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Theorem hgmapvvlem3 36734
Description: Lemma for hgmapvv 36735. 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 2467 . . . 4  |-  ( 0g
`  U )  =  ( 0g `  U
)
6 hdmapglem6.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
7 eqid 2467 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
8 eqid 2467 . . . . . 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 35918 . . . . 5  |-  ( ph  ->  E  e.  ( V 
\  { ( 0g
`  U ) } ) )
1110eldifad 3488 . . . 4  |-  ( ph  ->  E  e.  V )
121, 2, 3, 4, 5, 6, 11dochsnnz 36256 . . 3  |-  ( ph  ->  ( O `  { E } )  =/=  {
( 0g `  U
) } )
1311snssd 4172 . . . . 5  |-  ( ph  ->  { E }  C_  V )
14 eqid 2467 . . . . . 6  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
151, 3, 4, 14, 2dochlss 36160 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  { E }  C_  V )  ->  ( O `  { E } )  e.  (
LSubSp `  U ) )
166, 13, 15syl2anc 661 . . . 4  |-  ( ph  ->  ( O `  { E } )  e.  (
LSubSp `  U ) )
175, 14lssne0 17392 . . . 4  |-  ( ( O `  { E } )  e.  (
LSubSp `  U )  -> 
( ( O `  { E } )  =/= 
{ ( 0g `  U ) }  <->  E. k  e.  ( O `  { E } ) k  =/=  ( 0g `  U
) ) )
1816, 17syl 16 . . 3  |-  ( ph  ->  ( ( O `  { E } )  =/= 
{ ( 0g `  U ) }  <->  E. k  e.  ( O `  { E } ) k  =/=  ( 0g `  U
) ) )
1912, 18mpbid 210 . 2  |-  ( ph  ->  E. k  e.  ( O `  { E } ) k  =/=  ( 0g `  U
) )
20 eqid 2467 . . . . 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 1017 . . . . 5  |-  ( (
ph  /\  k  e.  ( O `  { E } )  /\  k  =/=  ( 0g `  U
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
261, 3, 4, 2dochssv 36161 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  { E }  C_  V )  ->  ( O `  { E } )  C_  V
)
276, 13, 26syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( O `  { E } )  C_  V
)
2827sselda 3504 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( O `  { E } ) )  -> 
k  e.  V )
29283adant3 1016 . . . . . 6  |-  ( (
ph  /\  k  e.  ( O `  { E } )  /\  k  =/=  ( 0g `  U
) )  ->  k  e.  V )
30 simp3 998 . . . . . 6  |-  ( (
ph  /\  k  e.  ( O `  { E } )  /\  k  =/=  ( 0g `  U
) )  ->  k  =/=  ( 0g `  U
) )
31 eldifsn 4152 . . . . . 6  |-  ( k  e.  ( V  \  { ( 0g `  U ) } )  <-> 
( k  e.  V  /\  k  =/=  ( 0g `  U ) ) )
3229, 30, 31sylanbrc 664 . . . . 5  |-  ( (
ph  /\  k  e.  ( O `  { E } )  /\  k  =/=  ( 0g `  U
) )  ->  k  e.  ( V  \  {
( 0g `  U
) } ) )
33 eqid 2467 . . . . 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 36725 . . . 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 999 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( O `  { E } )  /\  k  =/=  ( 0g `  U
) )  /\  (
( S `  k
) `  ( ( N `  ( ( S `  k ) `  k ) ) ( .s `  U ) k ) )  =  .1.  )  ->  ph )
4140, 6syl 16 . . . . 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 16 . . . . 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 35916 . . . . . . 7  |-  ( ph  ->  U  e.  LMod )
4540, 44syl 16 . . . . . 6  |-  ( ( ( ph  /\  k  e.  ( O `  { E } )  /\  k  =/=  ( 0g `  U
) )  /\  (
( S `  k
) `  ( ( N `  ( ( S `  k ) `  k ) ) ( .s `  U ) k ) )  =  .1.  )  ->  U  e.  LMod )
4640, 16syl 16 . . . . . 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 35915 . . . . . . . . 9  |-  ( ph  ->  U  e.  LVec )
4821lvecdrng 17546 . . . . . . . . 9  |-  ( U  e.  LVec  ->  R  e.  DivRing )
4947, 48syl 16 . . . . . . . 8  |-  ( ph  ->  R  e.  DivRing )
5040, 49syl 16 . . . . . . 7  |-  ( ( ( ph  /\  k  e.  ( O `  { E } )  /\  k  =/=  ( 0g `  U
) )  /\  (
( S `  k
) `  ( ( N `  ( ( S `  k ) `  k ) ) ( .s `  U ) k ) )  =  .1.  )  ->  R  e.  DivRing )
5129adantr 465 . . . . . . . 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 36714 . . . . . . 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 465 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( O `  { E } ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
