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Theorem lshpkrex 35240
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 2454 . . . . 5  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2454 . . . . 5  |-  ( LSpan `  W )  =  (
LSpan `  W )
3 eqid 2454 . . . . 5  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
4 eqid 2454 . . . . 5  |-  ( LSSum `  W )  =  (
LSSum `  W )
5 lshpkrex.h . . . . 5  |-  H  =  (LSHyp `  W )
6 lveclmod 17947 . . . . 5  |-  ( W  e.  LVec  ->  W  e. 
LMod )
71, 2, 3, 4, 5, 6islshpsm 35102 . . . 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 996 . . . 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 427 . 2  |-  ( ( W  e.  LVec  /\  U  e.  H )  ->  E. z  e.  ( Base `  W
) ( U (
LSSum `  W ) ( ( LSpan `  W ) `  { z } ) )  =  ( Base `  W ) )
11 eqid 2454 . . . . 5  |-  ( +g  `  W )  =  ( +g  `  W )
12 simp1l 1018 . . . . 5  |-  ( ( ( W  e.  LVec  /\  U  e.  H )  /\  z  e.  (
Base `  W )  /\  ( U ( LSSum `  W ) ( (
LSpan `  W ) `  { z } ) )  =  ( Base `  W ) )  ->  W  e.  LVec )
13 simp1r 1019 . . . . 5  |-  ( ( ( W  e.  LVec  /\  U  e.  H )  /\  z  e.  (
Base `  W )  /\  ( U ( LSSum `  W ) ( (
LSpan `  W ) `  { z } ) )  =  ( Base `  W ) )  ->  U  e.  H )
14 simp2 995 . . . . 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 996 . . . . 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 2454 . . . . 5  |-  (Scalar `  W )  =  (Scalar `  W )
17 eqid 2454 . . . . 5  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
18 eqid 2454 . . . . 5  |-  ( .s
`  W )  =  ( .s `  W
)
19 eqid 2454 . . . . 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 35238 . . . 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 35239 . . . 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 5848 . . . . . 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 2456 . . . . 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 3207 . . . 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 659 . . 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 2948 . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   E.wrex 2805   {csn 4016    |-> cmpt 4497   ` cfv 5570   iota_crio 6231  (class class class)co 6270   Basecbs 14716   +g cplusg 14784  Scalarcsca 14787   .scvsca 14788   LSSumclsm 16853   LSubSpclss 17773   LSpanclspn 17812   LVecclvec 17943  LSHypclsh 35097  LFnlclfn 35179  LKerclk 35207
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-tpos 6947  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-0g 14931  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-grp 16256  df-minusg 16257  df-sbg 16258  df-subg 16397  df-cntz 16554  df-lsm 16855  df-cmn 16999  df-abl 17000  df-mgp 17337  df-ur 17349  df-ring 17395  df-oppr 17467  df-dvdsr 17485  df-unit 17486  df-invr 17516  df-drng 17593  df-lmod 17709  df-lss 17774  df-lsp 17813  df-lvec 17944  df-lshyp 35099  df-lfl 35180  df-lkr 35208
This theorem is referenced by:  lshpset2N  35241  mapdordlem2  37761
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