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Theorem lshpkrex 29854
Description: There exists a functional whose kernel equals a given hyperplane. Part of Th. 1.27 of Barbu and Precupanu, Convexity and Optimization in Banach Spaces. (Contributed by NM, 17-Jul-2014.)
Hypotheses
Ref Expression
lshpkrex.h  |-  H  =  (LSHyp `  W )
lshpkrex.f  |-  F  =  (LFnl `  W )
lshpkrex.k  |-  K  =  (LKer `  W )
Assertion
Ref Expression
lshpkrex  |-  ( ( W  e.  LVec  /\  U  e.  H )  ->  E. g  e.  F  ( K `  g )  =  U )
Distinct variable groups:    g, F    g, K    U, g    g, W
Allowed substitution hint:    H( g)

Proof of Theorem lshpkrex
Dummy variables  z 
k  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2436 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2436 . . . . 5  |-  ( LSpan `  W )  =  (
LSpan `  W )
3 eqid 2436 . . . . 5  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
4 eqid 2436 . . . . 5  |-  ( LSSum `  W )  =  (
LSSum `  W )
5 lshpkrex.h . . . . 5  |-  H  =  (LSHyp `  W )
6 lveclmod 16171 . . . . 5  |-  ( W  e.  LVec  ->  W  e. 
LMod )
71, 2, 3, 4, 5, 6islshpsm 29716 . . . 4  |-  ( W  e.  LVec  ->  ( U  e.  H  <->  ( U  e.  ( LSubSp `  W )  /\  U  =/=  ( Base `  W )  /\  E. z  e.  ( Base `  W ) ( U ( LSSum `  W )
( ( LSpan `  W
) `  { z } ) )  =  ( Base `  W
) ) ) )
8 simp3 959 . . . 4  |-  ( ( U  e.  ( LSubSp `  W )  /\  U  =/=  ( Base `  W
)  /\  E. z  e.  ( Base `  W
) ( U (
LSSum `  W ) ( ( LSpan `  W ) `  { z } ) )  =  ( Base `  W ) )  ->  E. z  e.  ( Base `  W ) ( U ( LSSum `  W
) ( ( LSpan `  W ) `  {
z } ) )  =  ( Base `  W
) )
97, 8syl6bi 220 . . 3  |-  ( W  e.  LVec  ->  ( U  e.  H  ->  E. z  e.  ( Base `  W
) ( U (
LSSum `  W ) ( ( LSpan `  W ) `  { z } ) )  =  ( Base `  W ) ) )
109imp 419 . 2  |-  ( ( W  e.  LVec  /\  U  e.  H )  ->  E. z  e.  ( Base `  W
) ( U (
LSSum `  W ) ( ( LSpan `  W ) `  { z } ) )  =  ( Base `  W ) )
11 eqid 2436 . . . . 5  |-  ( +g  `  W )  =  ( +g  `  W )
12 simp1l 981 . . . . 5  |-  ( ( ( W  e.  LVec  /\  U  e.  H )  /\  z  e.  (
Base `  W )  /\  ( U ( LSSum `  W ) ( (
LSpan `  W ) `  { z } ) )  =  ( Base `  W ) )  ->  W  e.  LVec )
13 simp1r 982 . . . . 5  |-  ( ( ( W  e.  LVec  /\  U  e.  H )  /\  z  e.  (
Base `  W )  /\  ( U ( LSSum `  W ) ( (
LSpan `  W ) `  { z } ) )  =  ( Base `  W ) )  ->  U  e.  H )
14 simp2 958 . . . . 5  |-  ( ( ( W  e.  LVec  /\  U  e.  H )  /\  z  e.  (
Base `  W )  /\  ( U ( LSSum `  W ) ( (
LSpan `  W ) `  { z } ) )  =  ( Base `  W ) )  -> 
z  e.  ( Base `  W ) )
15 simp3 959 . . . . 5  |-  ( ( ( W  e.  LVec  /\  U  e.  H )  /\  z  e.  (
