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Theorem lkreqN 29360
Description: Proportional functionals have equal kernels. (Contributed by NM, 28-Mar-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
lkreq.s  |-  S  =  (Scalar `  W )
lkreq.r  |-  R  =  ( Base `  S
)
lkreq.o  |-  .0.  =  ( 0g `  S )
lkreq.f  |-  F  =  (LFnl `  W )
lkreq.k  |-  K  =  (LKer `  W )
lkreq.d  |-  D  =  (LDual `  W )
lkreq.t  |-  .x.  =  ( .s `  D )
lkreq.w  |-  ( ph  ->  W  e.  LVec )
lkreq.a  |-  ( ph  ->  A  e.  ( R 
\  {  .0.  }
) )
lkreq.h  |-  ( ph  ->  H  e.  F )
lkreq.g  |-  ( ph  ->  G  =  ( A 
.x.  H ) )
Assertion
Ref Expression
lkreqN  |-  ( ph  ->  ( K `  G
)  =  ( K `
 H ) )

Proof of Theorem lkreqN
StepHypRef Expression
1 lkreq.g . . . . . . . . 9  |-  ( ph  ->  G  =  ( A 
.x.  H ) )
21eqeq1d 2291 . . . . . . . 8  |-  ( ph  ->  ( G  =  ( 0g `  D )  <-> 
( A  .x.  H
)  =  ( 0g
`  D ) ) )
3 eqid 2283 . . . . . . . . . 10  |-  ( Base `  D )  =  (
Base `  D )
4 lkreq.t . . . . . . . . . 10  |-  .x.  =  ( .s `  D )
5 eqid 2283 . . . . . . . . . 10  |-  (Scalar `  D )  =  (Scalar `  D )
6 eqid 2283 . . . . . . . . . 10  |-  ( Base `  (Scalar `  D )
)  =  ( Base `  (Scalar `  D )
)
7 eqid 2283 . . . . . . . . . 10  |-  ( 0g
`  (Scalar `  D )
)  =  ( 0g
`  (Scalar `  D )
)
8 eqid 2283 . . . . . . . . . 10  |-  ( 0g
`  D )  =  ( 0g `  D
)
9 lkreq.d . . . . . . . . . . 11  |-  D  =  (LDual `  W )
10 lkreq.w . . . . . . . . . . 11  |-  ( ph  ->  W  e.  LVec )
119, 10lduallvec 29344 . . . . . . . . . 10  |-  ( ph  ->  D  e.  LVec )
12 lkreq.a . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  ( R 
\  {  .0.  }
) )
13 eldifi 3298 . . . . . . . . . . . 12  |-  ( A  e.  ( R  \  {  .0.  } )  ->  A  e.  R )
1412, 13syl 15 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  R )
15 lkreq.s . . . . . . . . . . . 12  |-  S  =  (Scalar `  W )
16 lkreq.r . . . . . . . . . . . 12  |-  R  =  ( Base `  S
)
1715, 16, 9, 5, 6, 10ldualsbase 29323 . . . . . . . . . . 11  |-  ( ph  ->  ( Base `  (Scalar `  D ) )  =  R )
1814, 17eleqtrrd 2360 . . . . . . . . . 10  |-  ( ph  ->  A  e.  ( Base `  (Scalar `  D )
) )
19 lkreq.f . . . . . . . . . . 11  |-  F  =  (LFnl `  W )
20 lkreq.h . . . . . . . . . . 11  |-  ( ph  ->  H  e.  F )
2119, 9, 3, 10, 20ldualelvbase 29317 . . . . . . . . . 10  |-  ( ph  ->  H  e.  ( Base `  D ) )
223, 4, 5, 6, 7, 8, 11, 18, 21lvecvs0or 15861 . . . . . . . . 9  |-  ( ph  ->  ( ( A  .x.  H )  =  ( 0g `  D )  <-> 
( A  =  ( 0g `  (Scalar `  D ) )  \/  H  =  ( 0g
`  D ) ) ) )
23 lkreq.o . . . . . . . . . . . . 13  |-  .0.  =  ( 0g `  S )
24 lveclmod 15859 . . . . . . . . . . . . . 14  |-  ( W  e.  LVec  ->  W  e. 
LMod )
2510, 24syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  W  e.  LMod )
2615, 23, 9, 5, 7, 25ldual0 29337 . . . . . . . . . . . 12  |-  ( ph  ->  ( 0g `  (Scalar `  D ) )  =  .0.  )
2726eqeq2d 2294 . . . . . . . . . . 11  |-  ( ph  ->  ( A  =  ( 0g `  (Scalar `  D ) )  <->  A  =  .0.  ) )
28 eldifsni 3750 . . . . . . . . . . . . . 14  |-  ( A  e.  ( R  \  {  .0.  } )  ->  A  =/=  .0.  )
2912, 28syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  A  =/=  .0.  )
3029a1d 22 . . . . . . . . . . . 12  |-  ( ph  ->  ( H  =/=  ( 0g `  D )  ->  A  =/=  .0.  ) )
3130necon4d 2509 . . . . . . . . . . 11  |-  ( ph  ->  ( A  =  .0. 
