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Theorem lkrlspeqN 30143
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 3321 . . . 4  |-  ( ph  ->  G  e.  ( N `
 { H }
) )
3 lkrlspeq.d . . . . . 6  |-  D  =  (LDual `  W )
4 lkrlspeq.w . . . . . . 7  |-  ( ph  ->  W  e.  LVec )
5 lveclmod 16216 . . . . . . 7  |-  ( W  e.  LVec  ->  W  e. 
LMod )
64, 5syl 16 . . . . . 6  |-  ( ph  ->  W  e.  LMod )
73, 6lduallmod 30125 . . . . 5  |-  ( ph  ->  D  e.  LMod )
8 lkrlspeq.f . . . . . 6  |-  F  =  (LFnl `  W )
9 eqid 2443 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
10 lkrlspeq.h . . . . . 6  |-  ( ph  ->  H  e.  F )
118, 3, 9, 4, 10ldualelvbase 30099 . . . . 5  |-  ( ph  ->  H  e.  ( Base `  D ) )
12 eqid 2443 . . . . . 6  |-  (Scalar `  D )  =  (Scalar `  D )
13 eqid 2443 . . . . . 6  |-  ( Base `  (Scalar `  D )
)  =  ( Base `  (Scalar `  D )
)
14 eqid 2443 . . . . . 6  |-  ( .s
`  D )  =  ( .s `  D
)
15 lkrlspeq.j . . . . . 6  |-  N  =  ( LSpan `  D )
1612, 13, 9, 14, 15lspsnel 16117 . . . . 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 644 . . . 4  |-  ( ph  ->  ( G  e.  ( N `  { H } )  <->  E. k  e.  ( Base `  (Scalar `  D ) ) G  =  ( k ( .s `  D ) H ) ) )
182, 17mpbid 203 . . 3  |-  ( ph  ->  E. k  e.  (
Base `  (Scalar `  D
) ) G  =  ( k ( .s
`  D ) H ) )
19 eqid 2443 . . . . 5  |-  (Scalar `  W )  =  (Scalar `  W )
20 eqid 2443 . . . . 5  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
2119, 20, 3, 12, 13, 4ldualsbase 30105 . . . 4  |-  ( ph  ->  ( Base `  (Scalar `  D ) )  =  ( Base `  (Scalar `  W ) ) )
2221rexeqdv 2918 . . 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 203 . 2  |-  ( ph  ->  E. k  e.  (
Base `  (Scalar `  W
) ) G  =  ( k ( .s
`  D ) H ) )
24 eqid 2443 . . . 4  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
25 lkrlspeq.l . . . 4  |-  L  =  (LKer `  W )
2643ad2ant1 979 . . . 4  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  ->  W  e.  LVec )
27 simp2 959 . . . . 5  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
k  e.  ( Base `  (Scalar `  W )
) )
28 simp3 960 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  ->  G  =  ( k
( .s `  D
) H ) )
29 eldifsni 3957 . . . . . . . . 9  |-  ( G  e.  ( ( N `
 { H }
)  \  {  .0.  } )  ->  G  =/=  .0.  )
301, 29syl 16 . . . . . . . 8  |-  ( ph  ->  G  =/=  .0.  )
31303ad2ant1 979 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  ->  G  =/=  .0.  )
3228, 31eqnetrrd 2628 . . . . . 6  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
( k ( .s
`  D ) H )  =/=  .0.  )
33 eqid 2443 . . . . . . . . . . . 12  |-  ( 0g
`  (Scalar `  D )
)  =  ( 0g
`  (Scalar `  D )
)
3419, 24, 3, 12, 33, 6ldual0 30119 . . . . . . . . . . 11  |-  ( ph  ->  ( 0g `  (Scalar `  D ) )  =  ( 0g `  (Scalar `  W ) ) )
35343ad2ant1 979 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
( 0g `  (Scalar `  D ) )  =  ( 0g `  (Scalar `  W ) ) )
3635eqeq2d 2454 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
( k  =  ( 0g `  (Scalar `  D ) )  <->  k  =  ( 0g `  (Scalar `  W ) ) ) )
37 orc 376 . . . . . . . . 9  |-  ( k  =  ( 0g `  (Scalar `  D ) )  ->  ( k  =  ( 0g `  (Scalar `  D ) )  \/  H  =  .0.  )
)
3836, 37syl6bir 222 . . . . . . . 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 30126 . . . . . . . . . 10  |-  ( ph  ->  D  e.  LVec )
41403ad2ant1 979 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  ->  D  e.  LVec )
42213ad2ant1 979 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
( Base `  (Scalar `  D
) )  =  (
