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Theorem hgmapvvlem1 32043
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 31227 . . . . 5  |-  ( ph  ->  U  e.  LMod )
5 hdmapglem6.r . . . . . 6  |-  R  =  (Scalar `  U )
65lmodrng 15887 . . . . 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 3277 . . . . . 6  |-  ( ph  ->  X  e.  B )
121, 2, 5, 8, 9, 3, 11hgmapcl 32009 . . . . 5  |-  ( ph  ->  ( G `  X
)  e.  B )
131, 2, 5, 8, 9, 3, 12hgmapcl 32009 . . . 4  |-  ( ph  ->  ( G `  ( G `  X )
)  e.  B )
14 hdmapglem6.y . . . . . 6  |-  ( ph  ->  Y  e.  ( B 
\  {  .0.  }
) )
1514eldifad 3277 . . . . 5  |-  ( ph  ->  Y  e.  B )
161, 2, 5, 8, 9, 3, 15hgmapcl 32009 . . . 4  |-  ( ph  ->  ( G `  Y
)  e.  B )
171, 2, 3dvhlvec 31226 . . . . . 6  |-  ( ph  ->  U  e.  LVec )
185lvecdrng 16106 . . . . . 6  |-  ( U  e.  LVec  ->  R  e.  DivRing )
1917, 18syl 16 . . . . 5  |-  ( ph  ->  R  e.  DivRing )
20 eldifsni 3873 . . . . . . 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 32024 . . . . . . 7  |-  ( ph  ->  ( ( G `  Y )  =  .0.  <->  Y  =  .0.  ) )
2423necon3bid 2587 . . . . . 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 15781 . . . . 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 15609 . . . 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 15784 . . . . 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 6038 . . 3  |-  ( ph  ->  ( ( G `  ( G `  X ) )  .X.  ( ( G `  Y )  .X.  ( N `  ( G `  Y )
) ) )  =  ( ( G `  ( G `  X ) )  .X.  .1.  )
)
368, 29, 32rngridm 15617 . . . 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 2426 . 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 5674 . . . . . 6  |-  ( ph  ->  ( G `  ( Y  .X.  ( G `  X ) ) )  =  ( G `  .1.  ) )
411, 2, 5, 8, 29, 9, 3, 15, 12hgmapmul 32015 . . . . . 6  |-  ( ph  ->  ( G `  ( Y  .X.  ( G `  X ) ) )  =  ( ( G `
 ( G `  X ) )  .X.  ( G `  Y ) ) )
4240, 41eqtr3d 2423 . . . . 5  |-  ( ph  ->  ( G `  .1.  )  =  ( ( G `  ( G `  X ) )  .X.  ( G `  Y ) ) )
43 hdmapglem6.cd . . . . . . 7  |-  ( ph  ->  ( ( S `  D ) `  C
)  =  .1.  )
4443fveq2d 5674 . . . . . 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 2389 . . . . . . 7  |-  ( +g  `  U )  =  ( +g  `  U )
49 eqid 2389 . . . . . . 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 32042 . . . . . 6  |-  ( ph  ->  ( G `  (
( S `  D
) `  C )
)  =  ( ( S `  C ) `
 D ) )
5544, 54eqtr3d 2423 . . . . 5  |-  ( ph  ->  ( G `  .1.  )  =  ( ( S `  C ) `  D ) )
5642, 55eqtr3d 2423 . . . 4  |-  ( ph  ->  ( ( G `  ( G `  X ) )  .X.  ( G `  Y ) )  =  ( ( S `  C ) `  D
) )
5739, 43eqtr4d 2424 . . . . 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 32041 . . . 4  |-  ( ph  ->  ( X  .X.  ( G `  Y )
)  =  ( ( S `  C ) `
 D ) )
5956, 58eqtr4d 2424 . . 3  |-  ( ph  ->  ( ( G `  ( G `  X ) )  .X.  ( G `  Y ) )  =  ( X  .X.  ( G `  Y )
) )
6059oveq1d 6037 . 2  |-  ( ph  ->  ( ( ( G `
 ( G `  X ) )  .X.  ( G `  Y ) )  .X.  ( N `  ( G `  Y