->  H  =  ( 0g `  D ) ) )
3227, 31sylbid 206 . . . . . . . . . 10  |-  ( ph  ->  ( A  =  ( 0g `  (Scalar `  D ) )  ->  H  =  ( 0g `  D ) ) )
33 idd 21 . . . . . . . . . 10  |-  ( ph  ->  ( H  =  ( 0g `  D )  ->  H  =  ( 0g `  D ) ) )
3432, 33jaod 369 . . . . . . . . 9  |-  ( ph  ->  ( ( A  =  ( 0g `  (Scalar `  D ) )  \/  H  =  ( 0g
`  D ) )  ->  H  =  ( 0g `  D ) ) )
3522, 34sylbid 206 . . . . . . . 8  |-  ( ph  ->  ( ( A  .x.  H )  =  ( 0g `  D )  ->  H  =  ( 0g `  D ) ) )
362, 35sylbid 206 . . . . . . 7  |-  ( ph  ->  ( G  =  ( 0g `  D )  ->  H  =  ( 0g `  D ) ) )
37 nne 2450 . . . . . . 7  |-  ( -.  H  =/=  ( 0g
`  D )  <->  H  =  ( 0g `  D ) )
3836, 37syl6ibr 218 . . . . . 6  |-  ( ph  ->  ( G  =  ( 0g `  D )  ->  -.  H  =/=  ( 0g `  D ) ) )
3938con3d 125 . . . . 5  |-  ( ph  ->  ( -.  -.  H  =/=  ( 0g `  D
)  ->  -.  G  =  ( 0g `  D ) ) )
4039orrd 367 . . . 4  |-  ( ph  ->  ( -.  H  =/=  ( 0g `  D
)  \/  -.  G  =  ( 0g `  D ) ) )
41 ianor 474 . . . 4  |-  ( -.  ( H  =/=  ( 0g `  D )  /\  G  =  ( 0g `  D ) )  <->  ( -.  H  =/=  ( 0g `  D )  \/  -.  G  =  ( 0g `  D ) ) )
4240, 41sylibr 203 . . 3  |-  ( ph  ->  -.  ( H  =/=  ( 0g `  D
)  /\  G  =  ( 0g `  D ) ) )
43 df-pss 3168 . . . . . 6  |-  ( ( K `  H ) 
C.  ( K `  G )  <->  ( ( K `  H )  C_  ( K `  G
)  /\  ( K `  H )  =/=  ( K `  G )
) )
44 lkreq.k . . . . . . 7  |-  K  =  (LKer `  W )
4519, 15, 16, 9, 4, 25, 14, 20ldualvscl 29329 . . . . . . . 8  |-  ( ph  ->  ( A  .x.  H
)  e.  F )
461, 45eqeltrd 2357 . . . . . . 7  |-  ( ph  ->  G  e.  F )
4719, 44, 9, 8, 10, 20, 46lkrpssN 29353 . . . . . 6  |-  ( ph  ->  ( ( K `  H )  C.  ( K `  G )  <->  ( H  =/=  ( 0g
`  D )  /\  G  =  ( 0g `  D ) ) ) )
4843, 47syl5rbbr 251 . . . . 5  |-  ( ph  ->  ( ( H  =/=  ( 0g `  D
)  /\  G  =  ( 0g `  D ) )  <->  ( ( K `
 H )  C_  ( K `  G )  /\  ( K `  H )  =/=  ( K `  G )
) ) )
4915, 16, 19, 44, 9, 4, 10, 20, 14lkrss 29358 . . . . . . 7  |-  ( ph  ->  ( K `  H
)  C_  ( K `  ( A  .x.  H
) ) )
501fveq2d 5529 . . . . . . 7  |-  ( ph  ->  ( K `  G
)  =  ( K `
 ( A  .x.  H ) ) )
5149, 50sseqtr4d 3215 . . . . . 6  |-  ( ph  ->  ( K `  H
)  C_  ( K `  G ) )
5251biantrurd 494 . . . . 5  |-  ( ph  ->  ( ( K `  H )  =/=  ( K `  G )  <->  ( ( K `  H
)  C_  ( K `  G )  /\  ( K `  H )  =/=  ( K `  G
) ) ) )
5348, 52bitr4d 247 . . . 4  |-  ( ph  ->  ( ( H  =/=  ( 0g `  D
)  /\  G  =  ( 0g `  D ) )  <->  ( K `  H )  =/=  ( K `  G )
) )
5453necon2bbid 2504 . . 3  |-  ( ph  ->  ( ( K `  H )  =  ( K `  G )  <->  -.  ( H  =/=  ( 0g `  D )  /\  G  =  ( 0g `  D ) ) ) )
5542, 54mpbird 223 . 2  |-  ( ph  ->  ( K `  H
)  =  ( K `
 G ) )
5655eqcomd 2288 1  |-  ( ph  ->  ( K `  G
)  =  ( K `
 H ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149    C_ wss 3152    C. wpss 3153   {csn 3640   ` cfv 5255  (class class class)co 5858   Basecbs 13148  Scalarcsca 13211   .scvsca 13212   0gc0g 13400   LModclmod 15627   LVecclvec 15855  LFnlclfn 29247  LKerclk 29275  LDualcld 29313
This theorem is referenced by:  lkrlspeqN  29361  lcdlkreq2N  31813
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-0g 13404  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cntz 14793  df-lsm 14947  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856  df-lshyp 29167  df-lfl 29248  df-lkr 29276  df-ldual 29314
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