Base `  W )  /\  ( U ( LSSum `  W ) ( (
LSpan `  W ) `  { z } ) )  =  ( Base `  W ) )  -> 
( U ( LSSum `  W ) ( (
LSpan `  W ) `  { z } ) )  =  ( Base `  W ) )
16 eqid 2436 . . . . 5  |-  (Scalar `  W )  =  (Scalar `  W )
17 eqid 2436 . . . . 5  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
18 eqid 2436 . . . . 5  |-  ( .s
`  W )  =  ( .s `  W
)
19 eqid 2436 . . . . 5  |-  ( x  e.  ( Base `  W
)  |->  ( iota_ k  e.  ( Base `  (Scalar `  W ) ) E. y  e.  U  x  =  ( y ( +g  `  W ) ( k ( .s
`  W ) z ) ) ) )  =  ( x  e.  ( Base `  W
)  |->  ( iota_ k  e.  ( Base `  (Scalar `  W ) ) E. y  e.  U  x  =  ( y ( +g  `  W ) ( k ( .s
`  W ) z ) ) ) )
20 lshpkrex.f . . . . 5  |-  F  =  (LFnl `  W )
211, 11, 2, 4, 5, 12, 13, 14, 15, 16, 17, 18, 19, 20lshpkrcl 29852 . . . 4  |-  ( ( ( W  e.  LVec  /\  U  e.  H )  /\  z  e.  (
Base `  W )  /\  ( U ( LSSum `  W ) ( (
LSpan `  W ) `  { z } ) )  =  ( Base `  W ) )  -> 
( x  e.  (
Base `  W )  |->  ( iota_ k  e.  (
Base `  (Scalar `  W
) ) E. y  e.  U  x  =  ( y ( +g  `  W ) ( k ( .s `  W
) z ) ) ) )  e.  F
)
22 lshpkrex.k . . . . 5  |-  K  =  (LKer `  W )
231, 11, 2, 4, 5, 12, 13, 14, 15, 16, 17, 18, 19, 22lshpkr 29853 . . . 4  |-  ( ( ( W  e.  LVec  /\  U  e.  H )  /\  z  e.  (
Base `  W )  /\  ( U ( LSSum `  W ) ( (
LSpan `  W ) `  { z } ) )  =  ( Base `  W ) )  -> 
( K `  (
x  e.  ( Base `  W )  |->  ( iota_ k  e.  ( Base `  (Scalar `  W ) ) E. y  e.  U  x  =  ( y ( +g  `  W ) ( k ( .s
`  W ) z ) ) ) ) )  =  U )
24 fveq2 5721 . . . . . 6  |-  ( g  =  ( x  e.  ( Base `  W
)  |->  ( iota_ k  e.  ( Base `  (Scalar `  W ) ) E. y  e.  U  x  =  ( y ( +g  `  W ) ( k ( .s
`  W ) z ) ) ) )  ->  ( K `  g )  =  ( K `  ( x  e.  ( Base `  W
)  |->  ( iota_ k  e.  ( Base `  (Scalar `  W ) ) E. y  e.  U  x  =  ( y ( +g  `  W ) ( k ( .s
`  W ) z ) ) ) ) ) )
2524eqeq1d 2444 . . . . 5  |-  ( g  =  ( x  e.  ( Base `  W
)  |->  ( iota_ k  e.  ( Base `  (Scalar `  W ) ) E. y  e.  U  x  =  ( y ( +g  `  W ) ( k ( .s
`  W ) z ) ) ) )  ->  ( ( K `
 g )  =  U  <->  ( K `  ( x  e.  ( Base `  W )  |->  (
iota_ k  e.  ( Base `  (Scalar `  W
) ) E. y  e.  U  x  =  ( y ( +g  `  W ) ( k ( .s `  W
) z ) ) ) ) )  =  U ) )
2625rspcev 3045 . . . 4  |-  ( ( ( x  e.  (
Base `  W )  |->  ( iota_ k  e.  (
Base `  (Scalar `  W
) ) E. y  e.  U  x  =  ( y ( +g  `  W ) ( k ( .s `  W
) z ) ) ) )  e.  F  /\  ( K `  (
x  e.  ( Base `  W )  |->  ( iota_ k  e.  ( Base `  (Scalar `  W ) ) E. y  e.  U  x  =  ( y ( +g  `  W ) ( k ( .s
`  W ) z ) ) ) ) )  =  U )  ->  E. g  e.  F  ( K `  g )  =  U )
2721, 23, 26syl2anc 643 . . 3  |-  ( ( ( W  e.  LVec  /\  U  e.  H )  /\  z  e.  (
