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Theorem hgmapvvlem3 35465
Description: Lemma for hgmapvv 35466. Eliminate  ( ( S `  D
) `  C )  =  .1. (Baer's f(h,k)=1). (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.  }
) )
Assertion
Ref Expression
hgmapvvlem3  |-  ( ph  ->  ( G `  ( G `  X )
)  =  X )

Proof of Theorem hgmapvvlem3
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 hdmapglem6.h . . . 4  |-  H  =  ( LHyp `  K
)
2 hdmapglem6.o . . . 4  |-  O  =  ( ( ocH `  K
) `  W )
3 hdmapglem6.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
4 hdmapglem6.v . . . 4  |-  V  =  ( Base `  U
)
5 eqid 2422 . . . 4  |-  ( 0g
`  U )  =  ( 0g `  U
)
6 hdmapglem6.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
7 eqid 2422 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
8 eqid 2422 . . . . . 6  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
9 hdmapglem6.e . . . . . 6  |-  E  = 
<. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  W ) ) >.
101, 7, 8, 3, 4, 5, 9, 6dvheveccl 34649 . . . . 5  |-  ( ph  ->  E  e.  ( V 
\  { ( 0g
`  U ) } ) )
1110eldifad 3448 . . . 4  |-  ( ph  ->  E  e.  V )
121, 2, 3, 4, 5, 6, 11dochsnnz 34987 . . 3  |-  ( ph  ->  ( O `  { E } )  =/=  {
( 0g `  U
) } )
1311snssd 4145 . . . . 5  |-  ( ph  ->  { E }  C_  V )
14 eqid 2422 . . . . . 6  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
151, 3, 4, 14, 2dochlss 34891 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  { E }  C_  V )  ->  ( O `  { E } )  e.  (
LSubSp `  U ) )
166, 13, 15syl2anc 665 . . . 4  |-  ( ph  ->  ( O `  { E } )  e.  (
LSubSp `  U ) )
175, 14lssne0 18173 . . . 4  |-  ( ( O `  { E } )  e.  (
LSubSp `  U )  -> 
( ( O `  { E } )  =/= 
{ ( 0g `  U ) }  <->  E. k  e.  ( O `  { E } ) k  =/=  ( 0g `  U
) ) )
1816, 17syl 17 . . 3  |-  ( ph  ->  ( ( O `  { E } )  =/= 
{ ( 0g `  U ) }  <->  E. k  e.  ( O `  { E } ) k  =/=  ( 0g `  U
) ) )
1912, 18mpbid 213 . 2  |-  ( ph  ->  E. k  e.  ( O `  { E } ) k  =/=  ( 0g `  U
) )
20 eqid 2422 . . . . 5  |-  ( .s
`  U )  =  ( .s `  U
)
21 hdmapglem6.r . . . . 5  |-  R  =  (Scalar `  U )
22 hdmapglem6.i . . . . 5  |-  .1.  =  ( 1r `  R )
23 hdmapglem6.n . . . . 5  |-  N  =  ( invr `  R
)
24 hdmapglem6.s . . . . 5  |-  S  =  ( (HDMap `  K
) `  W )
2563ad2ant1 1026 . . . . 5  |-  ( (
ph  /\  k  e.  ( O `  { E } )  /\  k  =/=  ( 0g `  U
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
261, 3, 4, 2dochssv 34892 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  { E }  C_  V )  ->  ( O `  { E } )  C_  V
)
276, 13, 26syl2anc 665 . . . . . . . 8  |-  ( ph  ->  ( O `  { E } )  C_  V
)
2827sselda 3464 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( O `  { E } ) )  -> 
k  e.  V )
29283adant3 1025 . . . . . 6  |-  ( (
ph  /\  k  e.  ( O `  { E } )  /\  k  =/=  ( 0g `  U
) )  ->  k  e.  V )
