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Theorem lclkrslem2 32237
Description: The set of functionals having closed kernels and majorizing the orthocomplement of a given subspace  Q is closed under scalar product. (Contributed by NM, 28-Jan-2015.)
Hypotheses
Ref Expression
lclkrslem1.h  |-  H  =  ( LHyp `  K
)
lclkrslem1.o  |-  ._|_  =  ( ( ocH `  K
) `  W )
lclkrslem1.u  |-  U  =  ( ( DVecH `  K
) `  W )
lclkrslem1.s  |-  S  =  ( LSubSp `  U )
lclkrslem1.f  |-  F  =  (LFnl `  U )
lclkrslem1.l  |-  L  =  (LKer `  U )
lclkrslem1.d  |-  D  =  (LDual `  U )
lclkrslem1.r  |-  R  =  (Scalar `  U )
lclkrslem1.b  |-  B  =  ( Base `  R
)
lclkrslem1.t  |-  .x.  =  ( .s `  D )
lclkrslem1.c  |-  C  =  { f  e.  F  |  ( (  ._|_  `  (  ._|_  `  ( L `
 f ) ) )  =  ( L `
 f )  /\  (  ._|_  `  ( L `  f ) )  C_  Q ) }
lclkrslem1.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
lclkrslem1.q  |-  ( ph  ->  Q  e.  S )
lclkrslem1.g  |-  ( ph  ->  G  e.  C )
lclkrslem2.p  |-  .+  =  ( +g  `  D )
lclkrslem2.e  |-  ( ph  ->  E  e.  C )
Assertion
Ref Expression
lclkrslem2  |-  ( ph  ->  ( E  .+  G
)  e.  C )
Distinct variable groups:    ._|_ , f    f, F    f, G    f, L    Q, f    .x. , f    f, E    .+ , f
Allowed substitution hints:    ph( f)    B( f)    C( f)    D( f)    R( f)    S( f)    U( f)    H( f)    K( f)    W( f)

Proof of Theorem lclkrslem2
StepHypRef Expression
1 lclkrslem1.h . . 3  |-  H  =  ( LHyp `  K
)
2 lclkrslem1.o . . 3  |-  ._|_  =  ( ( ocH `  K
) `  W )
3 lclkrslem1.u . . 3  |-  U  =  ( ( DVecH `  K
) `  W )
4 lclkrslem1.f . . 3  |-  F  =  (LFnl `  U )
5 lclkrslem1.l . . 3  |-  L  =  (LKer `  U )
6 lclkrslem1.d . . 3  |-  D  =  (LDual `  U )
7 lclkrslem2.p . . 3  |-  .+  =  ( +g  `  D )
8 eqid 2435 . . 3  |-  { f  e.  F  |  ( 
._|_  `  (  ._|_  `  ( L `  f )
) )  =  ( L `  f ) }  =  { f  e.  F  |  ( 
._|_  `  (  ._|_  `  ( L `  f )
) )  =  ( L `  f ) }
9 lclkrslem1.k . . 3  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
10 lclkrslem2.e . . . 4  |-  ( ph  ->  E  e.  C )
11 lclkrslem1.c . . . . . 6  |-  C  =  { f  e.  F  |  ( (  ._|_  `  (  ._|_  `  ( L `
 f ) ) )  =  ( L `
 f )  /\  (  ._|_  `  ( L `  f ) )  C_  Q ) }
1211, 8lcfls1c 32235 . . . . 5  |-  ( E  e.  C  <->  ( E  e.  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }  /\  (  ._|_  `  ( L `  E ) )  C_  Q ) )
1312simplbi 447 . . . 4  |-  ( E  e.  C  ->  E  e.  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) } )
1410, 13syl 16 . . 3  |-  ( ph  ->  E  e.  { f  e.  F  |  ( 
._|_  `  (  ._|_  `  ( L `  f )
) )  =  ( L `  f ) } )
15 lclkrslem1.g . . . 4  |-  ( ph  ->  G  e.  C )
1611, 8lcfls1c 32235 . . . . 5  |-  ( G  e.  C  <->  ( G  e.  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }  /\  (  ._|_  `  ( L `  G ) )  C_  Q ) )
1716simplbi 447 . . . 4  |-  ( G  e.  C  ->  G  e.  { f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) } )
1815, 17syl 16 . . 3  |-  ( ph  ->  G  e.  { f  e.  F  |  ( 