Base `  W )  /\  ( U ( LSSum `  W ) ( (
LSpan `  W ) `  { z } ) )  =  ( Base `  W ) )  ->  E. g  e.  F  ( K `  g )  =  U )
2827rexlimdv3a 2825 . 2  |-  ( ( W  e.  LVec  /\  U  e.  H )  ->  ( E. z  e.  ( Base `  W ) ( U ( LSSum `  W
) ( ( LSpan `  W ) `  {
z } ) )  =  ( Base `  W
)  ->  E. g  e.  F  ( K `  g )  =  U ) )
2910, 28mpd 15 1  |-  ( ( W  e.  LVec  /\  U  e.  H )  ->  E. g  e.  F  ( K `  g )  =  U )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   E.wrex 2699   {csn 3807    e. cmpt 4259   ` cfv 5447  (class class class)co 6074   iota_crio 6535   Basecbs 13462   +g cplusg 13522  Scalarcsca 13525   .scvsca 13526   LSSumclsm 15261   LSubSpclss 16001   LSpanclspn 16040   LVecclvec 16167  LSHypclsh 29711  LFnlclfn 29793  LKerclk 29821
This theorem is referenced by:  lshpset2N  29855  mapdordlem2  32373
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 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-cnex 9039  ax-resscn 9040  ax-1cn 9041  ax-icn 9042  ax-addcl 9043  ax-addrcl 9044  ax-mulcl 9045  ax-mulrcl 9046  ax-mulcom 9047  ax-addass 9048  ax-mulass 9049  ax-distr 9050  ax-i2m1 9051  ax-1ne0 9052  ax-1rid 9053  ax-rnegex 9054  ax-rrecex 9055  ax-cnre 9056  ax-pre-lttri 9057  ax-pre-lttrn 9058  ax-pre-ltadd 9059  ax-pre-mulgt0 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-reu 2705  df-rmo 2706  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-pss 3329  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-int 4044  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-tr 4296  df-eprel 4487  df-id 4491  df-po 4496  df-so 4497  df-fr 4534  df-we 4536  df-ord 4577  df-on 4578  df-lim 4579  df-suc 4580  df-om 4839  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-ov 6077  df-oprab 6078  df-mpt2 6079  df-1st 6342  df-2nd 6343  df-tpos 6472  df-riota 6542  df-recs 6626  df-rdg 6661  df-er 6898  df-map 7013  df-en 7103  df-dom 7104  df-sdom 7105  df-pnf 9115  df-mnf 9116  df-xr 9117  df-ltxr 9118  df-le 9119  df-sub 9286  df-neg 9287  df-nn 9994  df-2 10051  df-3 10052  df-ndx 13465  df-slot 13466  df-base 13467  df-sets 13468  df-ress 13469  df-plusg 13535  df-mulr 13536  df-0g 13720  df-mnd 14683  df-submnd 14732  df-grp 14805  df-minusg 14806  df-sbg 14807  df-subg 14934  df-cntz 15109  df-lsm 15263  df-cmn 15407  df-abl 15408  df-mgp 15642  df-rng 15656  df-ur 15658  df-oppr 15721  df-dvdsr 15739  df-unit 15740  df-invr 15770  df-drng 15830  df-lmod 15945  df-lss 16002  df-lsp 16041  df-lvec 16168  df-lshyp 29713  df-lfl 29794  df-lkr 29822
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