30 simp3 1007 . . . . . 6  |-  ( (
ph  /\  k  e.  ( O `  { E } )  /\  k  =/=  ( 0g `  U
) )  ->  k  =/=  ( 0g `  U
) )
31 eldifsn 4125 . . . . . 6  |-  ( k  e.  ( V  \  { ( 0g `  U ) } )  <-> 
( k  e.  V  /\  k  =/=  ( 0g `  U ) ) )
3229, 30, 31sylanbrc 668 . . . . 5  |-  ( (
ph  /\  k  e.  ( O `  { E } )  /\  k  =/=  ( 0g `  U
) )  ->  k  e.  ( V  \  {
( 0g `  U
) } ) )
33 eqid 2422 . . . . 5  |-  ( ( N `  ( ( S `  k ) `
 k ) ) ( .s `  U
) k )  =  ( ( N `  ( ( S `  k ) `  k
) ) ( .s
`  U ) k )
341, 3, 4, 20, 5, 21, 22, 23, 24, 25, 32, 33hdmapip1 35456 . . . 4  |-  ( (
ph  /\  k  e.  ( O `  { E } )  /\  k  =/=  ( 0g `  U
) )  ->  (
( S `  k
) `  ( ( N `  ( ( S `  k ) `  k ) ) ( .s `  U ) k ) )  =  .1.  )
35 hdmapglem6.q . . . . 5  |-  .x.  =  ( .s `  U )
36 hdmapglem6.b . . . . 5  |-  B  =  ( Base `  R
)
37 hdmapglem6.t . . . . 5  |-  .X.  =  ( .r `  R )
38 hdmapglem6.z . . . . 5  |-  .0.  =  ( 0g `  R )
39 hdmapglem6.g . . . . 5  |-  G  =  ( (HGMap `  K
) `  W )
40 simpl1 1008 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( O `  { E } )  /\  k  =/=  ( 0g `  U
) )  /\  (
( S `  k
) `  ( ( N `  ( ( S `  k ) `  k ) ) ( .s `  U ) k ) )  =  .1.  )  ->  ph )
4140, 6syl 17 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( O `  { E } )  /\  k  =/=  ( 0g `  U
) )  /\  (
( S `  k
) `  ( ( N `  ( ( S `  k ) `  k ) ) ( .s `  U ) k ) )  =  .1.  )  ->  ( K  e.  HL  /\  W  e.  H ) )
42 hdmapglem6.x . . . . . 6  |-  ( ph  ->  X  e.  ( B 
\  {  .0.  }
) )
4340, 42syl 17 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( O `  { E } )  /\  k  =/=  ( 0g `  U
) )  /\  (
( S `  k
) `  ( ( N `  ( ( S `  k ) `  k ) ) ( .s `  U ) k ) )  =  .1.  )  ->  X  e.  ( B  \  {  .0.  } ) )
441, 3, 6dvhlmod 34647 . . . . . . 7  |-  ( ph  ->  U  e.  LMod )
4540, 44syl 17 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( O `  { E } )  /\  k  =/=  ( 0g `  U
) )  /\  (
( S `  k
) `  ( ( N `  ( ( S `  k ) `  k ) ) ( .s `  U ) k ) )  =  .1.  )  ->  U  e.  LMod )
4640, 16syl 17 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( O `  { E } )  /\  k  =/=  ( 0g `  U
) )  /\  (
( S `  k
) `  ( ( N `  ( ( S `  k ) `  k ) ) ( .s `  U ) k ) )  =  .1.  )  ->  ( O `  { E } )  e.  (
LSubSp `  U ) )
471, 3, 6dvhlvec 34646 . . . . . . . . 9  |-  ( ph  ->  U  e.  LVec )
4821lvecdrng 18327 . . . . . . . . 9  |-  ( U  e.  LVec  ->  R  e.  DivRing )
4947, 48syl 17 . . . . . . . 8  |-  ( ph  ->  R  e.  DivRing )
5040, 49syl 17 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( O `  { E } )  /\  k  =/=  ( 0g `  U
) )  /\  (
( S `  k
) `  ( ( N `  ( ( S `  k ) `  k ) ) ( .s `  U ) k ) )  =  .1.  )  ->  R  e.  DivRing )
5129adantr 466 . . . . . . . 8  |-  ( ( ( ph  /\  k  e.  ( O `  { E } )  /\  k  =/=  ( 0g `  U
) )  /\  (
( S `  k
) `  ( ( N `  ( ( S `  k ) `  k ) ) ( .s `  U ) k ) )  =  .1.  )  ->  k  e.  V )
