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Theorem hgmapvvlem1 38050
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 37234 . . . . 5  |-  ( ph  ->  U  e.  LMod )
5 hdmapglem6.r . . . . . 6  |-  R  =  (Scalar `  U )
65lmodring 17715 . . . . 5  |-  ( U  e.  LMod  ->  R  e. 
Ring )
74, 6syl 16 . . . 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.  }
) )
1110eldifad 3473 . . . . . 6  |-  ( ph  ->  X  e.  B )
121, 2, 5, 8, 9, 3, 11hgmapcl 38016 . . . . 5  |-  ( ph  ->  ( G `  X
)  e.  B )
131, 2, 5, 8, 9, 3, 12hgmapcl 38016 . . . 4  |-  ( ph  ->  ( G `  ( G `  X )
)  e.  B )
14 hdmapglem6.y . . . . . 6  |-  ( ph  ->  Y  e.  ( B 
\  {  .0.  }
) )
1514eldifad 3473 . . . . 5  |-  ( ph  ->  Y  e.  B )
161, 2, 5, 8, 9, 3, 15hgmapcl 38016 . . . 4  |-  ( ph  ->  ( G `  Y
)  e.  B )
171, 2, 3dvhlvec 37233 . . . . . 6  |-  ( ph  ->  U  e.  LVec )
185lvecdrng 17946 . . . . . 6  |-  ( U  e.  LVec  ->  R  e.  DivRing )
1917, 18syl 16 . . . . 5  |-  ( ph  ->  R  e.  DivRing )
20 eldifsni 4142 . . . . . . 7  |-  ( Y  e.  ( B  \  {  .0.  } )  ->  Y  =/=  .0.  )
2114, 20syl 16 . . . . . 6  |-  ( ph  ->  Y  =/=  .0.  )
22 hdmapglem6.z . . . . . . . 8  |-  .0.  =  ( 0g `  R )
231, 2, 5, 8, 22, 9, 3, 15hgmapeq0 38031 . . . . . . 7  |-  ( ph  ->  ( ( G `  Y )  =  .0.  <->  Y  =  .0.  ) )
2423necon3bid 2712 . . . . . 6  |-  ( ph  ->  ( ( G `  Y )  =/=  .0.  <->  Y  =/=  .0.  ) )
2521, 24mpbird 232 . . . . 5  |-  ( ph  ->  ( G `  Y
)  =/=  .0.  )
26 hdmapglem6.n . . . . . 6  |-  N  =  ( invr `  R
)
278, 22, 26drnginvrcl 17608 . . . . 5  |-  ( ( R  e.  DivRing  /\  ( G `  Y )  e.  B  /\  ( G `  Y )  =/=  .0.  )  ->  ( N `  ( G `  Y ) )  e.  B )
2819, 16, 25, 27syl3anc 1226 . . . 4  |-  ( ph  ->  ( N `  ( G `  Y )
)  e.  B )
29 hdmapglem6.t . . . . 5  |-  .X.  =  ( .r `  R )
308, 29ringass 17410 . . . 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 )
) ) ) )
317, 13, 16, 28, 30syl13anc 1228 . . 3  |-  ( ph  ->  ( ( ( G `
 ( G `  X ) )  .X.  ( G `  Y ) )  .X.  ( N `  ( G `  Y
) ) )  =  ( ( G `  ( G `  X ) )  .X.  ( ( G `  Y )  .X.  ( N `  ( G `  Y )
) ) ) )
32 hdmapglem6.i . . . . . 6  |-  .1.  =  ( 1r `  R )
338, 22, 29, 32, 26drnginvrr 17611 . . . . 5  |-  ( ( R  e.  DivRing  /\  ( G `  Y )  e.  B  /\  ( G `  Y )  =/=  .0.  )  ->  (
( G `  Y
)  .X.  ( N `  ( G `  Y
) ) )  =  .1.  )
3419, 16, 25, 33syl3anc 1226 . . . 4  |-  ( ph  ->  ( ( G `  Y )  .X.  ( N `  ( G `  Y ) ) )  =  .1.  )
3534oveq2d 6286 . . 3  |-  ( ph  ->  ( ( G `  ( G `  X ) )  .X.  ( ( G `  Y )  .X.  ( N `  ( G `  Y )
) ) )  =  ( ( G `  ( G `  X ) )  .X.  .1.  )
)
368, 29, 32ringridm 17418 . . . 4  |-  ( ( R  e.  Ring  /\  ( G `  ( G `  X ) )  e.  B )  ->  (
( G `  ( G `  X )
)  .X.  .1.  )  =  ( G `  ( G `  X ) ) )
377, 13, 36syl2anc 659 . . 3  |-  ( ph  ->  ( ( G `  ( G `  X ) )  .X.  .1.  )  =  ( G `  ( G `  X ) ) )
3831, 35, 373eqtrrd 2500 . 2  |-  ( ph  ->  ( G `  ( G `  X )
)  =  ( ( ( G `  ( G `  X )
)  .X.  ( G `  Y ) )  .X.  ( N `  ( G `
 Y ) ) ) )
39 hdmapglem6.yx . . . . . . 7  |-  ( ph  ->  ( Y  .X.  ( G `  X )
)  =  .1.  )
4039fveq2d 5852 . . . . . 6  |-  ( ph  ->  ( G `  ( Y  .X.  ( G `  X ) ) )  =  ( G `  .1.  ) )
411, 2, 5, 8, 29, 9, 3, 15, 12hgmapmul 38022 . . . . . 6  |-  ( ph  ->  ( G `  ( Y  .X.  ( G `  X ) ) )  =  ( ( G `
 ( G `  X ) )  .X.  ( G `  Y ) ) )
4240, 41eqtr3d 2497 . . . . 5  |-  ( ph  ->  ( G `  .1.  )  =  ( ( G `  ( G `  X ) )  .X.  ( G `  Y ) ) )
43 hdmapglem6.cd . . . . . . 7  |-  ( ph  ->  ( ( S `  D ) `  C
)  =  .1.  )
4443fveq2d 5852 . . . . . 6  |-  ( ph  ->  ( G `  (
( S `  D
) `  C )
)  =  ( G `
 .1.  ) )
45 hdmapglem6.e . . . . . . 7  |-  E  = 
<. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  W ) ) >.