541, 3, 4, 5, 21, 38, 24, 53, 28hdmapip0 36724 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( O `  { E } ) )  -> 
( ( ( S `
 k ) `  k )  =  .0.  <->  k  =  ( 0g `  U ) ) )
5554necon3bid 2725 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( O `  { E } ) )  -> 
( ( ( S `
 k ) `  k )  =/=  .0.  <->  k  =/=  ( 0g `  U
) ) )
5655biimp3ar 1329 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( O `  { E } )  /\  k  =/=  ( 0g `  U
) )  ->  (
( S `  k
) `  k )  =/=  .0.  )
5756adantr 465 . . . . . . 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 17208 . . . . . . 7  |-  ( ( R  e.  DivRing  /\  (
( S `  k
) `  k )  e.  B  /\  (
( S `  k
) `  k )  =/=  .0.  )  ->  ( N `  ( ( S `  k ) `  k ) )  e.  B )
5950, 52, 57, 58syl3anc 1228 . . . . . 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 1000 . . . . . 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 17396 . . . . . 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 1229 . . . . 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 461 . . . . 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 36733 . . . 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 668 . . 3  |-  ( (
ph  /\  k  e.  ( O `  { E } )  /\  k  =/=  ( 0g `  U
) )  ->  ( G `  ( G `  X ) )  =  X )
6665rexlimdv3a 2957 . 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 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815    \ cdif 3473    C_ wss 3476   {csn 4027   <.cop 4033    _I cid 4790    |` cres 5001   ` cfv 5587  (class class class)co 6283   Basecbs 14489   .rcmulr 14555  Scalarcsca 14557   .scvsca 14558   0gc0g 14694   1rcur 16952   invrcinvr 17116   DivRingcdr 17191   LModclmod 17307   LSubSpclss 17373   LVecclvec 17543   HLchlt 34156   LHypclh 34789   LTrncltrn 34906   DVecHcdvh 35884   ocHcoch 36153  HDMapchdma 36599  HGMapchg 36692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-riotaBAD 33765
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-ot 4036  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-of 6523  df-om 6680  df-1st 6784  df-2nd 6785  df-tpos 6955  df-undef 7002  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-nn 10536  df-2 10593  df-3 10594  df-4 10595  df-5 10596  df-6 10597  df-n0 10795  df-z 10864  df-uz 11082  df-fz 11672  df-struct 14491  df-ndx 14492  df-slot 14493  df-base 14494  df-sets 14495  df-ress 14496  df-plusg 14567  df-mulr 14568  df-sca 14570  df-vsca 14571  df-0g 14696  df-mre 14840  df-mrc 14841  df-acs 14843  df-poset 15432  df-plt 15444  df-lub 15460  df-glb 15461  df-join 15462  df-meet 15463  df-p0 15525  df-p1 15526  df-lat 15532  df-clat 15594  df-mnd 15731  df-submnd 15784  df-grp 15864  df-minusg 15865  df-sbg 15866  df-subg 16000  df-cntz 16157  df-oppg 16183  df-lsm 16459  df-cmn 16603  df-abl 16604  df-mgp 16941  df-ur 16953  df-rng 16997  df-oppr 17068  df-dvdsr 17086  df-unit 17087  df-invr 17117  df-dvr 17128  df-drng 17193  df-lmod 17309  df-lss 17374  df-lsp 17413  df-lvec 17544  df-lsatoms 33782  df-lshyp 33783  df-lcv 33825  df-lfl 33864  df-lkr 33892  df-ldual 33930  df-oposet 33982  df-ol 33984  df-oml 33985  df-covers 34072  df-ats 34073  df-atl 34104  df-cvlat 34128  df-hlat 34157  df-llines 34303  df-lplanes 34304  df-lvols 34305  df-lines 34306  df-psubsp 34308  df-pmap 34309  df-padd 34601  df-lhyp 34793  df-laut 34794  df-ldil 34909  df-ltrn 34910  df-trl 34964  df-tgrp 35548  df-tendo 35560  df-edring 35562  df-dveca 35808  df-disoa 35835  df-dvech 35885  df-dib 35945  df-dic 35979  df-dih 36035  df-doch 36154  df-djh 36201  df-lcdual 36393  df-mapd 36431  df-hvmap 36563  df-hdmap1 36600  df-hdmap 36601  df-hgmap 36693
This theorem is referenced by:  hgmapvv  36735
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