521, 3, 4, 21, 36, 24, 41, 51, 51hdmapipcl 35445 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( O `  { E } )  /\  k  =/=  ( 0g `  U
) )  /\  (
( S `  k
) `  ( ( N `  ( ( S `  k ) `  k ) ) ( .s `  U ) k ) )  =  .1.  )  ->  (
( S `  k
) `  k )  e.  B )
536adantr 466 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( O `  { E } ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
541, 3, 4, 5, 21, 38, 24, 53, 28hdmapip0 35455 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( O `  { E } ) )  -> 
( ( ( S `
 k ) `  k )  =  .0.  <->  k  =  ( 0g `  U ) ) )
5554necon3bid 2678 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( O `  { E } ) )  -> 
( ( ( S `
 k ) `  k )  =/=  .0.  <->  k  =/=  ( 0g `  U
) ) )
5655biimp3ar 1365 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( O `  { E } )  /\  k  =/=  ( 0g `  U
) )  ->  (
( S `  k
) `  k )  =/=  .0.  )
5756adantr 466 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( O `  { E } )  /\  k  =/=  ( 0g `  U
) )  /\  (
( S `  k
) `  ( ( N `  ( ( S `  k ) `  k ) ) ( .s `  U ) k ) )  =  .1.  )  ->  (
( S `  k
) `  k )  =/=  .0.  )
5836, 38, 23drnginvrcl 17991 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  (
( S `  k
) `  k )  e.  B  /\  (
( S `  k
) `  k )  =/=  .0.  )  ->  ( N `  ( ( S `  k ) `  k ) )  e.  B )
5950, 52, 57, 58syl3anc 1264 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( O `  { E } )  /\  k  =/=  ( 0g `  U
) )  /\  (
( S `  k
) `  ( ( N `  ( ( S `  k ) `  k ) ) ( .s `  U ) k ) )  =  .1.  )  ->  ( N `  ( ( S `  k ) `  k ) )  e.  B )
60 simpl2 1009 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( O `  { E } )  /\  k  =/=  ( 0g `  U
) )  /\  (
( S `  k
) `  ( ( N `  ( ( S `  k ) `  k ) ) ( .s `  U ) k ) )  =  .1.  )  ->  k  e.  ( O `  { E } ) )
6121, 20, 36, 14lssvscl 18177 . . . . . 6  |-  ( ( ( U  e.  LMod  /\  ( O `  { E } )  e.  (
LSubSp `  U ) )  /\  ( ( N `
 ( ( S `
 k ) `  k ) )  e.  B  /\  k  e.  ( O `  { E } ) ) )  ->  ( ( N `
 ( ( S `
 k ) `  k ) ) ( .s `  U ) k )  e.  ( O `  { E } ) )
6245, 46, 59, 60, 61syl22anc 1265 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( O `  { E } )  /\  k  =/=  ( 0g `  U
) )  /\  (
( S `  k
) `  ( ( N `  ( ( S `  k ) `  k ) ) ( .s `  U ) k ) )  =  .1.  )  ->  (
( N `  (
( S `  k
) `  k )
) ( .s `  U ) k )  e.  ( O `  { E } ) )
63 simpr 462 . . . . 5  |-  ( ( ( ph  /\  k  e.  ( O `  { E } )  /\  k  =/=  ( 0g `  U
) )  /\  (
( S `  k
) `  ( ( N `  ( ( S `  k ) `  k ) ) ( .s `  U ) k ) )  =  .1.  )  ->  (
( S `  k
) `  ( ( N `  ( ( S `  k ) `  k ) ) ( .s `  U ) k ) )  =  .1.  )
641, 9, 2, 3, 4, 35, 21, 36, 37, 38, 22, 23, 24, 39, 41, 43, 62, 60, 63hgmapvvlem2 35464 . . . 4  |-  ( ( ( ph  /\  k  e.  ( O `  { E } )  /\  k  =/=  ( 0g `  U
) )  /\  (
( S `  k
) `  ( ( N `  ( ( S `  k ) `  k ) ) ( .s `  U ) k ) )  =  .1.  )  ->  ( G `  ( G `  X ) )  =  X )
