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Theorem lclkrs 35151
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 35145 proof. Do we achieve overall shortening by breaking them out as subtheorems? Or make lclkr 35145 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 3525 . . . 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 2461 . . . 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 34722 . . . 4  |-  ( ph  ->  U  e.  LMod )
125, 6, 7, 11ldualvbase 32736 . . 3  |-  ( ph  ->  ( Base `  D
)  =  F )
132, 4, 123sstr4d 3486 . 2  |-  ( ph  ->  C  C_  ( Base `  D ) )
14 eqid 2461 . . . . . 6  |-  (Scalar `  U )  =  (Scalar `  U )
15 eqid 2461 . . . . . 6  |-  ( 0g
`  (Scalar `  U )
)  =  ( 0g
`  (Scalar `  U )
)
16 eqid 2461 . . . . . 6  |-  ( Base `  U )  =  (
Base `  U )
1714, 15, 16, 5lfl0f 32679 . . . . 5  |-  ( U  e.  LMod  ->  ( (
Base `  U )  X.  { ( 0g `  (Scalar `  U ) ) } )  e.  F
)
1811, 17syl 17 . . . 4  |-  ( ph  ->  ( ( Base `  U
)  X.  { ( 0g `  (Scalar `  U ) ) } )  e.  F )
19 lclkrs.o . . . . . 6  |-  ._|_  =  ( ( ocH `  K
) `  W )
208, 9, 19, 16, 10dochoc1 34973 . . . . 5  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( Base `  U
) ) )  =  ( Base `  U
) )
21 eqidd 2462 . . . . . . . 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 32704 . . . . . . . . 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 671 . . . . . . . 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 240 . . . . . . 7  |-  ( ph  ->  ( L `  (
( Base `  U )  X.  { ( 0g `  (Scalar `  U ) ) } ) )  =  ( Base `  U
) )
2625fveq2d 5891 . . . . . 6  |-  ( ph  ->  (  ._|_  `  ( L `
 ( ( Base `  U )  X.  {
( 0g `  (Scalar `  U ) ) } ) ) )  =  (  ._|_  `  ( Base `  U ) ) )
2726fveq2d 5891 . . . . 5  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `  (
( Base `  U )  X.  { ( 0g `  (Scalar `  U ) ) } ) ) ) )  =  (  ._|_  `  (  ._|_  `  ( Base `  U ) ) ) )
2820, 27, 253eqtr4d 2505 . . . 4  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `  (
( Base `  U )  X.  { ( 0g `  (Scalar `  U ) ) } ) ) ) )  =  ( L `
 ( ( Base `  U )  X.  {
( 0g `  (Scalar `  U ) ) } ) ) )
29 eqid 2461 . . . . . . . 8  |-  ( 0g
`  U )  =  ( 0g `  U
)
308, 9, 19, 16, 29doch1 34971 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  ._|_  `  ( Base `  U ) )  =  { ( 0g `  U ) } )
3110, 30syl 17 . . . . . 6  |-  ( ph  ->  (  ._|_  `  ( Base `  U ) )  =  { ( 0g `  U ) } )
3226, 31eqtrd 2495 . . . . 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 18220 . . . . . 6  |-  ( ( U  e.  LMod  /\  R  e.  S )  ->  { ( 0g `  U ) }  C_  R )
3611, 33, 35syl2anc 671 . . . . 5  |-  ( ph  ->  { ( 0g `  U ) }  C_  R )
3732, 36eqsstrd 3477 . . . 4  |-  ( ph  ->  (  ._|_  `  ( L `
 ( ( Base `  U )  X.  {
( 0g `  (Scalar `  U ) ) } ) ) )  C_  R )
383lcfls1lem 35146 . . . 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 1198 . . 3  |-  ( ph  ->  ( ( Base `  U
)  X.  { ( 0g `  (Scalar `  U ) ) } )  e.  C )
40 ne0i 3748 . . 3  |-  ( ( ( Base `  U
)  X.  { ( 0g `  (Scalar `  U ) ) } )  e.  C  ->  C  =/=  (/) )
4139, 40syl 17 . 2  |-  ( ph  ->  C  =/=  (/) )
42 eqid 2461 . . . 4  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
43 eqid 2461 . . . 4  |-  ( .s
`  D )  =  ( .s `  D
)
4410adantr 471 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  D ) )  /\  a  e.  C  /\  b  e.  C )
)  ->  ( K  e.  HL  /\  W  e.  H ) )
4533adantr 471 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  D ) )  /\  a  e.  C  /\  b  e.  C )
)  ->  R  e.  S )
46 simpr3 1022 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  D ) )  /\  a  e.  C  /\  b  e.  C )
)  ->  b  e.  C )
47 eqid 2461 . . . 4  |-  ( +g  `  D )  =  ( +g  `  D )
