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Theorem lkrlspeqN 29666
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 3300 . . . 4  |-  ( ph  ->  G  e.  ( N `
 { H }
) )
3 lkrlspeq.d . . . . . 6  |-  D  =  (LDual `  W )
4 lkrlspeq.w . . . . . . 7  |-  ( ph  ->  W  e.  LVec )
5 lveclmod 16141 . . . . . . 7  |-  ( W  e.  LVec  ->  W  e. 
LMod )
64, 5syl 16 . . . . . 6  |-  ( ph  ->  W  e.  LMod )
73, 6lduallmod 29648 . . . . 5  |-  ( ph  ->  D  e.  LMod )
8 lkrlspeq.f . . . . . 6  |-  F  =  (LFnl `  W )
9 eqid 2412 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
10 lkrlspeq.h . . . . . 6  |-  ( ph  ->  H  e.  F )
118, 3, 9, 4, 10ldualelvbase 29622 . . . . 5  |-  ( ph  ->  H  e.  ( Base `  D ) )
12 eqid 2412 . . . . . 6  |-  (Scalar `  D )  =  (Scalar `  D )
13 eqid 2412 . . . . . 6  |-  ( Base `  (Scalar `  D )
)  =  ( Base `  (Scalar `  D )
)
14 eqid 2412 . . . . . 6  |-  ( .s
`  D )  =  ( .s `  D
)
15 lkrlspeq.j . . . . . 6  |-  N  =  ( LSpan `  D )
1612, 13, 9, 14, 15lspsnel 16042 . . . . 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 643 . . . 4  |-  ( ph  ->  ( G  e.  ( N `  { H } )  <->  E. k  e.  ( Base `  (Scalar `  D ) ) G  =  ( k ( .s `  D ) H ) ) )
182, 17mpbid 202 . . 3  |-  ( ph  ->  E. k  e.  (
Base `  (Scalar `  D
) ) G  =  ( k ( .s
`  D ) H ) )
19 eqid 2412 . . . . 5  |-  (Scalar `  W )  =  (Scalar `  W )
20 eqid 2412 . . . . 5  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
2119, 20, 3, 12, 13, 4ldualsbase 29628 . . . 4  |-  ( ph  ->  ( Base `  (Scalar `  D ) )  =  ( Base `  (Scalar `  W ) ) )
2221rexeqdv 2879 . . 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 202 . 2  |-  ( ph  ->  E. k  e.  (
Base `  (Scalar `  W
) ) G  =  ( k ( .s
`  D ) H ) )
24 eqid 2412 . . . 4  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
25 lkrlspeq.l . . . 4  |-  L  =  (LKer `  W )
2643ad2ant1 978 . . . 4  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  ->  W  e.  LVec )
27 simp2 958 . . . . 5  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
k  e.  ( Base `  (Scalar `  W )
) )
28 simp3 959 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  ->  G  =  ( k
( .s `  D
) H ) )
29 eldifsni 3896 . . . . . . . . 9  |-  ( G  e.  ( ( N `
 { H }
)  \  {  .0.  } )  ->  G  =/=  .0.  )
301, 29syl 16 . . . . . . . 8  |-  ( ph  ->  G  =/=  .0.  )
31303ad2ant1 978 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  ->  G  =/=  .0.  )
3228, 31eqnetrrd 2595 . . . . . 6  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
( k ( .s
`  D ) H )  =/=  .0.  )
33 eqid 2412 . . . . . . . . . . . 12  |-  ( 0g
`  (Scalar `  D )
)  =  ( 0g
`  (Scalar `  D )
)
3419, 24, 3, 12, 33, 6ldual0 29642 . . . . . . . . . . 11  |-  ( ph  ->  ( 0g `  (Scalar `  D ) )  =  ( 0g `  (Scalar `  W ) ) )
35343ad2ant1 978 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
( 0g `  (Scalar `  D ) )  =  ( 0g `  (Scalar `  W ) ) )
3635eqeq2d 2423 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
( k  =  ( 0g `  (Scalar `  D ) )  <->  k  =  ( 0g `  (Scalar `  W ) ) ) )
37 orc 375 . . . . . . . . 9  |-  ( k  =  ( 0g `  (Scalar `  D ) )  ->  ( k  =  ( 0g `  (Scalar `  D ) )  \/  H  =  .0.  )
)
3836, 37syl6bir 221 . . . . . . . 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 29649 . . . . . . . . . 10  |-  ( ph  ->  D  e.  LVec )
41403ad2ant1 978 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  ->  D  e.  LVec )
42213ad2ant1 978 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
( Base `  (Scalar `  D
) )  =  (
Base `  (Scalar `  W
) ) )
4327, 42eleqtrrd 2489 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
k  e.  ( Base `  (Scalar `  D )
) )
44113ad2ant1 978 . . . . . . . . 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 16143 . . . . . . . 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 226 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
( k  =  ( 0g `  (Scalar `  W ) )  -> 
( k ( .s
`  D ) H )  =  .0.  )
)
4746necon3d 2613 . . . . . 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 3895 . . . . 5  |-  ( k  e.  ( ( Base `  (Scalar `  W )
)  \  { ( 0g `  (Scalar `  W
) ) } )  <-> 
( k  e.  (
Base `  (Scalar `  W
) )  /\  k  =/=  ( 0g `  (Scalar `  W ) ) ) )
5027, 48, 49sylanbrc 646 . . . 4  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
k  e.  ( (
Base `  (Scalar `  W
) )  \  {
( 0g `  (Scalar `  W ) ) } ) )
51103ad2ant1 978 . . . 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 29665 . . 3  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
( L `  G
)  =  ( L `
 H ) )
5352rexlimdv3a 2800 . 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 177    \/ wo 358    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575   E.wrex 2675    \ cdif 3285   {csn 3782   ` cfv 5421  (class class class)co 6048   Basecbs 13432  Scalarcsca 13495   .scvsca 13496   0gc0g 13686   LModclmod 15913   LSpanclspn 16010   LVecclvec 16137  LFnlclfn 29552  LKerclk 29580  LDualcld 29618
This theorem is referenced by:  lcdlkreqN  32117
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-1st 6316  df-2nd 6317  df-tpos 6446  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-n0 10186  df-z 10247  df-uz 10453  df-fz 11008  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-mulr 13506  df-sca 13508  df-vsca 13509  df-0g 13690  df-mnd 14653  df-submnd 14702  df-grp 14775  df-minusg 14776  df-sbg 14777  df-subg 14904  df-cntz 15079  df-lsm 15233  df-cmn 15377  df-abl 15378  df-mgp 15612  df-rng 15626  df-ur 15628  df-oppr 15691  df-dvdsr 15709  df-unit 15710  df-invr 15740  df-drng 15800  df-lmod 15915  df-lss 15972  df-lsp 16011  df-lvec 16138  df-lshyp 29472  df-lfl 29553  df-lkr 29581  df-ldual 29619
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