Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lshpkrex Unicode version

Theorem lshpkrex 29930
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 2296 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2296 . . . . 5  |-  ( LSpan `  W )  =  (
LSpan `  W )
3 eqid 2296 . . . . 5  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
4 eqid 2296 . . . . 5  |-  ( LSSum `  W )  =  (
LSSum `  W )
5 lshpkrex.h . . . . 5  |-  H  =  (LSHyp `  W )
6 lveclmod 15875 . . . . 5  |-  ( W  e.  LVec  ->  W  e. 
LMod )
71, 2, 3, 4, 5, 6islshpsm 29792 . . . 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 957 . . . 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 219 . . 3  |-  ( W  e.  LVec  ->  ( U  e.  H  ->  E. z  e.  ( Base `  W
) ( U (
LSSum `  W ) ( ( LSpan `  W ) `  { z } ) )  =  ( Base `  W ) ) )
109imp 418 . 2  |-  ( ( W  e.  LVec  /\  U  e.  H )  ->  E. z  e.  ( Base `  W
) ( U (
LSSum `  W ) ( ( LSpan `  W ) `  { z } ) )  =  ( Base `  W ) )
11 eqid 2296 . . . . 5  |-  ( +g  `  W )  =  ( +g  `  W )
12 simp1l 979 . . . . 5  |-  ( ( ( W  e.  LVec  /\  U  e.  H )  /\  z  e.  (
Base `  W )  /\  ( U ( LSSum `  W ) ( (
LSpan `  W ) `  { z } ) )  =  ( Base `  W ) )  ->  W  e.  LVec )
13 simp1r 980 . . . . 5  |-  ( ( ( W  e.  LVec  /\  U  e.  H )  /\  z  e.  (
Base `  W )  /\  ( U ( LSSum `  W ) ( (
LSpan `  W ) `  { z } ) )  =  ( Base `  W ) )  ->  U  e.  H )
14 simp2 956 . . . . 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 957 . . . . 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 2296 . . . . 5  |-  (Scalar `  W )  =  (Scalar `  W )
17 eqid 2296 . . . . 5  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
18 eqid 2296 . . . . 5  |-  ( .s
`  W )  =  ( .s `  W
)
19 eqid 2296 . . . . 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 29928 . . . 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 29929 . . . 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 5541 . . . . . 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 2304 . . . . 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 2897 . . . 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 642 . . 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 2682 . 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 14 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557   {csn 3653    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   iota_crio 6313   Basecbs 13164   +g cplusg 13224  Scalarcsca 13227   .scvsca 13228   LSSumclsm 14961   LSubSpclss 15705   LSpanclspn 15744   LVecclvec 15871  LSHypclsh 29787  LFnlclfn 29869  LKerclk 29897
This theorem is referenced by:  lshpset2N  29931  mapdordlem2  32449
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-0g 13420  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-cntz 14809  df-lsm 14963  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-drng 15530  df-lmod 15645  df-lss 15706  df-lsp 15745  df-lvec 15872  df-lshyp 29789  df-lfl 29870  df-lkr 29898
  Copyright terms: Public domain W3C validator