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Theorem lcfl6lem 32223
Description: Lemma for lcfl6 32225. A functional  G (whose kernel is closed by dochsnkr 32197) is comletely determined by a vector  X in the orthocomplement in its kernel at which the functional value is 1. Note that the  \  {  .0.  } in the  X hypothesis is redundant by the last hypothesis but allows easier use of other theorems. (Contributed by NM, 3-Jan-2015.)
Hypotheses
Ref Expression
lcfl6lem.h  |-  H  =  ( LHyp `  K
)
lcfl6lem.o  |-  ._|_  =  ( ( ocH `  K
) `  W )
lcfl6lem.u  |-  U  =  ( ( DVecH `  K
) `  W )
lcfl6lem.v  |-  V  =  ( Base `  U
)
lcfl6lem.a  |-  .+  =  ( +g  `  U )
lcfl6lem.t  |-  .x.  =  ( .s `  U )
lcfl6lem.s  |-  S  =  (Scalar `  U )
lcfl6lem.i  |-  .1.  =  ( 1r `  S )
lcfl6lem.r  |-  R  =  ( Base `  S
)
lcfl6lem.z  |-  .0.  =  ( 0g `  U )
lcfl6lem.f  |-  F  =  (LFnl `  U )
lcfl6lem.l  |-  L  =  (LKer `  U )
lcfl6lem.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
lcfl6lem.g  |-  ( ph  ->  G  e.  F )
lcfl6lem.x  |-  ( ph  ->  X  e.  ( ( 
._|_  `  ( L `  G ) )  \  {  .0.  } ) )
lcfl6lem.y  |-  ( ph  ->  ( G `  X
)  =  .1.  )
Assertion
Ref Expression
lcfl6lem  |-  ( ph  ->  G  =  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { X } ) v  =  ( w  .+  (
k  .x.  X )
) ) ) )
Distinct variable groups:    v, k, w,  .+    .1. , k, w    ._|_ , k, v, w    R, k, v    S, k    .x. , k, v, w   
v, V    k, X, v, w    w,  .0.
Allowed substitution hints:    ph( w, v, k)    R( w)    S( w, v)    U( w, v, k)    .1. ( v)    F( w, v, k)    G( w, v, k)    H( w, v, k)    K( w, v, k)    L( w, v, k)    V( w, k)    W( w, v, k)    .0. ( v, k)

Proof of Theorem lcfl6lem
StepHypRef Expression
1 lcfl6lem.v . 2  |-  V  =  ( Base `  U
)
2 lcfl6lem.s . 2  |-  S  =  (Scalar `  U )
3 lcfl6lem.r . 2  |-  R  =  ( Base `  S
)
4 eqid 2435 . 2  |-  ( 0g
`  S )  =  ( 0g `  S
)
5 lcfl6lem.f . 2  |-  F  =  (LFnl `  U )
6 lcfl6lem.l . 2  |-  L  =  (LKer `  U )
7 lcfl6lem.h . . 3  |-  H  =  ( LHyp `  K
)
8 lcfl6lem.u . . 3  |-  U  =  ( ( DVecH `  K
) `  W )
9 lcfl6lem.k . . 3  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
107, 8, 9dvhlvec 31834 . 2  |-  ( ph  ->  U  e.  LVec )
117, 8, 9dvhlmod 31835 . . . . 5  |-  ( ph  ->  U  e.  LMod )
12 lcfl6lem.g . . . . 5  |-  ( ph  ->  G  e.  F )
131, 5, 6, 11, 12lkrssv 29821 . . . 4  |-  ( ph  ->  ( L `  G
)  C_  V )
14 lcfl6lem.o . . . . 5  |-  ._|_  =  ( ( ocH `  K
) `  W )
157, 8, 1, 14dochssv 32080 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( L `  G )  C_  V
)  ->  (  ._|_  `  ( L `  G
) )  C_  V
)
169, 13, 15syl2anc 643 . . 3  |-  ( ph  ->  (  ._|_  `  ( L `
 G ) ) 
C_  V )
17 lcfl6lem.x . . . 4  |-  ( ph  ->  X  e.  ( ( 
._|_  `  ( L `  G ) )  \  {  .0.  } ) )
1817eldifad 3324 . . 3  |-  ( ph  ->  X  e.  (  ._|_  `  ( L `  G
) ) )
1916, 18sseldd 3341 . 2  |-  ( ph  ->  X  e.  V )
20 lcfl6lem.z . . 3  |-  .0.  =  ( 0g `  U )
21 lcfl6lem.a . . 3  |-  .+  =  ( +g  `  U )
22 lcfl6lem.t . . 3  |-  .x.  =  ( .s `  U )
23 eqid 2435 . . 3  |-  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { X } ) v  =  ( w  .+  (
k  .x.  X )
) ) )  =  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { X }
) v  =  ( w  .+  ( k 
.x.  X ) ) ) )
24 eldifsni 3920 . . . . 5  |-  ( X  e.  ( (  ._|_  `  ( L `  G
