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Theorem lkrlspeqN 32656
Description: Condition for colinear functionals to have equal kernels. (Contributed by NM, 20-Mar-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
lkrlspeq.f  |-  F  =  (LFnl `  W )
lkrlspeq.l  |-  L  =  (LKer `  W )
lkrlspeq.d  |-  D  =  (LDual `  W )
lkrlspeq.o  |-  .0.  =  ( 0g `  D )
lkrlspeq.j  |-  N  =  ( LSpan `  D )
lkrlspeq.w  |-  ( ph  ->  W  e.  LVec )
lkrlspeq.h  |-  ( ph  ->  H  e.  F )
lkrlspeq.g  |-  ( ph  ->  G  e.  ( ( N `  { H } )  \  {  .0.  } ) )
Assertion
Ref Expression
lkrlspeqN  |-  ( ph  ->  ( L `  G
)  =  ( L `
 H ) )

Proof of Theorem lkrlspeqN
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 lkrlspeq.g . . . . 5  |-  ( ph  ->  G  e.  ( ( N `  { H } )  \  {  .0.  } ) )
21eldifad 3335 . . . 4  |-  ( ph  ->  G  e.  ( N `
 { H }
) )
3 lkrlspeq.d . . . . . 6  |-  D  =  (LDual `  W )
4 lkrlspeq.w . . . . . . 7  |-  ( ph  ->  W  e.  LVec )
5 lveclmod 17164 . . . . . . 7  |-  ( W  e.  LVec  ->  W  e. 
LMod )
64, 5syl 16 . . . . . 6  |-  ( ph  ->  W  e.  LMod )
73, 6lduallmod 32638 . . . . 5  |-  ( ph  ->  D  e.  LMod )
8 lkrlspeq.f . . . . . 6  |-  F  =  (LFnl `  W )
9 eqid 2438 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
10 lkrlspeq.h . . . . . 6  |-  ( ph  ->  H  e.  F )
118, 3, 9, 4, 10ldualelvbase 32612 . . . . 5  |-  ( ph  ->  H  e.  ( Base `  D ) )
12 eqid 2438 . . . . . 6  |-  (Scalar `  D )  =  (Scalar `  D )
13 eqid 2438 . . . . . 6  |-  ( Base `  (Scalar `  D )
)  =  ( Base `  (Scalar `  D )
)
14 eqid 2438 . . . . . 6  |-  ( .s
`  D )  =  ( .s `  D
)
15 lkrlspeq.j . . . . . 6  |-  N  =  ( LSpan `  D )
1612, 13, 9, 14, 15lspsnel 17061 . . . . 5  |-  ( ( D  e.  LMod  /\  H  e.  ( Base `  D
) )  ->  ( G  e.  ( N `  { H } )  <->  E. k  e.  ( Base `  (Scalar `  D
) ) G  =  ( k ( .s
`  D ) H ) ) )
177, 11, 16syl2anc 661 . . . 4  |-  ( ph  ->  ( G  e.  ( N `  { H } )  <->  E. k  e.  ( Base `  (Scalar `  D ) ) G  =  ( k ( .s `  D ) H ) ) )
182, 17mpbid 210 . . 3  |-  ( ph  ->  E. k  e.  (
Base `  (Scalar `  D
) ) G  =  ( k ( .s
`  D ) H ) )
19 eqid 2438 . . . . 5  |-  (Scalar `  W )  =  (Scalar `  W )
20 eqid 2438 . . . . 5  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
2119, 20, 3, 12, 13, 4ldualsbase 32618 . . . 4  |-  ( ph  ->  ( Base `  (Scalar `  D ) )  =  ( Base `  (Scalar `  W ) ) )
2221rexeqdv 2919 . . 3  |-  ( ph  ->  ( E. k  e.  ( Base `  (Scalar `  D ) ) G  =  ( k ( .s `  D ) H )  <->  E. k  e.  ( Base `  (Scalar `  W ) ) G  =  ( k ( .s `  D ) H ) ) )
2318, 22mpbid 210 . 2  |-  ( ph  ->  E. k  e.  (
Base `  (Scalar `  W
) ) G  =  ( k ( .s
`  D ) H ) )
24 eqid 2438 . . . 4  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
25 lkrlspeq.l . . . 4  |-  L  =  (LKer `  W )
2643ad2ant1 1009 . . . 4  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  ->  W  e.  LVec )
27 simp2 989 . . . . 5  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
k  e.  ( Base `  (Scalar `  W )
) )
28 simp3 990 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  ->  G  =  ( k
( .s `  D
) H ) )
29 eldifsni 3996 . . . . . . . . 9  |-  ( G  e.  ( ( N `
 { H }
)  \  {  .0.  } )  ->  G  =/=  .0.  )
301, 29syl 16 . . . . . . . 8  |-  ( ph  ->  G  =/=  .0.  )
31303ad2ant1 1009 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  ->  G  =/=  .0.  )
3228, 31eqnetrrd 2623 . . . . . 6  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
( k ( .s
`  D ) H )  =/=  .0.  )
33 eqid 2438 . . . . . . . . . . . 12  |-  ( 0g
`  (Scalar `  D )
)  =  ( 0g
`  (Scalar `  D )
)
3419, 24, 3, 12, 33, 6ldual0 32632 . . . . . . . . . . 11  |-  ( ph  ->  ( 0g `  (Scalar `  D ) )  =  ( 0g `  (Scalar `  W ) ) )