48 simpr2 1021 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  D ) )  /\  a  e.  C  /\  b  e.  C )
)  ->  a  e.  C )
49 simpr1 1020 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  D ) )  /\  a  e.  C  /\  b  e.  C )
)  ->  x  e.  ( Base `  (Scalar `  D
) ) )
50 eqid 2461 . . . . . . . 8  |-  (Scalar `  D )  =  (Scalar `  D )
51 eqid 2461 . . . . . . . 8  |-  ( Base `  (Scalar `  D )
)  =  ( Base `  (Scalar `  D )
)
5214, 42, 6, 50, 51, 11ldualsbase 32743 . . . . . . 7  |-  ( ph  ->  ( Base `  (Scalar `  D ) )  =  ( Base `  (Scalar `  U ) ) )
5352adantr 471 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  D ) )  /\  a  e.  C  /\  b  e.  C )
)  ->  ( Base `  (Scalar `  D )
)  =  ( Base `  (Scalar `  U )
) )
5449, 53eleqtrd 2541 . . . . 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 35149 . . . 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 35150 . . 3  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  D ) )  /\  a  e.  C  /\  b  e.  C )
)  ->  ( (
x ( .s `  D ) a ) ( +g  `  D
) b )  e.  C )
5756ralrimivvva 2821 . 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 18206 . 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 1198 1  |-  ( ph  ->  C  e.  T )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1454    e. wcel 1897    =/= wne 2632   A.wral 2748   {crab 2752    C_ wss 3415   (/)c0 3742   {csn 3979    X. cxp 4850   ` cfv 5600  (class class class)co 6314   Basecbs 15169   +g cplusg 15238  Scalarcsca 15241   .scvsca 15242   0gc0g 15386   LModclmod 18139   LSubSpclss 18203  LFnlclfn 32667  LKerclk 32695  LDualcld 32733   HLchlt 32960   LHypclh 33593   DVecHcdvh 34690   ocHcoch 34959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641  ax-riotaBAD 32569
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-fal 1460  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-iin 4294  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-of 6557  df-om 6719  df-1st 6819  df-2nd 6820  df-tpos 6998  df-undef 7045  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-oadd 7211  df-er 7388  df-map 7499  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-nn 10637  df-2 10695  df-3 10696  df-4 10697  df-5 10698  df-6 10699  df-n0 10898  df-z 10966  df-uz 11188  df-fz 11813  df-struct 15171  df-ndx 15172  df-slot 15173  df-base 15174  df-sets 15175  df-ress 15176  df-plusg 15251  df-mulr 15252  df-sca 15254  df-vsca 15255  df-0g 15388  df-mre 15540  df-mrc 15541  df-acs 15543  df-preset 16221  df-poset 16239  df-plt 16252  df-lub 16268  df-glb 16269  df-join 16270  df-meet 16271  df-p0 16333  df-p1 16334  df-lat 16340  df-clat 16402  df-mgm 16536  df-sgrp 16575  df-mnd 16585  df-submnd 16631  df-grp 16721  df-minusg 16722  df-sbg 16723  df-subg 16862  df-cntz 17019  df-oppg 17045  df-lsm 17336  df-cmn 17480  df-abl 17481  df-mgp 17772  df-ur 17784  df-ring 17830  df-oppr 17899  df-dvdsr 17917  df-unit 17918  df-invr 17948  df-dvr 17959  df-drng 18025  df-lmod 18141  df-lss 18204  df-lsp 18243  df-lvec 18374  df-lsatoms 32586  df-lshyp 32587  df-lcv 32629  df-lfl 32668  df-lkr 32696  df-ldual 32734  df-oposet 32786  df-ol 32788  df-oml 32789  df-covers 32876  df-ats 32877  df-atl 32908  df-cvlat 32932  df-hlat 32961  df-llines 33107  df-lplanes 33108  df-lvols 33109  df-lines 33110  df-psubsp 33112  df-pmap 33113  df-padd 33405  df-lhyp 33597  df-laut 33598  df-ldil 33713  df-ltrn 33714  df-trl 33769  df-tgrp 34354  df-tendo 34366  df-edring 34368  df-dveca 34614  df-disoa 34641  df-dvech 34691  df-dib 34751  df-dic 34785  df-dih 34841  df-doch 34960  df-djh 35007
This theorem is referenced by:  lclkrs2  35152  mapddlssN  35252
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