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Theorem lclkrs 32034
Description: The set of functionals having closed kernels and majorizing the orthocomplement of a given subspace  R is a subspace of the dual space. TODO: This proof repeats large parts of the lclkr 32028 proof. Do we achieve overall shortening by breaking them out as subtheorems? Or make lclkr 32028 a special case of this? (Contributed by NM, 29-Jan-2015.)
Hypotheses
Ref Expression
lclkrs.h  |-  H  =  ( LHyp `  K
)
lclkrs.o  |-  ._|_  =  ( ( ocH `  K
) `  W )
lclkrs.u  |-  U  =  ( ( DVecH `  K
) `  W )
lclkrs.s  |-  S  =  ( LSubSp `  U )
lclkrs.f  |-  F  =  (LFnl `  U )
lclkrs.l  |-  L  =  (LKer `  U )
lclkrs.d  |-  D  =  (LDual `  U )
lclkrs.t  |-  T  =  ( LSubSp `  D )
lclkrs.c  |-  C  =  { f  e.  F  |  ( (  ._|_  `  (  ._|_  `  ( L `
 f ) ) )  =  ( L `
 f )  /\  (  ._|_  `  ( L `  f ) )  C_  R ) }
lclkrs.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
lclkrs.r  |-  ( ph  ->  R  e.  S )
Assertion
Ref Expression
lclkrs  |-  ( ph  ->  C  e.  T )
Distinct variable groups:    D, f    f, F    f, L    R, f    U, f    ._|_ , f
Allowed substitution hints:    ph( f)    C( f)    S( f)    T( f)    H( f)    K( f)    W( f)

Proof of Theorem lclkrs
Dummy variables  a 
b  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrab2 3396 . . . 4  |-  { f  e.  F  |  ( (  ._|_  `  (  ._|_  `  ( L `  f
) ) )  =  ( L `  f
)  /\  (  ._|_  `  ( L `  f
) )  C_  R
) }  C_  F
21a1i 11 . . 3  |-  ( ph  ->  { f  e.  F  |  ( (  ._|_  `  (  ._|_  `  ( L `
 f ) ) )  =  ( L `
 f )  /\  (  ._|_  `  ( L `  f ) )  C_  R ) }  C_  F )
3 lclkrs.c . . . 4  |-  C  =  { f  e.  F  |  ( (  ._|_  `  (  ._|_  `  ( L `
 f ) ) )  =  ( L `
 f )  /\  (  ._|_  `  ( L `  f ) )  C_  R ) }
43a1i 11 . . 3  |-  ( ph  ->  C  =  { f  e.  F  |  ( (  ._|_  `  (  ._|_  `  ( L `  f
) ) )  =  ( L `  f
)  /\  (  ._|_  `  ( L `  f
) )  C_  R
) } )
5 lclkrs.f . . . 4  |-  F  =  (LFnl `  U )
6 lclkrs.d . . . 4  |-  D  =  (LDual `  U )
7 eqid 2412 . . . 4  |-  ( Base `  D )  =  (
Base `  D )
8 lclkrs.h . . . . 5  |-  H  =  ( LHyp `  K
)
9 lclkrs.u . . . . 5  |-  U  =  ( ( DVecH `  K
) `  W )
10 lclkrs.k . . . . 5  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
118, 9, 10dvhlmod 31605 . . . 4  |-  ( ph  ->  U  e.  LMod )
125, 6, 7, 11ldualvbase 29621 . . 3  |-  ( ph  ->  ( Base `  D
)  =  F )
132, 4, 123sstr4d 3359 . 2  |-  ( ph  ->  C  C_  ( Base `  D ) )
14 eqid 2412 . . . . . 6  |-  (Scalar `  U )  =  (Scalar `  U )
15 eqid 2412 . . . . . 6  |-  ( 0g
`  (Scalar `  U )
)  =  ( 0g
`  (Scalar `  U )
)
16 eqid 2412 . . . . . 6  |-  ( Base `  U )  =  (
Base `  U )
1714, 15, 16, 5lfl0f 29564 . . . . 5  |-  ( U  e.  LMod  ->  ( (
Base `  U )  X.  { ( 0g `  (Scalar `  U ) ) } )  e.  F
)
1811, 17syl 16 . . . 4  |-  ( ph  ->  ( ( Base `  U
)  X.  { ( 0g `  (Scalar `  U ) ) } )  e.  F )
19 lclkrs.o . . . . . 6  |-  ._|_  =  ( ( ocH `  K
) `  W )
208, 9, 19, 16, 10dochoc1 31856 . . . . 5  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( Base `  U
) ) )  =  ( Base `  U
) )
21 eqidd 2413 . . . . . . . 8  |-  ( ph  ->  ( ( Base `  U
)  X.  { ( 0g `  (Scalar `  U ) ) } )  =  ( (
Base `  U )  X.  { ( 0g `  (Scalar `  U ) ) } ) )
22 lclkrs.l . . . . . . . . . 10  |-  L  =  (LKer `  U )
2314, 15, 16, 5, 22lkr0f 29589 . . . . . . . . 9  |-  ( ( U  e.  LMod  /\  (
( Base `  U )  X.  { ( 0g `  (Scalar `  U ) ) } )  e.  F
)  ->  ( ( L `  ( ( Base `  U )  X. 