) ) )  =  ( ( X  .X.  ( G `  Y ) )  .X.  ( N `  ( G `  Y
) ) ) )
618, 29rngass 15609 . . . 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 6038 . . 3  |-  ( ph  ->  ( X  .X.  (
( G `  Y
)  .X.  ( N `  ( G `  Y
) ) ) )  =  ( X  .X.  .1.  ) )
648, 29, 32rngridm 15617 . . . 4  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  ( X  .X.  .1.  )  =  X )
657, 11, 64syl2anc 643 . . 3  |-  ( ph  ->  ( X  .X.  .1.  )  =  X )
6662, 63, 653eqtrd 2425 . 2  |-  ( ph  ->  ( ( X  .X.  ( G `  Y ) )  .X.  ( N `  ( G `  Y
) ) )  =  X )
6738, 60, 663eqtrd 2425 1  |-  ( ph  ->  ( G `  ( G `  X )
)  =  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2552    \ cdif 3262   {csn 3759   <.cop 3762    _I cid 4436    |` cres 4822   ` cfv 5396  (class class class)co 6022   Basecbs 13398   +g cplusg 13458   .rcmulr 13459  Scalarcsca 13461   .scvsca 13462   0gc0g 13652   -gcsg 14617   Ringcrg 15589   1rcur 15591   invrcinvr 15705   DivRingcdr 15764   LModclmod 15879   LVecclvec 16103   HLchlt 29467   LHypclh 30100   LTrncltrn 30217   DVecHcdvh 31195   ocHcoch 31464  HDMapchdma 31910  HGMapchg 32003
This theorem is referenced by:  hgmapvvlem2  32044
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-ot 3769  df-uni 3960  df-int 3995  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-of 6246  df-1st 6290  df-2nd 6291  df-tpos 6417  df-undef 6481  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-map 6958  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-3 9993  df-4 9994  df-5 9995  df-6 9996  df-n0 10156  df-z 10217  df-uz 10423  df-fz 10978  df-struct 13400  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-mulr 13472  df-sca 13474  df-vsca 13475  df-0g 13656  df-mre 13740  df-mrc 13741  df-acs 13743  df-poset 14332  df-plt 14344  df-lub 14360  df-glb 14361  df-join 14362  df-meet 14363  df-p0 14397  df-p1 14398  df-lat 14404  df-clat 14466  df-mnd 14619  df-submnd 14668  df-grp 14741  df-minusg 14742  df-sbg 14743  df-subg 14870  df-cntz 15045  df-oppg 15071  df-lsm 15199  df-cmn 15343  df-abl 15344  df-mgp 15578  df-rng 15592  df-ur 15594  df-oppr 15657  df-dvdsr 15675  df-unit 15676  df-invr 15706  df-dvr 15717  df-drng 15766  df-lmod 15881  df-lss 15938  df-lsp 15977  df-lvec 16104  df-lsatoms 29093  df-lshyp 29094  df-lcv 29136  df-lfl 29175  df-lkr 29203  df-ldual 29241  df-oposet 29293  df-ol 29295  df-oml 29296  df-covers 29383  df-ats 29384  df-atl 29415  df-cvlat 29439  df-hlat 29468  df-llines 29614  df-lplanes 29615  df-lvols 29616  df-lines 29617  df-psubsp 29619  df-pmap 29620  df-padd 29912  df-lhyp 30104  df-laut 30105  df-ldil 30220  df-ltrn 30221  df-trl 30275  df-tgrp 30859  df-tendo 30871  df-edring 30873  df-dveca 31119  df-disoa 31146  df-dvech 31196  df-dib 31256  df-dic 31290  df-dih 31346  df-doch 31465  df-djh 31512  df-lcdual 31704  df-mapd 31742  df-hvmap 31874  df-hdmap1 31911  df-hdmap 31912  df-hgmap 32004
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