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Theorem lcfl6lem 31613
Description: Lemma for lcfl6 31615. A functional  G (whose kernel is closed by dochsnkr 31587) 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 2387 . 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 31224 . 2  |-  ( ph  ->  U  e.  LVec )
117, 8, 9dvhlmod 31225 . . . . 5  |-  ( ph  ->  U  e.  LMod )
12 lcfl6lem.g . . . . 5  |-  ( ph  ->  G  e.  F )
131, 5, 6, 11, 12lkrssv 29211 . . . 4  |-  ( ph  ->  ( L `  G
)  C_  V )
14 lcfl6lem.o . . . . 5  |-  ._|_  =  ( ( ocH `  K
) `  W )
157, 8, 1, 14dochssv 31470 . . . 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 3275 . . 3  |-  ( ph  ->  X  e.  (  ._|_  `  ( L `  G
) ) )
1916, 18sseldd 3292 . 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 2387 . . 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 3871 . . . . 5  |-  ( X  e.  ( (  ._|_  `  ( L `  G
) )  \  {  .0.  } )  ->  X  =/=  .0.  )
2517, 24syl 16 . . . 4  |-  ( ph  ->  X  =/=  .0.  )
26 eldifsn 3870 . . . 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 31590 . 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 31587 . . 3  |-  ( ph  ->  ( L `  G
)  =  (  ._|_  `  { X } ) )
307, 14, 8, 1, 20, 21, 22, 6, 2, 3, 23, 9, 27dochsnkr2 31588 . . 3  |-  ( ph  ->  ( L `  (
v  e.  V  |->  (
iota_ k  e.  R E. w  e.  (  ._|_  `  { X }
) v  =  ( w  .+  ( k 
.x.  X ) ) ) ) )  =  (  ._|_  `  { X } ) )
3129, 30eqtr4d 2422 . 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 31591 . . 3  |-  ( ph  ->  ( ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { X } ) v  =  ( w  .+  (
k  .x.  X )
) ) ) `  X )  =  .1.  )
3532, 34eqtr4d 2422 . 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 31592 . 2  |-  ( ph  ->  ( G `  X
)  =/=  ( 0g
`  S ) )
371, 2, 3, 4, 5, 6, 10, 19, 12, 28, 31, 35, 36eqlkr3 29216 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 1649    e. wcel 1717    =/= wne 2550   E.wrex 2650    \ cdif 3260    C_ wss 3263   {csn 3757    e. cmpt 4207   ` cfv 5394  (class class class)co 6020   iota_crio 6478   Basecbs 13396   +g cplusg 13456  Scalarcsca 13459   .scvsca 13460   0gc0g 13650   1rcur 15589  LFnlclfn 29172  LKerclk 29200   HLchlt 29465   LHypclh 30098   DVecHcdvh 31193   ocHcoch 31462
This theorem is referenced by:  lcfl6  31615
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 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-tpos 6415  df-undef 6479  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-n0 10154  df-z 10215  df-uz 10421  df-fz 10976  df-struct 13398  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-mulr 13470  df-sca 13472  df-vsca 13473  df-0g 13654  df-poset 14330  df-plt 14342  df-lub 14358  df-glb 14359  df-join 14360  df-meet 14361  df-p0 14395  df-p1 14396  df-lat 14402  df-clat 14464  df-mnd 14617  df-submnd 14666  df-grp 14739  df-minusg 14740  df-sbg 14741  df-subg 14868  df-cntz 15043  df-lsm 15197  df-cmn 15341  df-abl 15342  df-mgp 15576  df-rng 15590  df-ur 15592  df-oppr 15655  df-dvdsr 15673  df-unit 15674  df-invr 15704  df-dvr 15715  df-drng 15764  df-lmod 15879  df-lss 15936  df-lsp 15975  df-lvec 16102  df-lsatoms 29091  df-lshyp 29092  df-lfl 29173  df-lkr 29201  df-oposet 29291  df-ol 29293  df-oml 29294  df-covers 29381  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466  df-llines 29612  df-lplanes 29613  df-lvols 29614  df-lines 29615  df-psubsp 29617  df-pmap 29618  df-padd 29910  df-lhyp 30102  df-laut 30103  df-ldil 30218  df-ltrn 30219  df-trl 30273  df-tgrp 30857  df-tendo 30869  df-edring 30871  df-dveca 31117  df-disoa 31144  df-dvech 31194  df-dib 31254  df-dic 31288  df-dih 31344  df-doch 31463  df-djh 31510
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