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Theorem lshpkrex 29284
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 2380 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2380 . . . . 5  |-  ( LSpan `  W )  =  (
LSpan `  W )
3 eqid 2380 . . . . 5  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
4 eqid 2380 . . . . 5  |-  ( LSSum `  W )  =  (
LSSum `  W )
5 lshpkrex.h . . . . 5  |-  H  =  (LSHyp `  W )
6 lveclmod 16098 . . . . 5  |-  ( W  e.  LVec  ->  W  e. 
LMod )
71, 2, 3, 4, 5, 6islshpsm 29146 . . . 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 2380 . . . . 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 2380 . . . . 5  |-  (Scalar `  W )  =  (Scalar `  W )
17 eqid 2380 . . . . 5  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
18 eqid 2380 . . . . 5  |-  ( .s
`  W )  =  ( .s `  W
)
19 eqid 2380 . . . . 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 29282 . . . 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 29283 . . . 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 5661 . . . . . 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 2388 . . . . 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 2988 . . . 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 2768 . 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 1649    e. wcel 1717    =/= wne 2543   E.wrex 2643   {csn 3750    e. cmpt 4200   ` cfv 5387  (class class class)co 6013   iota_crio 6471   Basecbs 13389   +g cplusg 13449  Scalarcsca 13452   .scvsca 13453   LSSumclsm 15188   LSubSpclss 15928   LSpanclspn 15967   LVecclvec 16094  LSHypclsh 29141  LFnlclfn 29223  LKerclk 29251
This theorem is referenced by:  lshpset2N  29285  mapdordlem2  31803
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 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-tpos 6408  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-3 9984  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-ress 13396  df-plusg 13462  df-mulr 13463  df-0g 13647  df-mnd 14610  df-submnd 14659  df-grp 14732  df-minusg 14733  df-sbg 14734  df-subg 14861  df-cntz 15036  df-lsm 15190  df-cmn 15334  df-abl 15335  df-mgp 15569  df-rng 15583  df-ur 15585  df-oppr 15648  df-dvdsr 15666  df-unit 15667  df-invr 15697  df-drng 15757  df-lmod 15872  df-lss 15929  df-lsp 15968  df-lvec 16095  df-lshyp 29143  df-lfl 29224  df-lkr 29252
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