) )  \  {  .0.  } )  ->  X  =/=  .0.  )
2517, 24syl 16 . . . 4  |-  ( ph  ->  X  =/=  .0.  )
26 eldifsn 3919 . . . 4  |-  ( X  e.  ( V  \  {  .0.  } )  <->  ( X  e.  V  /\  X  =/= 
.0.  ) )
2719, 25, 26sylanbrc 646 . . 3  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
287, 14, 8, 1, 20, 21, 22, 5, 2, 3, 23, 9, 27dochflcl 32200 . 2  |-  ( ph  ->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { X }
) v  =  ( w  .+  ( k 
.x.  X ) ) ) )  e.  F
)
297, 14, 8, 1, 20, 5, 6, 9, 12, 17dochsnkr 32197 . . 3  |-  ( ph  ->  ( L `  G
)  =  (  ._|_  `  { X } ) )
307, 14, 8, 1, 20, 21, 22, 6, 2, 3, 23, 9, 27dochsnkr2 32198 . . 3  |-  ( ph  ->  ( L `  (
v  e.  V  |->  (
iota_ k  e.  R E. w  e.  (  ._|_  `  { X }
) v  =  ( w  .+  ( k 
.x.  X ) ) ) ) )  =  (  ._|_  `  { X } ) )
3129, 30eqtr4d 2470 . 2  |-  ( ph  ->  ( L `  G
)  =  ( L `
 ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { X } ) v  =  ( w  .+  (
k  .x.  X )
) ) ) ) )
32 lcfl6lem.y . . 3  |-  ( ph  ->  ( G `  X
)  =  .1.  )
33 lcfl6lem.i . . . 4  |-  .1.  =  ( 1r `  S )
347, 14, 8, 1, 21, 22, 20, 2, 3, 33, 9, 27, 23dochfl1 32201 . . 3  |-  ( ph  ->  ( ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { X } ) v  =  ( w  .+  (
k  .x.  X )
) ) ) `  X )  =  .1.  )
3532, 34eqtr4d 2470 . 2  |-  ( ph  ->  ( G `  X
)  =  ( ( v  e.  V  |->  (
iota_ k  e.  R E. w  e.  (  ._|_  `  { X }
) v  =  ( w  .+  ( k 
.x.  X ) ) ) ) `  X
) )
367, 14, 8, 1, 2, 4, 20, 5, 6, 9, 12, 17dochfln0 32202 . 2  |-  ( ph  ->  ( G `  X
)  =/=  ( 0g
`  S ) )
371, 2, 3, 4, 5, 6, 10, 19, 12, 28, 31, 35, 36eqlkr3 29826 1  |-  ( ph  ->  G  =  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { X } ) v  =  ( w  .+  (
k  .x.  X )
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698    \ cdif 3309    C_ wss 3312   {csn 3806    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   iota_crio 6534   Basecbs 13461   +g cplusg 13521  Scalarcsca 13524   .scvsca 13525   0gc0g 13715   1rcur 15654  LFnlclfn 29782  LKerclk 29810   HLchlt 30075   LHypclh 30708   DVecHcdvh 31803   ocHcoch 32072
This theorem is referenced by:  lcfl6  32225
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-fal 1329  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-tpos 6471  df-undef 6535  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-sca 13537  df-vsca 13538  df-0g 13719  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-mnd 14682  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-subg 14933  df-cntz 15108  df-lsm 15262  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-ur 15657  df-oppr 15720  df-dvdsr 15738  df-unit 15739  df-invr 15769  df-dvr 15780  df-drng 15829  df-lmod 15944  df-lss 16001  df-lsp 16040  df-lvec 16167  df-lsatoms 29701  df-lshyp 29702  df-lfl 29783  df-lkr 29811  df-oposet 29901  df-ol 29903  df-oml 29904  df-covers 29991  df-ats 29992  df-atl 30023  df-cvlat 30047  df-hlat 30076  df-llines 30222  df-lplanes 30223  df-lvols 30224  df-lines 30225  df-psubsp 30227  df-pmap 30228  df-padd 30520  df-lhyp 30712  df-laut 30713  df-ldil 30828  df-ltrn 30829  df-trl 30883  df-tgrp 31467  df-tendo 31479  df-edring 31481  df-dveca 31727  df-disoa 31754  df-dvech 31804  df-dib 31864  df-dic 31898  df-dih 31954  df-doch 32073  df-djh 32120
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