{ ( 0g `  (Scalar `  U ) ) } ) )  =  ( Base `  U
)  <->  ( ( Base `  U )  X.  {
( 0g `  (Scalar `  U ) ) } )  =  ( (
Base `  U )  X.  { ( 0g `  (Scalar `  U ) ) } ) ) )
2411, 18, 23syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( ( L `  ( ( Base `  U
)  X.  { ( 0g `  (Scalar `  U ) ) } ) )  =  (
Base `  U )  <->  ( ( Base `  U
)  X.  { ( 0g `  (Scalar `  U ) ) } )  =  ( (
Base `  U )  X.  { ( 0g `  (Scalar `  U ) ) } ) ) )
2521, 24mpbird 224 . . . . . . 7  |-  ( ph  ->  ( L `  (
( Base `  U )  X.  { ( 0g `  (Scalar `  U ) ) } ) )  =  ( Base `  U
) )
2625fveq2d 5699 . . . . . 6  |-  ( ph  ->  (  ._|_  `  ( L `
 ( ( Base `  U )  X.  {
( 0g `  (Scalar `  U ) ) } ) ) )  =  (  ._|_  `  ( Base `  U ) ) )
2726fveq2d 5699 . . . . 5  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `  (
( Base `  U )  X.  { ( 0g `  (Scalar `  U ) ) } ) ) ) )  =  (  ._|_  `  (  ._|_  `  ( Base `  U ) ) ) )
2820, 27, 253eqtr4d 2454 . . . 4  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `  (
( Base `  U )  X.  { ( 0g `  (Scalar `  U ) ) } ) ) ) )  =  ( L `
 ( ( Base `  U )  X.  {
( 0g `  (Scalar `  U ) ) } ) ) )
29 eqid 2412 . . . . . . . 8  |-  ( 0g
`  U )  =  ( 0g `  U
)
308, 9, 19, 16, 29doch1 31854 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  ._|_  `  ( Base `  U ) )  =  { ( 0g `  U ) } )
3110, 30syl 16 . . . . . 6  |-  ( ph  ->  (  ._|_  `  ( Base `  U ) )  =  { ( 0g `  U ) } )
3226, 31eqtrd 2444 . . . . 5  |-  ( ph  ->  (  ._|_  `  ( L `
 ( ( Base `  U )  X.  {
( 0g `  (Scalar `  U ) ) } ) ) )  =  { ( 0g `  U ) } )
33 lclkrs.r . . . . . 6  |-  ( ph  ->  R  e.  S )
34 lclkrs.s . . . . . . 7  |-  S  =  ( LSubSp `  U )
3529, 34lss0ss 15988 . . . . . 6  |-  ( ( U  e.  LMod  /\  R  e.  S )  ->  { ( 0g `  U ) }  C_  R )
3611, 33, 35syl2anc 643 . . . . 5  |-  ( ph  ->  { ( 0g `  U ) }  C_  R )
3732, 36eqsstrd 3350 . . . 4  |-  ( ph  ->  (  ._|_  `  ( L `
 ( ( Base `  U )  X.  {
( 0g `  (Scalar `  U ) ) } ) ) )  C_  R )
383lcfls1lem 32029 . . . 4  |-  ( ( ( Base `  U
)  X.  { ( 0g `  (Scalar `  U ) ) } )  e.  C  <->  ( (
( Base `  U )  X.  { ( 0g `  (Scalar `  U ) ) } )  e.  F  /\  (  ._|_  `  (  ._|_  `  ( L `  ( ( Base `  U
)  X.  { ( 0g `  (Scalar `  U ) ) } ) ) ) )  =  ( L `  ( ( Base `  U
)  X.  { ( 0g `  (Scalar `  U ) ) } ) )  /\  (  ._|_  `  ( L `  ( ( Base `  U
)  X.  { ( 0g `  (Scalar `  U ) ) } ) ) )  C_  R ) )
3918, 28, 37, 38syl3anbrc 1138 . . 3  |-  ( ph  ->  ( ( Base `  U
)  X.  { ( 0g `  (Scalar `  U ) ) } )  e.  C )
40 ne0i 3602 . . 3  |-  ( ( ( Base `  U
)  X.  { ( 0g `  (Scalar `  U ) ) } )  e.  C  ->  C  =/=  (/) )
4139, 40syl 16 . 2  |-  ( ph  ->  C  =/=  (/) )
42 eqid 2412 . . . 4  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
43 eqid 2412 . . . 4  |-  ( .s
`  D )  =  ( .s `  D
)
4410adantr 452 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  D ) )  /\  a  e.  C  /\  b  e.  C )
)  ->  ( K  e.  HL  /\  W  e.  H ) )
4533adantr 452 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  D ) )  /\  a  e.  C  /\  b  e.  C )
)  ->  R  e.  S )
46 simpr3 965 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  D ) )  /\  a  e.  C  /\  b  e.  C )
)  ->  b  e.  C )
47 eqid 2412 . . . 4  |-  ( +g  `  D )  =  ( +g  `  D )
48 simpr2 964 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  D ) )  /\  a  e.  C  /\  b  e.  C )
)  ->  a  e.  C )
49 simpr1 963 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  D ) )  /\  a  e.  C  /\  b  e.  C )
)  ->  x  e.  ( Base `  (Scalar `  D
