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Theorem lkrlspeqN 33968
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 3488 . . . 4  |-  ( ph  ->  G  e.  ( N `
 { H }
) )
3 lkrlspeq.d . . . . . 6  |-  D  =  (LDual `  W )
4 lkrlspeq.w . . . . . . 7  |-  ( ph  ->  W  e.  LVec )
5 lveclmod 17535 . . . . . . 7  |-  ( W  e.  LVec  ->  W  e. 
LMod )
64, 5syl 16 . . . . . 6  |-  ( ph  ->  W  e.  LMod )
73, 6lduallmod 33950 . . . . 5  |-  ( ph  ->  D  e.  LMod )
8 lkrlspeq.f . . . . . 6  |-  F  =  (LFnl `  W )
9 eqid 2467 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
10 lkrlspeq.h . . . . . 6  |-  ( ph  ->  H  e.  F )
118, 3, 9, 4, 10ldualelvbase 33924 . . . . 5  |-  ( ph  ->  H  e.  ( Base `  D ) )
12 eqid 2467 . . . . . 6  |-  (Scalar `  D )  =  (Scalar `  D )
13 eqid 2467 . . . . . 6  |-  ( Base `  (Scalar `  D )
)  =  ( Base `  (Scalar `  D )
)
14 eqid 2467 . . . . . 6  |-  ( .s
`  D )  =  ( .s `  D
)
15 lkrlspeq.j . . . . . 6  |-  N  =  ( LSpan `  D )
1612, 13, 9, 14, 15lspsnel 17432 . . . . 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 2467 . . . . 5  |-  (Scalar `  W )  =  (Scalar `  W )
20 eqid 2467 . . . . 5  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
2119, 20, 3, 12, 13, 4ldualsbase 33930 . . . 4  |-  ( ph  ->  ( Base `  (Scalar `  D ) )  =  ( Base `  (Scalar `  W ) ) )
2221rexeqdv 3065 . . 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 2467 . . . 4  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
25 lkrlspeq.l . . . 4  |-  L  =  (LKer `  W )
2643ad2ant1 1017 . . . 4  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  ->  W  e.  LVec )
27 simp2 997 . . . . 5  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
k  e.  ( Base `  (Scalar `  W )
) )
28 simp3 998 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  ->  G  =  ( k
( .s `  D
) H ) )
29 eldifsni 4153 . . . . . . . . 9  |-  ( G  e.  ( ( N `
 { H }
)  \  {  .0.  } )  ->  G  =/=  .0.  )
301, 29syl 16 . . . . . . . 8  |-  ( ph  ->  G  =/=  .0.  )
31303ad2ant1 1017 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  ->  G  =/=  .0.  )
3228, 31eqnetrrd 2761 . . . . . 6  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
( k ( .s
`  D ) H )  =/=  .0.  )
33 eqid 2467 . . . . . . . . . . . 12  |-  ( 0g
`  (Scalar `  D )
)  =  ( 0g
`  (Scalar `  D )
)
3419, 24, 3, 12, 33, 6ldual0 33944 . . . . . . . . . . 11  |-  ( ph  ->  ( 0g `  (Scalar `  D ) )  =  ( 0g `  (Scalar `  W ) ) )
35343ad2ant1 1017 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
( 0g `  (Scalar `  D ) )  =  ( 0g `  (Scalar `  W ) ) )
3635eqeq2d 2481 . . . . . . . . 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 33951 . . . . . . . . . 10  |-  ( ph  ->  D  e.  LVec )
41403ad2ant1 1017 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  ->  D  e.  LVec )
42213ad2ant1 1017 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
( Base `  (Scalar `  D
) )  =  (
Base `  (Scalar `  W
) ) )
4327, 42eleqtrrd 2558 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
k  e.  ( Base `  (Scalar `  D )
) )
44113ad2ant1 1017 . . . . . . . . 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 17537 . . . . . . . 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 2691 . . . . . 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 4152 . . . . 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 1017 . . . 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 33967 . . 3  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
( L `  G
)  =  ( L `
 H ) )
5352rexlimdv3a 2957 . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815    \ cdif 3473   {csn 4027   ` cfv 5586  (class class class)co 6282   Basecbs 14486  Scalarcsca 14554   .scvsca 14555   0gc0g 14691   LModclmod 17295   LSpanclspn 17400   LVecclvec 17531  LFnlclfn 33854  LKerclk 33882  LDualcld 33920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-tpos 6952  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-sca 14567  df-vsca 14568  df-0g 14693  df-mnd 15728  df-submnd 15778  df-grp 15858  df-minusg 15859  df-sbg 15860  df-subg 15993  df-cntz 16150  df-lsm 16452  df-cmn 16596  df-abl 16597  df-mgp 16932  df-ur 16944  df-rng 16988  df-oppr 17056  df-dvdsr 17074  df-unit 17075  df-invr 17105  df-drng 17181  df-lmod 17297  df-lss 17362  df-lsp 17401  df-lvec 17532  df-lshyp 33774  df-lfl 33855  df-lkr 33883  df-ldual 33921
This theorem is referenced by:  lcdlkreqN  36419
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