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Theorem lcfl6lem 32310
Description: Lemma for lcfl6 32312. A functional  G (whose kernel is closed by dochsnkr 32284) 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 2296 . 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 31921 . 2  |-  ( ph  ->  U  e.  LVec )
117, 8, 9dvhlmod 31922 . . . . 5  |-  ( ph  ->  U  e.  LMod )
12 lcfl6lem.g . . . . 5  |-  ( ph  ->  G  e.  F )
131, 5, 6, 11, 12lkrssv 29908 . . . 4  |-  ( ph  ->  ( L `  G
)  C_  V )
14 lcfl6lem.o . . . . 5  |-  ._|_  =  ( ( ocH `  K
) `  W )
157, 8, 1, 14dochssv 32167 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( L `  G )  C_  V
)  ->  (  ._|_  `  ( L `  G
) )  C_  V
)
169, 13, 15syl2anc 642 . . 3  |-  ( ph  ->  (  ._|_  `  ( L `
 G ) ) 
C_  V )
17 lcfl6lem.x . . . 4  |-  ( ph  ->  X  e.  ( ( 
._|_  `  ( L `  G ) )  \  {  .0.  } ) )
18 eldifi 3311 . . . 4  |-  ( X  e.  ( (  ._|_  `  ( L `  G
) )  \  {  .0.  } )  ->  X  e.  (  ._|_  `  ( L `  G )
) )
1917, 18syl 15 . . 3  |-  ( ph  ->  X  e.  (  ._|_  `  ( L `  G
) ) )
2016, 19sseldd 3194 . 2  |-  ( ph  ->  X  e.  V )
21 lcfl6lem.z . . 3  |-  .0.  =  ( 0g `  U )
22 lcfl6lem.a . . 3  |-  .+  =  ( +g  `  U )
23 lcfl6lem.t . . 3  |-  .x.  =  ( .s `  U )
24 eqid 2296 . . 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 ) ) ) )
25 eldifsni 3763 . . . . 5  |-  ( X  e.  ( (  ._|_  `  ( L `  G
) )  \  {  .0.  } )  ->  X  =/=  .0.  )
2617, 25syl 15 . . . 4  |-  ( ph  ->  X  =/=  .0.  )
27 eldifsn 3762 . . . 4  |-  ( X  e.  ( V  \  {  .0.  } )  <->  ( X  e.  V  /\  X  =/= 
.0.  ) )
2820, 26, 27sylanbrc 645 . . 3  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
297, 14, 8, 1, 21, 22, 23, 5, 2, 3, 24, 9, 28dochflcl 32287 . 2  |-  ( ph  ->  ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { X }
) v  =  ( w  .+  ( k 
.x.  X ) ) ) )  e.  F
)
307, 14, 8, 1, 21, 5, 6, 9, 12, 17dochsnkr 32284 . . 3  |-  ( ph  ->  ( L `  G
)  =  (  ._|_  `  { X } ) )
317, 14, 8, 1, 21, 22, 23, 6, 2, 3, 24, 9, 28dochsnkr2 32285 . . 3  |-  ( ph  ->  ( L `  (
v  e.  V  |->  (
iota_ k  e.  R E. w  e.  (  ._|_  `  { X }
) v  =  ( w  .+  ( k 
.x.  X ) ) ) ) )  =  (  ._|_  `  { X } ) )
3230, 31eqtr4d 2331 . 2  |-  ( ph  ->  ( L `  G
)  =  ( L `
 ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { X } ) v  =  ( w  .+  (
k  .x.  X )
) ) ) ) )
33 lcfl6lem.y . . 3  |-  ( ph  ->  ( G `  X
)  =  .1.  )
34 lcfl6lem.i . . . 4  |-  .1.  =  ( 1r `  S )
357, 14, 8, 1, 22, 23, 21, 2, 3, 34, 9, 28, 24dochfl1 32288 . . 3  |-  ( ph  ->  ( ( v  e.  V  |->  ( iota_ k  e.  R E. w  e.  (  ._|_  `  { X } ) v  =  ( w  .+  (
k  .x.  X )
) ) ) `  X )  =  .1.  )
3633, 35eqtr4d 2331 . 2  |-  ( ph  ->  ( G `  X
)  =  ( ( v  e.  V  |->  (
iota_ k  e.  R E. w  e.  (  ._|_  `  { X }
) v  =  ( w  .+  ( k 
.x.  X ) ) ) ) `  X
) )
377, 14, 8, 1, 2, 4, 21, 5, 6, 9, 12, 17dochfln0 32289 . 2  |-  ( ph  ->  ( G `  X
)  =/=  ( 0g
`  S ) )
381, 2, 3, 4, 5, 6, 10, 20, 12, 29, 32, 36, 37eqlkr3 29913 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 358    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557    \ cdif 3162    C_ wss 3165   {csn 3653    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   iota_crio 6313   Basecbs 13164   +g cplusg 13224  Scalarcsca 13227   .scvsca 13228   0gc0g 13416   1rcur 15355  LFnlclfn 29869  LKerclk 29897   HLchlt 30162   LHypclh 30795   DVecHcdvh 31890   ocHcoch 32159
This theorem is referenced by:  lcfl6  32312
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-undef 6314  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-0g 13420  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-cntz 14809  df-lsm 14963  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-dvr 15481  df-drng 15530  df-lmod 15645  df-lss 15706  df-lsp 15745  df-lvec 15872  df-lsatoms 29788  df-lshyp 29789  df-lfl 29870  df-lkr 29898  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310  df-lvols 30311  df-lines 30312  df-psubsp 30314  df-pmap 30315  df-padd 30607  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916  df-trl 30970  df-tgrp 31554  df-tendo 31566  df-edring 31568  df-dveca 31814  df-disoa 31841  df-dvech 31891  df-dib 31951  df-dic 31985  df-dih 32041  df-doch 32160  df-djh 32207
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