._|_  `  (  ._|_  `  ( L `  f )
) )  =  ( L `  f ) } )
191, 2, 3, 4, 5, 6, 7, 8, 9, 14, 18lclkrlem2 32231 . 2  |-  ( ph  ->  ( E  .+  G
)  e.  { f  e.  F  |  ( 
._|_  `  (  ._|_  `  ( L `  f )
) )  =  ( L `  f ) } )
20 eqid 2435 . . . . 5  |-  ( Base `  U )  =  (
Base `  U )
211, 3, 9dvhlmod 31809 . . . . 5  |-  ( ph  ->  U  e.  LMod )
2211lcfls1lem 32233 . . . . . . . 8  |-  ( E  e.  C  <->  ( E  e.  F  /\  (  ._|_  `  (  ._|_  `  ( L `  E )
) )  =  ( L `  E )  /\  (  ._|_  `  ( L `  E )
)  C_  Q )
)
2310, 22sylib 189 . . . . . . 7  |-  ( ph  ->  ( E  e.  F  /\  (  ._|_  `  (  ._|_  `  ( L `  E ) ) )  =  ( L `  E )  /\  (  ._|_  `  ( L `  E ) )  C_  Q ) )
2423simp1d 969 . . . . . 6  |-  ( ph  ->  E  e.  F )
2511lcfls1lem 32233 . . . . . . . 8  |-  ( G  e.  C  <->  ( G  e.  F  /\  (  ._|_  `  (  ._|_  `  ( L `  G )
) )  =  ( L `  G )  /\  (  ._|_  `  ( L `  G )
)  C_  Q )
)
2615, 25sylib 189 . . . . . . 7  |-  ( ph  ->  ( G  e.  F  /\  (  ._|_  `  (  ._|_  `  ( L `  G ) ) )  =  ( L `  G )  /\  (  ._|_  `  ( L `  G ) )  C_  Q ) )
2726simp1d 969 . . . . . 6  |-  ( ph  ->  G  e.  F )
284, 6, 7, 21, 24, 27ldualvaddcl 29829 . . . . 5  |-  ( ph  ->  ( E  .+  G
)  e.  F )
2920, 4, 5, 21, 28lkrssv 29795 . . . 4  |-  ( ph  ->  ( L `  ( E  .+  G ) ) 
C_  ( Base `  U
) )
304, 5, 6, 7, 21, 24, 27lkrin 29863 . . . 4  |-  ( ph  ->  ( ( L `  E )  i^i  ( L `  G )
)  C_  ( L `  ( E  .+  G
) ) )
311, 3, 20, 2dochss 32064 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( L `  ( E  .+  G ) )  C_  ( Base `  U )  /\  (
( L `  E
)  i^i  ( L `  G ) )  C_  ( L `  ( E 
.+  G ) ) )  ->  (  ._|_  `  ( L `  ( E  .+  G ) ) )  C_  (  ._|_  `  ( ( L `  E )  i^i  ( L `  G )
) ) )
329, 29, 30, 31syl3anc 1184 . . 3  |-  ( ph  ->  (  ._|_  `  ( L `
 ( E  .+  G ) ) ) 
C_  (  ._|_  `  (
( L `  E
)  i^i  ( L `  G ) ) ) )
33 eqid 2435 . . . . . 6  |-  ( (
DIsoH `  K ) `  W )  =  ( ( DIsoH `  K ) `  W )
34 eqid 2435 . . . . . 6  |-  ( (joinH `  K ) `  W
)  =  ( (joinH `  K ) `  W
)
3523simp2d 970 . . . . . . 7  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `  E
) ) )  =  ( L `  E
) )
361, 33, 2, 3, 4, 5, 9, 24lcfl5a 32196 . . . . . . 7  |-  ( ph  ->  ( (  ._|_  `  (  ._|_  `  ( L `  E ) ) )  =  ( L `  E )  <->  ( L `  E )  e.  ran  ( ( DIsoH `  K
) `  W )
) )
3735, 36mpbid 202 . . . . . 6  |-  ( ph  ->  ( L `  E
)  e.  ran  (
( DIsoH `  K ) `  W ) )
3826simp2d 970 . . . . . . 7  |-  ( ph  ->  (  ._|_  `  (  ._|_  `  ( L `  G
) ) )  =  ( L `  G
) )
391, 33, 2, 3, 4, 5, 9, 27lcfl5a 32196 . . . . . . 7  |-  ( ph  ->  ( (  ._|_  `  (  ._|_  `  ( L `  G ) ) )  =  ( L `  G )  <->  ( L `  G )  e.  ran  ( ( DIsoH `  K
) `  W )
) )
4038, 39mpbid 202 . . . . . 6  |-  ( ph  ->  ( L `  G
)  e.  ran  (
( DIsoH `  K ) `  W ) )
411, 33, 3, 20, 2, 34, 9, 37, 40dochdmm1 32109 . . . . 5  |-  ( ph  ->  (  ._|_  `  ( ( L `  E )  i^i  ( L `  G ) ) )  =  ( (  ._|_  `  ( L `  E
) ) ( (joinH `  K ) `  W