6534, 64mpdan 672 . . 3  |-  ( (
ph  /\  k  e.  ( O `  { E } )  /\  k  =/=  ( 0g `  U
) )  ->  ( G `  ( G `  X ) )  =  X )
6665rexlimdv3a 2916 . 2  |-  ( ph  ->  ( E. k  e.  ( O `  { E } ) k  =/=  ( 0g `  U
)  ->  ( G `  ( G `  X
) )  =  X ) )
6719, 66mpd 15 1  |-  ( ph  ->  ( G `  ( G `  X )
)  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   E.wrex 2772    \ cdif 3433    C_ wss 3436   {csn 3998   <.cop 4004    _I cid 4763    |` cres 4855   ` cfv 5601  (class class class)co 6305   Basecbs 15120   .rcmulr 15190  Scalarcsca 15192   .scvsca 15193   0gc0g 15337   1rcur 17734   invrcinvr 17898   DivRingcdr 17974   LModclmod 18090   LSubSpclss 18154   LVecclvec 18324   HLchlt 32885   LHypclh 33518   LTrncltrn 33635   DVecHcdvh 34615   ocHcoch 34884  HDMapchdma 35330  HGMapchg 35423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623  ax-riotaBAD 32494
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-ot 4007  df-uni 4220  df-int 4256  df-iun 4301  df-iin 4302  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-tpos 6984  df-undef 7031  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-oadd 7197  df-er 7374  df-map 7485  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-2 10675  df-3 10676  df-4 10677  df-5 10678  df-6 10679  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-struct 15122  df-ndx 15123  df-slot 15124  df-base 15125  df-sets 15126  df-ress 15127  df-plusg 15202  df-mulr 15203  df-sca 15205  df-vsca 15206  df-0g 15339  df-mre 15491  df-mrc 15492  df-acs 15494  df-preset 16172  df-poset 16190  df-plt 16203  df-lub 16219  df-glb 16220  df-join 16221  df-meet 16222  df-p0 16284  df-p1 16285  df-lat 16291  df-clat 16353  df-mgm 16487  df-sgrp 16526  df-mnd 16536  df-submnd 16582  df-grp 16672  df-minusg 16673  df-sbg 16674  df-subg 16813  df-cntz 16970  df-oppg 16996  df-lsm 17287  df-cmn 17431  df-abl 17432  df-mgp 17723  df-ur 17735  df-ring 17781  df-oppr 17850  df-dvdsr 17868  df-unit 17869  df-invr 17899  df-dvr 17910  df-drng 17976  df-lmod 18092  df-lss 18155  df-lsp 18194  df-lvec 18325  df-lsatoms 32511  df-lshyp 32512  df-lcv 32554  df-lfl 32593  df-lkr 32621  df-ldual 32659  df-oposet 32711  df-ol 32713  df-oml 32714  df-covers 32801  df-ats 32802  df-atl 32833  df-cvlat 32857  df-hlat 32886  df-llines 33032  df-lplanes 33033  df-lvols 33034  df-lines 33035  df-psubsp 33037  df-pmap 33038  df-padd 33330  df-lhyp 33522  df-laut 33523  df-ldil 33638  df-ltrn 33639  df-trl 33694  df-tgrp 34279  df-tendo 34291  df-edring 34293  df-dveca 34539  df-disoa 34566  df-dvech 34616  df-dib 34676  df-dic 34710  df-dih 34766  df-doch 34885  df-djh 34932  df-lcdual 35124  df-mapd 35162  df-hvmap 35294  df-hdmap1 35331  df-hdmap 35332  df-hgmap 35424
This theorem is referenced by:  hgmapvv  35466
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