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Theorem lclkrs 31656
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 31650 proof. Do we achieve overall shortening by breaking them out as subtheorems? Or make lclkr 31650 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 3373 . . . 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 2389 . . . 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 31227 . . . 4  |-  ( ph  ->  U  e.  LMod )
125, 6, 7, 11ldualvbase 29243 . . 3  |-  ( ph  ->  ( Base `  D
)  =  F )
132, 4, 123sstr4d 3336 . 2  |-  ( ph  ->  C  C_  ( Base `  D ) )
14 eqid 2389 . . . . . 6  |-  (Scalar `  U )  =  (Scalar `  U )
15 eqid 2389 . . . . . 6  |-  ( 0g
`  (Scalar `  U )
)  =  ( 0g
`  (Scalar `  U )
)
16 eqid 2389 . . . . . 6  |-  ( Base `  U )  =  (
Base `  U )
1714, 15, 16, 5lfl0f 29186 . . . . 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 31478 . . . . 5  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( Base `  U
) ) )  =  ( Base `  U
) )
21 eqidd 2390 . . . . . . . 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 29211 . . . . . . . . 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 5674 . . . . . 6  |-  ( ph  ->  (  ._|_  `  ( L `
 ( ( Base `  U )  X.  {
( 0g `  (Scalar `  U ) ) } ) ) )  =  (  ._|_  `  ( Base `  U ) ) )
2726fveq2d 5674 . . . . 5  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `  (
( Base `  U )  X.  { ( 0g `  (Scalar `  U ) ) } ) ) ) )  =  (  ._|_  `  (  ._|_  `  ( Base `  U ) ) ) )
2820, 27, 253eqtr4d 2431 . . . 4  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `  (
( Base `  U )  X.  { ( 0g `  (Scalar `  U ) ) } ) ) ) )  =  ( L `
 ( ( Base `  U )  X.  {
( 0g `  (Scalar `  U ) ) } ) ) )
29 eqid 2389 . . . . . . . 8  |-  ( 0g
`  U )  =  ( 0g `  U
)
308, 9, 19, 16, 29doch1 31476 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  ._|_  `  ( Base `  U ) )  =  { ( 0g `  U ) } )
3110, 30syl 16 . . . . . 6  |-  ( ph  ->  (  ._|_  `  ( Base `  U ) )  =  { ( 0g `  U ) } )
3226, 31eqtrd 2421 . . . . 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 15954 . . . . . 6  |-  ( ( U  e.  LMod  /\  R  e.  S )  ->  { ( 0g `  U ) }  C_  R )
3611, 33, 35syl2anc 643 . . . . 5  |-  ( ph  ->  { ( 0g `  U ) }  C_  R )
3732, 36eqsstrd 3327 . . . 4  |-  ( ph  ->  (  ._|_  `  ( L `
 ( ( Base `  U )  X.  {
( 0g `  (Scalar `  U ) ) } ) ) )  C_  R )
383lcfls1lem 31651 . . . 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 3579 . . 3  |-  ( ( ( Base `  U
)  X.  { ( 0g `  (Scalar `  U ) ) } )  e.  C  ->  C  =/=  (/) )
4139, 40syl 16 . 2  |-  ( ph  ->  C  =/=  (/) )
42 eqid 2389 . . . 4  |-  ( Base `  (Scalar `  U )
)  =  ( Base `  (Scalar `  U )
)
43 eqid 2389 . . . 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 2389 . . . 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 2389 . . . . . . . 8  |-  (Scalar `  D )  =  (Scalar `  D )
51 eqid 2389 . . . . . . . 8  |-  ( Base `  (Scalar `  D )
)  =  ( Base `  (Scalar `  D )
)
5214, 42, 6, 50, 51, 11ldualsbase 29250 . . . . . . 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 2465 . . . . 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 31654 . . . 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 31655 . . 3  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  D ) )  /\  a  e.  C  /\  b  e.  C )
)  ->  ( (
x ( .s `  D ) a ) ( +g  `  D
) b )  e.  C )
5756ralrimivvva 2744 . 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 15940 . 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 1717    =/= wne 2552   A.wral 2651   {crab 2655    C_ wss 3265   (/)c0 3573   {csn 3759    X. cxp 4818   ` cfv 5396  (class class class)co 6022   Basecbs 13398   +g cplusg 13458  Scalarcsca 13461   .scvsca 13462   0gc0g 13652   LModclmod 15879   LSubSpclss 15937  LFnlclfn 29174  LKerclk 29202  LDualcld 29240   HLchlt 29467   LHypclh 30100   DVecHcdvh 31195   ocHcoch 31464
This theorem is referenced by:  lclkrs2  31657  mapddlssN  31757
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 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-of 6246  df-1st 6290  df-2nd 6291  df-tpos 6417  df-undef 6481  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-map 6958  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-3 9993  df-4 9994  df-5 9995  df-6 9996  df-n0 10156  df-z 10217  df-uz 10423  df-fz 10978  df-struct 13400  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-mulr 13472  df-sca 13474  df-vsca 13475  df-0g 13656  df-mre 13740  df-mrc 13741  df-acs 13743  df-poset 14332  df-plt 14344  df-lub 14360  df-glb 14361  df-join 14362  df-meet 14363  df-p0 14397  df-p1 14398  df-lat 14404  df-clat 14466  df-mnd 14619  df-submnd 14668  df-grp 14741  df-minusg 14742  df-sbg 14743  df-subg 14870  df-cntz 15045  df-oppg 15071  df-lsm 15199  df-cmn 15343  df-abl 15344  df-mgp 15578  df-rng 15592  df-ur 15594  df-oppr 15657  df-dvdsr 15675  df-unit 15676  df-invr 15706  df-dvr 15717  df-drng 15766  df-lmod 15881  df-lss 15938  df-lsp 15977  df-lvec 16104  df-lsatoms 29093  df-lshyp 29094  df-lcv 29136  df-lfl 29175  df-lkr 29203  df-ldual 29241  df-oposet 29293  df-ol 29295  df-oml 29296  df-covers 29383  df-ats 29384  df-atl 29415  df-cvlat 29439  df-hlat 29468  df-llines 29614  df-lplanes 29615  df-lvols 29616  df-lines 29617  df-psubsp 29619  df-pmap 29620  df-padd 29912  df-lhyp 30104  df-laut 30105  df-ldil 30220  df-ltrn 30221  df-trl 30275  df-tgrp 30859  df-tendo 30871  df-edring 30873  df-dveca 31119  df-disoa 31146  df-dvech 31196  df-dib 31256  df-dic 31290  df-dih 31346  df-doch 31465  df-djh 31512
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