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

Theorem lshpkrex 33082
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 2452 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2452 . . . . 5  |-  ( LSpan `  W )  =  (
LSpan `  W )
3 eqid 2452 . . . . 5  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
4 eqid 2452 . . . . 5  |-  ( LSSum `  W )  =  (
LSSum `  W )
5 lshpkrex.h . . . . 5  |-  H  =  (LSHyp `  W )
6 lveclmod 17305 . . . . 5  |-  ( W  e.  LVec  ->  W  e. 
LMod )
71, 2, 3, 4, 5, 6islshpsm 32944 . . . 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 990 . . . 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 228 . . 3  |-  ( W  e.  LVec  ->  ( U  e.  H  ->  E. z  e.  ( Base `  W
) ( U (
LSSum `  W ) ( ( LSpan `  W ) `  { z } ) )  =  ( Base `  W ) ) )
109imp 429 . 2  |-  ( ( W  e.  LVec  /\  U  e.  H )  ->  E. z  e.  ( Base `  W
) ( U (
LSSum `  W ) ( ( LSpan `  W ) `  { z } ) )  =  ( Base `  W ) )
11 eqid 2452 . . . . 5  |-  ( +g  `  W )  =  ( +g  `  W )
12 simp1l 1012 . . . . 5  |-  ( ( ( W  e.  LVec  /\  U  e.  H )  /\  z  e.  (
Base `  W )  /\  ( U ( LSSum `  W ) ( (
LSpan `  W ) `  { z } ) )  =  ( Base `  W ) )  ->  W  e.  LVec )
13 simp1r 1013 . . . . 5  |-  ( ( ( W  e.  LVec  /\  U  e.  H )  /\  z  e.  (
Base `  W )  /\  ( U ( LSSum `  W ) ( (
LSpan `  W ) `  { z } ) )  =  ( Base `  W ) )  ->  U  e.  H )
14 simp2 989 . . . . 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 990 . . . . 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 2452 . . . . 5  |-  (Scalar `  W )  =  (Scalar `  W )
17 eqid 2452 . . . . 5  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
18 eqid 2452 . . . . 5  |-  ( .s
`  W )  =  ( .s `  W
)
19 eqid 2452 . . . . 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 33080 . . . 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 33081 . . . 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 5794 . . . . . 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 2454 . . . . 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 3173 . . . 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 661 . . 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 2943 . 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2645   E.wrex 2797   {csn 3980    |-> cmpt 4453   ` cfv 5521   iota_crio 6155  (class class class)co 6195   Basecbs 14287   +g cplusg 14352  Scalarcsca 14355   .scvsca 14356   LSSumclsm 16249   LSubSpclss 17131   LSpanclspn 17170   LVecclvec 17301  LSHypclsh 32939  LFnlclfn 33021  LKerclk 33049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-int 4232  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-tpos 6850  df-recs 6937  df-rdg 6971  df-er 7206  df-map 7321  df-en 7416  df-dom 7417  df-sdom 7418  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-2 10486  df-3 10487  df-ndx 14290  df-slot 14291  df-base 14292  df-sets 14293  df-ress 14294  df-plusg 14365  df-mulr 14366  df-0g 14494  df-mnd 15529  df-submnd 15579  df-grp 15659  df-minusg 15660  df-sbg 15661  df-subg 15792  df-cntz 15949  df-lsm 16251  df-cmn 16395  df-abl 16396  df-mgp 16709  df-ur 16721  df-rng 16765  df-oppr 16833  df-dvdsr 16851  df-unit 16852  df-invr 16882  df-drng 16952  df-lmod 17068  df-lss 17132  df-lsp 17171  df-lvec 17302  df-lshyp 32941  df-lfl 33022  df-lkr 33050
This theorem is referenced by:  lshpset2N  33083  mapdordlem2  35601
  Copyright terms: Public domain W3C validator