) (  ._|_  `  ( L `  G )
) ) )
42 eqid 2435 . . . . . . 7  |-  ( LSSum `  U )  =  (
LSSum `  U )
4320, 4, 5, 21, 24lkrssv 29795 . . . . . . . 8  |-  ( ph  ->  ( L `  E
)  C_  ( Base `  U ) )
441, 33, 3, 20, 2dochcl 32052 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( L `  E )  C_  ( Base `  U ) )  ->  (  ._|_  `  ( L `  E )
)  e.  ran  (
( DIsoH `  K ) `  W ) )
459, 43, 44syl2anc 643 . . . . . . 7  |-  ( ph  ->  (  ._|_  `  ( L `
 E ) )  e.  ran  ( (
DIsoH `  K ) `  W ) )
461, 33, 2, 3, 42, 4, 5, 9, 45, 27dochkrsm 32157 . . . . . 6  |-  ( ph  ->  ( (  ._|_  `  ( L `  E )
) ( LSSum `  U
) (  ._|_  `  ( L `  G )
) )  e.  ran  ( ( DIsoH `  K
) `  W )
)
47 lclkrslem1.s . . . . . . 7  |-  S  =  ( LSubSp `  U )
481, 3, 20, 47, 2dochlss 32053 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( L `  E )  C_  ( Base `  U ) )  ->  (  ._|_  `  ( L `  E )
)  e.  S )
499, 43, 48syl2anc 643 . . . . . . 7  |-  ( ph  ->  (  ._|_  `  ( L `
 E ) )  e.  S )
5020, 4, 5, 21, 27lkrssv 29795 . . . . . . . 8  |-  ( ph  ->  ( L `  G
)  C_  ( Base `  U ) )
511, 3, 20, 47, 2dochlss 32053 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( L `  G )  C_  ( Base `  U ) )  ->  (  ._|_  `  ( L `  G )
)  e.  S )
529, 50, 51syl2anc 643 . . . . . . 7  |-  ( ph  ->  (  ._|_  `  ( L `
 G ) )  e.  S )
531, 3, 20, 47, 42, 33, 34, 9, 49, 52djhlsmcl 32113 . . . . . 6  |-  ( ph  ->  ( ( (  ._|_  `  ( L `  E
) ) ( LSSum `  U ) (  ._|_  `  ( L `  G
) ) )  e. 
ran  ( ( DIsoH `  K ) `  W
)  <->  ( (  ._|_  `  ( L `  E
) ) ( LSSum `  U ) (  ._|_  `  ( L `  G
) ) )  =  ( (  ._|_  `  ( L `  E )
) ( (joinH `  K ) `  W
) (  ._|_  `  ( L `  G )
) ) ) )
5446, 53mpbid 202 . . . . 5  |-  ( ph  ->  ( (  ._|_  `  ( L `  E )
) ( LSSum `  U
) (  ._|_  `  ( L `  G )
) )  =  ( (  ._|_  `  ( L `
 E ) ) ( (joinH `  K
) `  W )
(  ._|_  `  ( L `  G ) ) ) )
5541, 54eqtr4d 2470 . . . 4  |-  ( ph  ->  (  ._|_  `  ( ( L `  E )  i^i  ( L `  G ) ) )  =  ( (  ._|_  `  ( L `  E
) ) ( LSSum `  U ) (  ._|_  `  ( L `  G
) ) ) )
5623simp3d 971 . . . . 5  |-  ( ph  ->  (  ._|_  `  ( L `
 E ) ) 
C_  Q )
5726simp3d 971 . . . . 5  |-  ( ph  ->  (  ._|_  `  ( L `
 G ) ) 
C_  Q )
5847lsssssubg 16024 . . . . . . . 8  |-  ( U  e.  LMod  ->  S  C_  (SubGrp `  U ) )
5921, 58syl 16 . . . . . . 7  |-  ( ph  ->  S  C_  (SubGrp `  U
) )
6059, 49sseldd 3341 . . . . . 6  |-  ( ph  ->  (  ._|_  `  ( L `
 E ) )  e.  (SubGrp `  U
) )
6159, 52sseldd 3341 . . . . . 6  |-  ( ph  ->  (  ._|_  `  ( L `
 G ) )  e.  (SubGrp `  U
) )
62 lclkrslem1.q . . . . . . 7  |-  ( ph  ->  Q  e.  S )
6359, 62sseldd 3341 . . . . . 6  |-  ( ph  ->  Q  e.  (SubGrp `  U ) )
6442lsmlub 15287 . . . . . 6  |-  ( ( (  ._|_  `  ( L `
 E ) )  e.  (SubGrp `  U
)  /\  (  ._|_  `  ( L `  G
) )  e.  (SubGrp `  U )  /\  Q  e.  (SubGrp `  U )
)  ->  ( (
(  ._|_  `  ( L `  E ) )  C_  Q  /\  (  ._|_  `  ( L `  G )
)  C_  Q )  <->  ( (  ._|_  `  ( L `
 E ) ) ( LSSum `  U )
(  ._|_  `  ( L `  G ) ) ) 
