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Theorem lkrlspeqN 32782
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 3428 . . . 4  |-  ( ph  ->  G  e.  ( N `
 { H }
) )
3 lkrlspeq.d . . . . . 6  |-  D  =  (LDual `  W )
4 lkrlspeq.w . . . . . . 7  |-  ( ph  ->  W  e.  LVec )
5 lveclmod 18378 . . . . . . 7  |-  ( W  e.  LVec  ->  W  e. 
LMod )
64, 5syl 17 . . . . . 6  |-  ( ph  ->  W  e.  LMod )
73, 6lduallmod 32764 . . . . 5  |-  ( ph  ->  D  e.  LMod )
8 lkrlspeq.f . . . . . 6  |-  F  =  (LFnl `  W )
9 eqid 2462 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
10 lkrlspeq.h . . . . . 6  |-  ( ph  ->  H  e.  F )
118, 3, 9, 4, 10ldualelvbase 32738 . . . . 5  |-  ( ph  ->  H  e.  ( Base `  D ) )
12 eqid 2462 . . . . . 6  |-  (Scalar `  D )  =  (Scalar `  D )
13 eqid 2462 . . . . . 6  |-  ( Base `  (Scalar `  D )
)  =  ( Base `  (Scalar `  D )
)
14 eqid 2462 . . . . . 6  |-  ( .s
`  D )  =  ( .s `  D
)
15 lkrlspeq.j . . . . . 6  |-  N  =  ( LSpan `  D )
1612, 13, 9, 14, 15lspsnel 18275 . . . . 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 671 . . . 4  |-  ( ph  ->  ( G  e.  ( N `  { H } )  <->  E. k  e.  ( Base `  (Scalar `  D ) ) G  =  ( k ( .s `  D ) H ) ) )
182, 17mpbid 215 . . 3  |-  ( ph  ->  E. k  e.  (
Base `  (Scalar `  D
) ) G  =  ( k ( .s
`  D ) H ) )
19 eqid 2462 . . . . 5  |-  (Scalar `  W )  =  (Scalar `  W )
20 eqid 2462 . . . . 5  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
2119, 20, 3, 12, 13, 4ldualsbase 32744 . . . 4  |-  ( ph  ->  ( Base `  (Scalar `  D ) )  =  ( Base `  (Scalar `  W ) ) )
2221rexeqdv 3006 . . 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 215 . 2  |-  ( ph  ->  E. k  e.  (
Base `  (Scalar `  W
) ) G  =  ( k ( .s
`  D ) H ) )
24 eqid 2462 . . . 4  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
25 lkrlspeq.l . . . 4  |-  L  =  (LKer `  W )
2643ad2ant1 1035 . . . 4  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  ->  W  e.  LVec )
27 simp2 1015 . . . . 5  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
k  e.  ( Base `  (Scalar `  W )
) )
28 simp3 1016 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  ->  G  =  ( k
( .s `  D
) H ) )
29 eldifsni 4111 . . . . . . . . 9  |-  ( G  e.  ( ( N `
 { H }
)  \  {  .0.  } )  ->  G  =/=  .0.  )
301, 29syl 17 . . . . . . . 8  |-  ( ph  ->  G  =/=  .0.  )
31303ad2ant1 1035 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  ->  G  =/=  .0.  )
3228, 31eqnetrrd 2704 . . . . . 6  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
( k ( .s
`  D ) H )  =/=  .0.  )
33 eqid 2462 . . . . . . . . . . . 12  |-  ( 0g
`  (Scalar `  D )
)  =  ( 0g
`  (Scalar `  D )
)
3419, 24, 3, 12, 33, 6ldual0 32758 . . . . . . . . . . 11  |-  ( ph  ->  ( 0g `  (Scalar `  D ) )  =  ( 0g `  (Scalar `  W ) ) )
35343ad2ant1 1035 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
( 0g `  (Scalar `  D ) )  =  ( 0g `  (Scalar `  W ) ) )
3635eqeq2d 2472 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
( k  =  ( 0g `  (Scalar `  D ) )  <->  k  =  ( 0g `  (Scalar `  W ) ) ) )
37 orc 391 . . . . . . . . 9  |-  ( k  =  ( 0g `  (Scalar `  D ) )  ->  ( k  =  ( 0g `  (Scalar `  D ) )  \/  H  =  .0.  )
)
3836, 37syl6bir 237 . . . . . . . 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 32765 . . . . . . . . . 10  |-  ( ph  ->  D  e.  LVec )
41403ad2ant1 1035 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  ->  D  e.  LVec )
42213ad2ant1 1035 . . . . . . . . . 10  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
( Base `  (Scalar `  D
) )  =  (
Base `  (Scalar `  W
) ) )
4327, 42eleqtrrd 2543 . . . . . . . . 9  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
k  e.  ( Base `  (Scalar `  D )
) )
44113ad2ant1 1035 . . . . . . . . 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 18380 . . . . . . . 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 242 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
( k  =  ( 0g `  (Scalar `  W ) )  -> 
( k ( .s
`  D ) H )  =  .0.  )
)
4746necon3d 2657 . . . . . 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 4110 . . . . 5  |-  ( k  e.  ( ( Base `  (Scalar `  W )
)  \  { ( 0g `  (Scalar `  W
) ) } )  <-> 
( k  e.  (
Base `  (Scalar `  W
) )  /\  k  =/=  ( 0g `  (Scalar `  W ) ) ) )
5027, 48, 49sylanbrc 675 . . . 4  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
k  e.  ( (
Base `  (Scalar `  W
) )  \  {
( 0g `  (Scalar `  W ) ) } ) )
51103ad2ant1 1035 . . . 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 32781 . . 3  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) )  /\  G  =  ( k ( .s `  D ) H ) )  -> 
( L `  G
)  =  ( L `
 H ) )
5352rexlimdv3a 2893 . 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 189    \/ wo 374    /\ w3a 991    = wceq 1455    e. wcel 1898    =/= wne 2633   E.wrex 2750    \ cdif 3413   {csn 3980   ` cfv 5601  (class class class)co 6315   Basecbs 15170  Scalarcsca 15242   .scvsca 15243   0gc0g 15387   LModclmod 18140   LSpanclspn 18243   LVecclvec 18374  LFnlclfn 32668  LKerclk 32696  LDualcld 32734
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-of 6558  df-om 6720  df-1st 6820  df-2nd 6821  df-tpos 6999  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-oadd 7212  df-er 7389  df-map 7500  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-nn 10638  df-2 10696  df-3 10697  df-4 10698  df-5 10699  df-6 10700  df-n0 10899  df-z 10967  df-uz 11189  df-fz 11814  df-struct 15172  df-ndx 15173  df-slot 15174  df-base 15175  df-sets 15176  df-ress 15177  df-plusg 15252  df-mulr 15253  df-sca 15255  df-vsca 15256  df-0g 15389  df-mgm 16537  df-sgrp 16576  df-mnd 16586  df-submnd 16632  df-grp 16722  df-minusg 16723  df-sbg 16724  df-subg 16863  df-cntz 17020  df-lsm 17337  df-cmn 17481  df-abl 17482  df-mgp 17773  df-ur 17785  df-ring 17831  df-oppr 17900  df-dvdsr 17918  df-unit 17919  df-invr 17949  df-drng 18026  df-lmod 18142  df-lss 18205  df-lsp 18244  df-lvec 18375  df-lshyp 32588  df-lfl 32669  df-lkr 32697  df-ldual 32735
This theorem is referenced by:  lcdlkreqN  35235
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