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Theorem hgmapvvlem1 32413
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 31597 . . . . 5  |-  ( ph  ->  U  e.  LMod )
5 hdmapglem6.r . . . . . 6  |-  R  =  (Scalar `  U )
65lmodrng 15917 . . . . 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 3296 . . . . . 6  |-  ( ph  ->  X  e.  B )
121, 2, 5, 8, 9, 3, 11hgmapcl 32379 . . . . 5  |-  ( ph  ->  ( G `  X
)  e.  B )
131, 2, 5, 8, 9, 3, 12hgmapcl 32379 . . . 4  |-  ( ph  ->  ( G `  ( G `  X )
)  e.  B )
14 hdmapglem6.y . . . . . 6  |-  ( ph  ->  Y  e.  ( B 
\  {  .0.  }
) )
1514eldifad 3296 . . . . 5  |-  ( ph  ->  Y  e.  B )
161, 2, 5, 8, 9, 3, 15hgmapcl 32379 . . . 4  |-  ( ph  ->  ( G `  Y
)  e.  B )
171, 2, 3dvhlvec 31596 . . . . . 6  |-  ( ph  ->  U  e.  LVec )
185lvecdrng 16136 . . . . . 6  |-  ( U  e.  LVec  ->  R  e.  DivRing )
1917, 18syl 16 . . . . 5  |-  ( ph  ->  R  e.  DivRing )
20 eldifsni 3892 . . . . . . 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 32394 . . . . . . 7  |-  ( ph  ->  ( ( G `  Y )  =  .0.  <->  Y  =  .0.  ) )
2423necon3bid 2606 . . . . . 6  |-  ( ph  ->  ( ( G `  Y )  =/=  .0.  <->  Y  =/=  .0.  ) )
2521, 24mpbird 224 . . . . 5  |-  ( ph  ->  ( G `  Y
)  =/=  .0.  )
26 hdmapglem6.n . . . . . 6  |-  N  =  ( invr `  R
)
278, 22, 26drnginvrcl 15811 . . . . 5  |-  ( ( R  e.  DivRing  /\  ( G `  Y )  e.  B  /\  ( G `  Y )  =/=  .0.  )  ->  ( N `  ( G `  Y ) )  e.  B )
2819, 16, 25, 27syl3anc 1184 . . . 4  |-  ( ph  ->  ( N `  ( G `  Y )
)  e.  B )
29 hdmapglem6.t . . . . 5  |-  .X.  =  ( .r `  R )
308, 29rngass 15639 . . . 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 1186 . . 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 15814 . . . . 5  |-  ( ( R  e.  DivRing  /\  ( G `  Y )  e.  B  /\  ( G `  Y )  =/=  .0.  )  ->  (
( G `  Y
)  .X.  ( N `  ( G `  Y
) ) )  =  .1.  )
3419, 16, 25, 33syl3anc 1184 . . . 4  |-  ( ph  ->  ( ( G `  Y )  .X.  ( N `  ( G `  Y ) ) )  =  .1.  )
3534oveq2d 6060 . . 3  |-  ( ph  ->  ( ( G `  ( G `  X ) )  .X.  ( ( G `  Y )  .X.  ( N `  ( G `  Y )
) ) )  =  ( ( G `  ( G `  X ) )  .X.  .1.  )
)
368, 29, 32rngridm 15647 . . . 4  |-  ( ( R  e.  Ring  /\  ( G `  ( G `  X ) )  e.  B )  ->  (
( G `  ( G `  X )
)  .X.  .1.  )  =  ( G `  ( G `  X ) ) )
377, 13, 36syl2anc 643 . . 3  |-  ( ph  ->  ( ( G `  ( G `  X ) )  .X.  .1.  )  =  ( G `  ( G `  X ) ) )
3831, 35, 373eqtrrd 2445 . 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 5695 . . . . . 6  |-  ( ph  ->  ( G `  ( Y  .X.  ( G `  X ) ) )  =  ( G `  .1.  ) )
411, 2, 5, 8, 29, 9, 3, 15, 12hgmapmul 32385 . . . . . 6  |-  ( ph  ->  ( G `  ( Y  .X.  ( G `  X ) ) )  =  ( ( G `
 ( G `  X ) )  .X.  ( G `  Y ) ) )
4240, 41eqtr3d 2442 . . . . 5  |-  ( ph  ->  ( G `  .1.  )  =  ( ( G `  ( G `  X ) )  .X.  ( G `  Y ) ) )
43 hdmapglem6.cd . . . . . . 7  |-  ( ph  ->  ( ( S `  D ) `  C
)  =  .1.  )
4443fveq2d 5695 . . . . . 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 2408 . . . . . . 7  |-  ( +g  `  U )  =  ( +g  `  U )
49 eqid 2408 . . . . . . 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 32412 . . . . . 6  |-  ( ph  ->  ( G `  (
( S `  D
) `  C )
)  =  ( ( S `  C ) `
 D ) )
5544, 54eqtr3d 2442 . . . . 5  |-  ( ph  ->  ( G `  .1.  )  =  ( ( S `  C ) `  D ) )
5642, 55eqtr3d 2442 . . . 4  |-  ( ph  ->  ( ( G `  ( G `  X ) )  .X.  ( G `  Y ) )  =  ( ( S `  C ) `  D
) )
5739, 43eqtr4d 2443 . . . . 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 32411 . . . 4  |-  ( ph  ->  ( X  .X.  ( G `  Y )