) ) )
50 eqid 2412 . . . . . . . 8  |-  (Scalar `  D )  =  (Scalar `  D )
51 eqid 2412 . . . . . . . 8  |-  ( Base `  (Scalar `  D )
)  =  ( Base `  (Scalar `  D )
)
5214, 42, 6, 50, 51, 11ldualsbase 29628 . . . . . . 7  |-  ( ph  ->  ( Base `  (Scalar `  D ) )  =  ( Base `  (Scalar `  U ) ) )
5352adantr 452 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  D ) )  /\  a  e.  C  /\  b  e.  C )
)  ->  ( Base `  (Scalar `  D )
)  =  ( Base `  (Scalar `  U )
) )
5449, 53eleqtrd 2488 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  D ) )  /\  a  e.  C  /\  b  e.  C )
)  ->  x  e.  ( Base `  (Scalar `  U
) ) )
558, 19, 9, 34, 5, 22, 6, 14, 42, 43, 3, 44, 45, 48, 54lclkrslem1 32032 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  D ) )  /\  a  e.  C  /\  b  e.  C )
)  ->  ( x
( .s `  D
) a )  e.  C )
568, 19, 9, 34, 5, 22, 6, 14, 42, 43, 3, 44, 45, 46, 47, 55lclkrslem2 32033 . . 3  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  D ) )  /\  a  e.  C  /\  b  e.  C )
)  ->  ( (
x ( .s `  D ) a ) ( +g  `  D
) b )  e.  C )
5756ralrimivvva 2767 . 2  |-  ( ph  ->  A. x  e.  (
Base `  (Scalar `  D
) ) A. a  e.  C  A. b  e.  C  ( (
x ( .s `  D ) a ) ( +g  `  D
) b )  e.  C )
58 lclkrs.t . . 3  |-  T  =  ( LSubSp `  D )
5950, 51, 7, 47, 43, 58islss 15974 . 2  |-  ( C  e.  T  <->  ( C  C_  ( Base `  D
)  /\  C  =/=  (/) 
/\  A. x  e.  (
Base `  (Scalar `  D
) ) A. a  e.  C  A. b  e.  C  ( (
x ( .s `  D ) a ) ( +g  `  D
) b )  e.  C ) )
6013, 41, 57, 59syl3anbrc 1138 1  |-  ( ph  ->  C  e.  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575   A.wral 2674   {crab 2678    C_ wss 3288   (/)c0 3596   {csn 3782    X. cxp 4843   ` cfv 5421  (class class class)co 6048   Basecbs 13432   +g cplusg 13492  Scalarcsca 13495   .scvsca 13496   0gc0g 13686   LModclmod 15913   LSubSpclss 15971  LFnlclfn 29552  LKerclk 29580  LDualcld 29618   HLchlt 29845   LHypclh 30478   DVecHcdvh 31573   ocHcoch 31842
This theorem is referenced by:  lclkrs2  32035  mapddlssN  32135
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-1st 6316  df-2nd 6317  df-tpos 6446  df-undef 6510  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-n0 10186  df-z 10247  df-uz 10453  df-fz 11008  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-mulr 13506  df-sca 13508  df-vsca 13509  df-0g 13690  df-mre 13774  df-mrc 13775  df-acs 13777  df-poset 14366  df-plt 14378  df-lub 14394  df-glb 14395  df-join 14396  df-meet 14397  df-p0 14431  df-p1 14432  df-lat 14438  df-clat 14500  df-mnd 14653  df-submnd 14702  df-grp 14775  df-minusg 14776  df-sbg 14777  df-subg 14904  df-cntz 15079  df-oppg 15105  df-lsm 15233  df-cmn 15377  df-abl 15378  df-mgp 15612  df-rng 15626  df-ur 15628  df-oppr 15691  df-dvdsr 15709  df-unit 15710  df-invr 15740  df-dvr 15751  df-drng 15800  df-lmod 15915  df-lss 15972  df-lsp 16011  df-lvec 16138  df-lsatoms 29471  df-lshyp 29472  df-lcv 29514  df-lfl 29553  df-lkr 29581  df-ldual 29619  df-oposet 29671  df-ol 29673  df-oml 29674  df-covers 29761  df-ats 29762  df-atl 29793  df-cvlat 29817  df-hlat 29846  df-llines 29992  df-lplanes 29993  df-lvols 29994  df-lines 29995  df-psubsp 29997  df-pmap 29998  df-padd 30290  df-lhyp 30482  df-laut 30483  df-ldil 30598  df-ltrn 30599  df-trl 30653  df-tgrp 31237  df-tendo 31249  df-edring 31251  df-dveca 31497  df-disoa 31524  df-dvech 31574  df-dib 31634  df-dic 31668  df-dih 31724  df-doch 31843  df-djh 31890
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