46 hdmapglem6.o . . . . . . 7  |-  O  =  ( ( ocH `  K
) `  W )
47 hdmapglem6.v . . . . . . 7  |-  V  =  ( Base `  U
)
48 eqid 2454 . . . . . . 7  |-  ( +g  `  U )  =  ( +g  `  U )
49 eqid 2454 . . . . . . 7  |-  ( -g `  U )  =  (
-g `  U )
50 hdmapglem6.q . . . . . . 7  |-  .x.  =  ( .s `  U )
51 hdmapglem6.s . . . . . . 7  |-  S  =  ( (HDMap `  K
) `  W )
52 hdmapglem6.c . . . . . . 7  |-  ( ph  ->  C  e.  ( O `
 { E }
) )
53 hdmapglem6.d . . . . . . 7  |-  ( ph  ->  D  e.  ( O `
 { E }
) )
541, 45, 46, 2, 47, 48, 49, 50, 5, 8, 29, 22, 51, 9, 3, 52, 53, 15, 11hdmapglem5 38049 . . . . . 6  |-  ( ph  ->  ( G `  (
( S `  D
) `  C )
)  =  ( ( S `  C ) `
 D ) )
5544, 54eqtr3d 2497 . . . . 5  |-  ( ph  ->  ( G `  .1.  )  =  ( ( S `  C ) `  D ) )
5642, 55eqtr3d 2497 . . . 4  |-  ( ph  ->  ( ( G `  ( G `  X ) )  .X.  ( G `  Y ) )  =  ( ( S `  C ) `  D
) )
5739, 43eqtr4d 2498 . . . . 5  |-  ( ph  ->  ( Y  .X.  ( G `  X )
)  =  ( ( S `  D ) `
 C ) )
581, 45, 46, 2, 47, 48, 49, 50, 5, 8, 29, 22, 51, 9, 3, 52, 53, 15, 11, 57hdmapinvlem4 38048 . . . 4  |-  ( ph  ->  ( X  .X.  ( G `  Y )
)  =  ( ( S `  C ) `
 D ) )
5956, 58eqtr4d 2498 . . 3  |-  ( ph  ->  ( ( G `  ( G `  X ) )  .X.  ( G `  Y ) )  =  ( X  .X.  ( G `  Y )
) )
6059oveq1d 6285 . 2  |-  ( ph  ->  ( ( ( G `
 ( G `  X ) )  .X.  ( G `  Y ) )  .X.  ( N `  ( G `  Y
) ) )  =  ( ( X  .X.  ( G `  Y ) )  .X.  ( N `  ( G `  Y
) ) ) )
618, 29ringass 17410 . . . 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
) ) ) ) )
627, 11, 16, 28, 61syl13anc 1228 . . 3  |-  ( ph  ->  ( ( X  .X.  ( G `  Y ) )  .X.  ( N `  ( G `  Y
) ) )  =  ( X  .X.  (
( G `  Y
)  .X.  ( N `  ( G `  Y
) ) ) ) )
6334oveq2d 6286 . . 3  |-  ( ph  ->  ( X  .X.  (
( G `  Y
)  .X.  ( N `  ( G `  Y
) ) ) )  =  ( X  .X.  .1.  ) )
648, 29, 32ringridm 17418 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( X  .X.  .1.  )  =  X )
657, 11, 64syl2anc 659 . . 3  |-  ( ph  ->  ( X  .X.  .1.  )  =  X )
6662, 63, 653eqtrd 2499 . 2  |-  ( ph  ->  ( ( X  .X.  ( G `  Y ) )  .X.  ( N `  ( G `  Y
) ) )  =  X )
6738, 60, 663eqtrd 2499 1  |-  ( ph  ->  ( G `  ( G `  X )
)  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649    \ cdif 3458   {csn 4016   <.cop 4022    _I cid 4779    |` cres 4990   ` cfv 5570  (class class class)co 6270   Basecbs 14716   +g cplusg 14784   .rcmulr 14785  Scalarcsca 14787   .scvsca 14788   0gc0g 14929   -gcsg 16254   1rcur 17348   Ringcrg 17393   invrcinvr 17515   DivRingcdr 17591   LModclmod 17707   LVecclvec 17943   HLchlt 35472   LHypclh 36105   LTrncltrn 36222   DVecHcdvh 37202   ocHcoch 37471  HDMapchdma 37917  HGMapchg 38010
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-hvmap 37881  df-hdmap1 37918  df-hdmap 37919  df-hgmap 38011
This theorem is referenced by:  hgmapvvlem2  38051
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