35343ad2ant1 1009 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
( 0g `  (Scalar `  D ) )  =  ( 0g `  (Scalar `  W ) ) )
3635eqeq2d 2449 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
( k  =  ( 0g `  (Scalar `  D ) )  <->  k  =  ( 0g `  (Scalar `  W ) ) ) )
37 orc 385 . . . . . . . . 9  |-  ( k  =  ( 0g `  (Scalar `  D ) )  ->  ( k  =  ( 0g `  (Scalar `  D ) )  \/  H  =  .0.  )
)
3836, 37syl6bir 229 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
( k  =  ( 0g `  (Scalar `  W ) )  -> 
( k  =  ( 0g `  (Scalar `  D ) )  \/  H  =  .0.  )
) )
39 lkrlspeq.o . . . . . . . . 9  |-  .0.  =  ( 0g `  D )
403, 4lduallvec 32639 . . . . . . . . . 10  |-  ( ph  ->  D  e.  LVec )
41403ad2ant1 1009 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  ->  D  e.  LVec )
42213ad2ant1 1009 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
( Base `  (Scalar `  D
) )  =  (
Base `  (Scalar `  W
) ) )
4327, 42eleqtrrd 2515 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
k  e.  ( Base `  (Scalar `  D )
) )
44113ad2ant1 1009 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  ->  H  e.  ( Base `  D ) )
459, 14, 12, 13, 33, 39, 41, 43, 44lvecvs0or 17166 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
( ( k ( .s `  D ) H )  =  .0.  <->  ( k  =  ( 0g
`  (Scalar `  D )
)  \/  H  =  .0.  ) ) )
4638, 45sylibrd 234 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
( k  =  ( 0g `  (Scalar `  W ) )  -> 
( k ( .s
`  D ) H )  =  .0.  )
)
4746necon3d 2641 . . . . . 6  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
( ( k ( .s `  D ) H )  =/=  .0.  ->  k  =/=  ( 0g
`  (Scalar `  W )
) ) )
4832, 47mpd 15 . . . . 5  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
k  =/=  ( 0g
`  (Scalar `  W )
) )
49 eldifsn 3995 . . . . 5  |-  ( k  e.  ( ( Base `  (Scalar `  W )
)  \  { ( 0g `  (Scalar `  W
) ) } )  <-> 
( k  e.  (
Base `  (Scalar `  W
) )  /\  k  =/=  ( 0g `  (Scalar `  W ) ) ) )
5027, 48, 49sylanbrc 664 . . . 4  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
k  e.  ( (
Base `  (Scalar `  W
) )  \  {
( 0g `  (Scalar `  W ) ) } ) )
51103ad2ant1 1009 . . . 4  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  ->  H  e.  F )
5219, 20, 24, 8, 25, 3, 14, 26, 50, 51, 28lkreqN 32655 . . 3  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
( L `  G
)  =  ( L `
 H ) )
5352rexlimdv3a 2838 . 2  |-  ( ph  ->  ( E. k  e.  ( Base `  (Scalar `  W ) ) G  =  ( k ( .s `  D ) H )  ->  ( L `  G )  =  ( L `  H ) ) )
5423, 53mpd 15 1  |-  ( ph  ->  ( L `  G
)  =  ( L `
 H ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601   E.wrex 2711    \ cdif 3320   {csn 3872   ` cfv 5413  (class class class)co 6086   Basecbs 14166  Scalarcsca 14233   .scvsca 14234   0gc0g 14370   LModclmod 16926   LSpanclspn 17029   LVecclvec 17160  LFnlclfn 32542  LKerclk 32570  LDualcld 32608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-tpos 6740  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-n0 10572  df-z 10639  df-uz 10854  df-fz 11430  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-sca 14246  df-vsca 14247  df-0g 14372  df-mnd 15407  df-submnd 15457  df-grp 15536  df-minusg 15537  df-sbg 15538  df-subg 15669  df-cntz 15826  df-lsm 16126  df-cmn 16270  df-abl 16271  df-mgp 16580  df-ur 16592  df-rng 16635  df-oppr 16703  df-dvdsr 16721  df-unit 16722  df-invr 16752  df-drng 16812  df-lmod 16928  df-lss 16991  df-lsp 17030  df-lvec 17161  df-lshyp 32462  df-lfl 32543  df-lkr 32571  df-ldual 32609
This theorem is referenced by:  lcdlkreqN  35107
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