C_  Q ) )
6560, 61, 63, 64syl3anc 1184 . . . . 5  |-  ( ph  ->  ( ( (  ._|_  `  ( L `  E
) )  C_  Q  /\  (  ._|_  `  ( L `  G )
)  C_  Q )  <->  ( (  ._|_  `  ( L `
 E ) ) ( LSSum `  U )
(  ._|_  `  ( L `  G ) ) ) 
C_  Q ) )
6656, 57, 65mpbi2and 888 . . . 4  |-  ( ph  ->  ( (  ._|_  `  ( L `  E )
) ( LSSum `  U
) (  ._|_  `  ( L `  G )
) )  C_  Q
)
6755, 66eqsstrd 3374 . . 3  |-  ( ph  ->  (  ._|_  `  ( ( L `  E )  i^i  ( L `  G ) ) ) 
C_  Q )
6832, 67sstrd 3350 . 2  |-  ( ph  ->  (  ._|_  `  ( L `
 ( E  .+  G ) ) ) 
C_  Q )
6911, 8lcfls1c 32235 . 2  |-  ( ( E  .+  G )  e.  C  <->  ( ( E  .+  G )  e. 
{ f  e.  F  |  (  ._|_  `  (  ._|_  `  ( L `  f ) ) )  =  ( L `  f ) }  /\  (  ._|_  `  ( L `  ( E  .+  G
) ) )  C_  Q ) )
7019, 68, 69sylanbrc 646 1  |-  ( ph  ->  ( E  .+  G
)  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {crab 2701    i^i cin 3311    C_ wss 3312   ran crn 4871   ` cfv 5446  (class class class)co 6073   Basecbs 13459   +g cplusg 13519  Scalarcsca 13522   .scvsca 13523  SubGrpcsubg 14928   LSSumclsm 15258   LModclmod 15940   LSubSpclss 15998  LFnlclfn 29756  LKerclk 29784  LDualcld 29822   HLchlt 30049   LHypclh 30682   DVecHcdvh 31777   DIsoHcdih 31927   ocHcoch 32046  joinHcdjh 32093
This theorem is referenced by:  lclkrs  32238
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-fal 1329  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-tpos 6471  df-undef 6535  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-nn 9991  df-2 10048  df-3 10049  df-4 10050  df-5 10051  df-6 10052  df-n0 10212  df-z 10273  df-uz 10479  df-fz 11034  df-struct 13461  df-ndx 13462  df-slot 13463  df-base 13464  df-sets 13465  df-ress 13466  df-plusg 13532  df-mulr 13533  df-sca 13535  df-vsca 13536  df-0g 13717  df-mre 13801  df-mrc 13802  df-acs 13804  df-poset 14393  df-plt 14405  df-lub 14421  df-glb 14422  df-join 14423  df-meet 14424  df-p0 14458  df-p1 14459  df-lat 14465  df-clat 14527  df-mnd 14680  df-submnd 14729  df-grp 14802  df-minusg 14803  df-sbg 14804  df-subg 14931  df-cntz 15106  df-oppg 15132  df-lsm 15260  df-cmn 15404  df-abl 15405  df-mgp 15639  df-rng 15653  df-ur 15655  df-oppr 15718  df-dvdsr 15736  df-unit 15737  df-invr 15767  df-dvr 15778  df-drng 15827  df-lmod 15942  df-lss 15999  df-lsp 16038  df-lvec 16165  df-lsatoms 29675  df-lshyp 29676  df-lcv 29718  df-lfl 29757  df-lkr 29785  df-ldual 29823  df-oposet 29875  df-ol 29877  df-oml 29878  df-covers 29965  df-ats 29966  df-atl 29997  df-cvlat 30021  df-hlat 30050  df-llines 30196  df-lplanes 30197  df-lvols 30198  df-lines 30199  df-psubsp 30201  df-pmap 30202  df-padd 30494  df-lhyp 30686  df-laut 30687  df-ldil 30802  df-ltrn 30803  df-trl 30857  df-tgrp 31441  df-tendo 31453  df-edring 31455  df-dveca 31701  df-disoa 31728  df-dvech 31778  df-dib 31838  df-dic 31872  df-dih 31928  df-doch 32047  df-djh 32094
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