)  =  ( ( S `  C ) `
 D ) )
5956, 58eqtr4d 2443 . . 3  |-  ( ph  ->  ( ( G `  ( G `  X ) )  .X.  ( G `  Y ) )  =  ( X  .X.  ( G `  Y )
) )
6059oveq1d 6059 . 2  |-  ( ph  ->  ( ( ( G `
 ( G `  X ) )  .X.  ( G `  Y ) )  .X.  ( N `  ( G `  Y
) ) )  =  ( ( X  .X.  ( G `  Y ) )  .X.  ( N `  ( G `  Y
) ) ) )
618, 29rngass 15639 . . . 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 1186 . . 3  |-  ( ph  ->  ( ( X  .X.  ( G `  Y ) )  .X.  ( N `  ( G `  Y
) ) )  =  ( X  .X.  (
( G `  Y
)  .X.  ( N `  ( G `  Y
) ) ) ) )
6334oveq2d 6060 . . 3  |-  ( ph  ->  ( X  .X.  (
( G `  Y
)  .X.  ( N `  ( G `  Y
) ) ) )  =  ( X  .X.  .1.  ) )
648, 29, 32rngridm 15647 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( X  .X.  .1.  )  =  X )
657, 11, 64syl2anc 643 . . 3  |-  ( ph  ->  ( X  .X.  .1.  )  =  X )
6662, 63, 653eqtrd 2444 . 2  |-  ( ph  ->  ( ( X  .X.  ( G `  Y ) )  .X.  ( N `  ( G `  Y
) ) )  =  X )
6738, 60, 663eqtrd 2444 1  |-  ( ph  ->  ( G `  ( G `  X )
)  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2571    \ cdif 3281   {csn 3778   <.cop 3781    _I cid 4457    |` cres 4843   ` cfv 5417  (class class class)co 6044   Basecbs 13428   +g cplusg 13488   .rcmulr 13489  Scalarcsca 13491   .scvsca 13492   0gc0g 13682   -gcsg 14647   Ringcrg 15619   1rcur 15621   invrcinvr 15735   DivRingcdr 15794   LModclmod 15909   LVecclvec 16133   HLchlt 29837   LHypclh 30470   LTrncltrn 30587   DVecHcdvh 31565   ocHcoch 31834  HDMapchdma 32280  HGMapchg 32373
This theorem is referenced by:  hgmapvvlem2  32414
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-fal 1326  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-ot 3788  df-uni 3980  df-int 4015  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-of 6268  df-1st 6312  df-2nd 6313  df-tpos 6442  df-undef 6506  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-er 6868  df-map 6983  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-nn 9961  df-2 10018  df-3 10019  df-4 10020  df-5 10021  df-6 10022  df-n0 10182  df-z 10243  df-uz 10449  df-fz 11004  df-struct 13430  df-ndx 13431  df-slot 13432  df-base 13433  df-sets 13434  df-ress 13435  df-plusg 13501  df-mulr 13502  df-sca 13504  df-vsca 13505  df-0g 13686  df-mre 13770  df-mrc 13771  df-acs 13773  df-poset 14362  df-plt 14374  df-lub 14390  df-glb 14391  df-join 14392  df-meet 14393  df-p0 14427  df-p1 14428  df-lat 14434  df-clat 14496  df-mnd 14649  df-submnd 14698  df-grp 14771  df-minusg 14772  df-sbg 14773  df-subg 14900  df-cntz 15075  df-oppg 15101  df-lsm 15229  df-cmn 15373  df-abl 15374  df-mgp 15608  df-rng 15622  df-ur 15624  df-oppr 15687  df-dvdsr 15705  df-unit 15706  df-invr 15736  df-dvr 15747  df-drng 15796  df-lmod 15911  df-lss 15968  df-lsp 16007  df-lvec 16134  df-lsatoms 29463  df-lshyp 29464  df-lcv 29506  df-lfl 29545  df-lkr 29573  df-ldual 29611  df-oposet 29663  df-ol 29665  df-oml 29666  df-covers 29753  df-ats 29754  df-atl 29785  df-cvlat 29809  df-hlat 29838  df-llines 29984  df-lplanes 29985  df-lvols 29986  df-lines 29987  df-psubsp 29989  df-pmap 29990  df-padd 30282  df-lhyp 30474  df-laut 30475  df-ldil 30590  df-ltrn 30591  df-trl 30645  df-tgrp 31229  df-tendo 31241  df-edring 31243  df-dveca 31489  df-disoa 31516  df-dvech 31566  df-dib 31626  df-dic 31660  df-dih 31716  df-doch 31835  df-djh 31882  df-lcdual 32074  df-mapd 32112  df-hvmap 32244  df-hdmap1 32281  df-hdmap 32282  df-hgmap 32374
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