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 15527 . . . . 5  |-  ( ( R  e.  DivRing  /\  ( G `  Y )  e.  B  /\  ( G `  Y )  =/=  .0.  )  ->  (
( G `  Y
)  .X.  ( N `  ( G `  Y
) ) )  =  .1.  )
3621, 18, 27, 35syl3anc 1184 . . . 4  |-  ( ph  ->  ( ( G `  Y )  .X.  ( N `  ( G `  Y ) ) )  =  .1.  )
3736oveq2d 5836 . . 3  |-  ( ph  ->  ( ( G `  ( G `  X ) )  .X.  ( ( G `  Y )  .X.  ( N `  ( G `  Y )
) ) )  =  ( ( G `  ( G `  X ) )  .X.  .1.  )
)
388, 31, 34rngridm 15360 . . . 4  |-  ( ( R  e.  Ring  /\  ( G `  ( G `  X ) )  e.  B )  ->  (
( G `  ( G `  X )
)  .X.  .1.  )  =  ( G `  ( G `  X ) ) )
397, 14, 38syl2anc 644 . . 3  |-  ( ph  ->  ( ( G `  ( G `  X ) )  .X.  .1.  )  =  ( G `  ( G `  X ) ) )
4033, 37, 393eqtrrd 2322 . 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 5490 . . . . . 6  |-  ( ph  ->  ( G `  ( Y  .X.  ( G `  X ) ) )  =  ( G `  .1.  ) )
431, 2, 5, 8, 31, 9, 3, 17, 13hgmapmul 31356 . . . . . 6  |-  ( ph  ->  ( G `  ( Y  .X.  ( G `  X ) ) )  =  ( ( G `
 ( G `  X ) )  .X.  ( G `  Y ) ) )
4442, 43eqtr3d 2319 . . . . 5  |-  ( ph  ->  ( G `  .1.  )  =  ( ( G `  ( G `  X ) )  .X.  ( G `  Y ) ) )
45 hdmapglem6.cd . . . . . . 7  |-  ( ph  ->  ( ( S `  D ) `  C
)  =  .1.  )
4645fveq2d 5490 . . . . . 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 2285 . . . . . . 7  |-  ( +g  `  U )  =  ( +g  `  U )
51 eqid 2285 . . . . . . 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 31383 . . . . . 6  |-  ( ph  ->  ( G `  (
( S `  D
) `  C )
)  =  ( ( S `  C ) `
 D ) )
5746, 56eqtr3d 2319 . . . . 5  |-  ( ph  ->  ( G `  .1.  )  =  ( ( S `  C ) `  D ) )
5844, 57eqtr3d 2319 . . . 4  |-  ( ph  ->  ( ( G `  ( G `  X ) )  .X.  ( G `  Y ) )  =  ( ( S `  C ) `  D
) )
5941, 45eqtr4d 2320 . . . . 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 31382 . . . 4  |-  ( ph  ->  ( X  .X.  ( G `  Y )
)  =  ( ( S `  C ) `
 D ) )
6158, 60eqtr4d 2320 . . 3  |-  ( ph  ->  ( ( G `  ( G `  X ) )  .X.  ( G `  Y ) )  =  ( X  .X.  ( G `  Y )
) )
6261oveq1d 5835 . 2  |-  ( ph  ->  ( ( ( G `
 ( G `  X ) )  .X.  ( G `  Y ) )  .X.  ( N `  ( G `  Y
) ) )  =  ( ( X  .X.  ( G `  Y ) )  .X.  ( N `  ( G `  Y
) ) ) )
638, 31rngass 15352 . . . 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 1186 . . 3  |-  ( ph  ->  ( ( X  .X.  ( G `  Y ) )  .X.  ( N `  ( G `  Y
) ) )  =  ( X  .X.  (
( G `  Y
)  .X.  ( N `  ( G `  Y
) ) ) ) )
6536oveq2d 5836 . . 3  |-  ( ph  ->  ( X  .X.  (
( G `  Y
)  .X.  ( N `  ( G `  Y
) ) ) )  =  ( X  .X.  .1.  ) )
668, 31, 34rngridm 15360 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( X  .X.  .1.  )  =  X )
677, 12, 66syl2anc 644 . . 3  |-  ( ph  ->  ( X  .X.  .1.  )  =  X )