Base `  (Scalar `  W
) ) )
4327, 42eleqtrrd 2520 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
k  e.  ( Base `  (Scalar `  D )
) )
44113ad2ant1 979 . . . . . . . . 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 16218 . . . . . . . 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 227 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
( k  =  ( 0g `  (Scalar `  W ) )  -> 
( k ( .s
`  D ) H )  =  .0.  )
)
4746necon3d 2646 . . . . . 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 3956 . . . . 5  |-  ( k  e.  ( ( Base `  (Scalar `  W )
)  \  { ( 0g `  (Scalar `  W
) ) } )  <-> 
( k  e.  (
Base `  (Scalar `  W
) )  /\  k  =/=  ( 0g `  (Scalar `  W ) ) ) )
5027, 48, 49sylanbrc 647 . . . 4  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
k  e.  ( (
Base `  (Scalar `  W
) )  \  {
( 0g `  (Scalar `  W ) ) } ) )
51103ad2ant1 979 . . . 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 30142 . . 3  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
( L `  G
)  =  ( L `
 H ) )
5352rexlimdv3a 2839 . 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 set class
Syntax hints:    -> wi 4    <-> wb 178    \/ wo 359    /\ w3a 937    = wceq 1654    e. wcel 1728    =/= wne 2606   E.wrex 2713    \ cdif 3306   {csn 3843   ` cfv 5489  (class class class)co 6117   Basecbs 13507  Scalarcsca 13570   .scvsca 13571   0gc0g 13761   LModclmod 15988   LSpanclspn 16085   LVecclvec 16212  LFnlclfn 30029  LKerclk 30057  LDualcld 30095
This theorem is referenced by:  lcdlkreqN  32594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-rep 4351  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736  ax-cnex 9084  ax-resscn 9085  ax-1cn 9086  ax-icn 9087  ax-addcl 9088  ax-addrcl 9089  ax-mulcl 9090  ax-mulrcl 9091  ax-mulcom 9092  ax-addass 9093  ax-mulass 9094  ax-distr 9095  ax-i2m1 9096  ax-1ne0 9097  ax-1rid 9098  ax-rnegex 9099  ax-rrecex 9100  ax-cnre 9101  ax-pre-lttri 9102  ax-pre-lttrn 9103  ax-pre-ltadd 9104  ax-pre-mulgt0 9105
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2717  df-rex 2718  df-reu 2719  df-rmo 2720  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-pss 3325  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-tp 3851  df-op 3852  df-uni 4045  df-int 4080  df-iun 4124  df-br 4244  df-opab 4298  df-mpt 4299  df-tr 4334  df-eprel 4529  df-id 4533  df-po 4538  df-so 4539  df-fr 4576  df-we 4578  df-ord 4619  df-on 4620  df-lim 4621  df-suc 4622  df-om 4881  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-of 6341  df-1st 6385  df-2nd 6386  df-tpos 6515  df-riota 6585  df-recs 6669  df-rdg 6704  df-1o 6760  df-oadd 6764  df-er 6941  df-map 7056  df-en 7146  df-dom 7147  df-sdom 7148  df-fin 7149  df-pnf 9160  df-mnf 9161  df-xr 9162  df-ltxr 9163  df-le 9164  df-sub 9331  df-neg 9332  df-nn 10039  df-2 10096  df-3 10097  df-4 10098  df-5 10099  df-6 10100  df-n0 10260  df-z 10321  df-uz 10527  df-fz 11082  df-struct 13509  df-ndx 13510  df-slot 13511  df-base 13512  df-sets 13513  df-ress 13514  df-plusg 13580  df-mulr 13581  df-sca 13583  df-vsca 13584  df-0g 13765  df-mnd 14728  df-submnd 14777  df-grp 14850  df-minusg 14851  df-sbg 14852  df-subg 14979  df-cntz 15154  df-lsm 15308  df-cmn 15452  df-abl 15453  df-mgp 15687  df-rng 15701  df-ur 15703  df-oppr 15766  df-dvdsr 15784  df-unit 15785  df-invr 15815  df-drng 15875  df-lmod 15990  df-lss 16047  df-lsp 16086  df-lvec 16213  df-lshyp 29949  df-lfl 30030  df-lkr 30058  df-ldual 30096
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