6864, 65, 673eqtrd 2321 . 2  |-  ( ph  ->  ( ( X  .X.  ( G `  Y ) )  .X.  ( N `  ( G `  Y
) ) )  =  X )
6940, 62, 683eqtrd 2321 1  |-  ( ph  ->  ( G `  ( G `  X )
)  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1624    e. wcel 1685    =/= wne 2448    \ cdif 3151   {csn 3642   <.cop 3645    _I cid 4304    |` cres 4691   ` cfv 5222  (class class class)co 5820   Basecbs 13143   +g cplusg 13203   .rcmulr 13204  Scalarcsca 13206   .scvsca 13207   0gc0g 13395   -gcsg 14360   Ringcrg 15332   1rcur 15334   invrcinvr 15448   DivRingcdr 15507   LModclmod 15622   LVecclvec 15850   HLchlt 28808   LHypclh 29441   LTrncltrn 29558   DVecHcdvh 30536   ocHcoch 30805  HDMapchdma 31251  HGMapchg 31344
This theorem is referenced by:  hgmapvvlem2  31385
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-fal 1313  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-ot 3652  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-of 6040  df-1st 6084  df-2nd 6085  df-tpos 6196  df-iota 6253  df-undef 6292  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-oadd 6479  df-er 6656  df-map 6770  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-nn 9743  df-2 9800  df-3 9801  df-4 9802  df-5 9803  df-6 9804  df-n0 9962  df-z 10021  df-uz 10227  df-fz 10778  df-struct 13145  df-ndx 13146  df-slot 13147  df-base 13148  df-sets 13149  df-ress 13150  df-plusg 13216  df-mulr 13217  df-sca 13219  df-vsca 13220  df-0g 13399  df-mre 13483  df-mrc 13484  df-acs 13486  df-poset 14075  df-plt 14087  df-lub 14103  df-glb 14104  df-join 14105  df-meet 14106  df-p0 14140  df-p1 14141  df-lat 14147  df-clat 14209  df-mnd 14362  df-submnd 14411  df-grp 14484  df-minusg 14485  df-sbg 14486  df-subg 14613  df-cntz 14788  df-oppg 14814  df-lsm 14942  df-cmn 15086  df-abl 15087  df-mgp 15321  df-rng 15335  df-ur 15337  df-oppr 15400  df-dvdsr 15418  df-unit 15419  df-invr 15449  df-dvr 15460  df-drng 15509  df-lmod 15624  df-lss 15685  df-lsp 15724  df-lvec 15851  df-lsatoms 28434  df-lshyp 28435  df-lcv 28477  df-lfl 28516  df-lkr 28544  df-ldual 28582  df-oposet 28634  df-ol 28636  df-oml 28637  df-covers 28724  df-ats 28725  df-atl 28756  df-cvlat 28780  df-hlat 28809  df-llines 28955  df-lplanes 28956  df-lvols 28957  df-lines 28958  df-psubsp 28960  df-pmap 28961  df-padd 29253  df-lhyp 29445  df-laut 29446  df-ldil 29561  df-ltrn 29562  df-trl 29616  df-tgrp 30200  df-tendo 30212  df-edring 30214  df-dveca 30460  df-disoa 30487  df-dvech 30537  df-dib 30597  df-dic 30631  df-dih 30687  df-doch 30806  df-djh 30853  df-lcdual 31045  df-mapd 31083  df-hvmap 31215  df-hdmap1 31252  df-hdmap 31253  df-hgmap 31345
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