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Theorem lkreqN 30066
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 2450 . . . . . . . 8  |-  ( ph  ->  ( G  =  ( 0g `  D )  <-> 
( A  .x.  H
)  =  ( 0g
`  D ) ) )
3 eqid 2442 . . . . . . . . . 10  |-  ( Base `  D )  =  (
Base `  D )
4 lkreq.t . . . . . . . . . 10  |-  .x.  =  ( .s `  D )
5 eqid 2442 . . . . . . . . . 10  |-  (Scalar `  D )  =  (Scalar `  D )
6 eqid 2442 . . . . . . . . . 10  |-  ( Base `  (Scalar `  D )
)  =  ( Base `  (Scalar `  D )
)
7 eqid 2442 . . . . . . . . . 10  |-  ( 0g
`  (Scalar `  D )
)  =  ( 0g
`  (Scalar `  D )
)
8 eqid 2442 . . . . . . . . . 10  |-  ( 0g
`  D )  =  ( 0g `  D
)
9 lkreq.d . . . . . . . . . . 11  |-  D  =  (LDual `  W )
10 lkreq.w . . . . . . . . . . 11  |-  ( ph  ->  W  e.  LVec )
119, 10lduallvec 30050 . . . . . . . . . 10  |-  ( ph  ->  D  e.  LVec )
12 lkreq.a . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  ( R 
\  {  .0.  }
) )
1312eldifad 3318 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  R )
14 lkreq.s . . . . . . . . . . . 12  |-  S  =  (Scalar `  W )
15 lkreq.r . . . . . . . . . . . 12  |-  R  =  ( Base `  S
)
1614, 15, 9, 5, 6, 10ldualsbase 30029 . . . . . . . . . . 11  |-  ( ph  ->  ( Base `  (Scalar `  D ) )  =  R )
1713, 16eleqtrrd 2519 . . . . . . . . . 10  |-  ( ph  ->  A  e.  ( Base `  (Scalar `  D )
) )
18 lkreq.f . . . . . . . . . . 11  |-  F  =  (LFnl `  W )
19 lkreq.h . . . . . . . . . . 11  |-  ( ph  ->  H  e.  F )
2018, 9, 3, 10, 19ldualelvbase 30023 . . . . . . . . . 10  |-  ( ph  ->  H  e.  ( Base `  D ) )
213, 4, 5, 6, 7, 8, 11, 17, 20lvecvs0or 16211 . . . . . . . . 9  |-  ( ph  ->  ( ( A  .x.  H )  =  ( 0g `  D )  <-> 
( A  =  ( 0g `  (Scalar `  D ) )  \/  H  =  ( 0g
`  D ) ) ) )
22 lkreq.o . . . . . . . . . . . . 13  |-  .0.  =  ( 0g `  S )
23 lveclmod 16209 . . . . . . . . . . . . . 14  |-  ( W  e.  LVec  ->  W  e. 
LMod )
2410, 23syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  W  e.  LMod )
2514, 22, 9, 5, 7, 24ldual0 30043 . . . . . . . . . . . 12  |-  ( ph  ->  ( 0g `  (Scalar `  D ) )  =  .0.  )
2625eqeq2d 2453 . . . . . . . . . . 11  |-  ( ph  ->  ( A  =  ( 0g `  (Scalar `  D ) )  <->  A  =  .0.  ) )
27 eldifsni 3952 . . . . . . . . . . . . . 14  |-  ( A  e.  ( R  \  {  .0.  } )  ->  A  =/=  .0.  )
2812, 27syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  A  =/=  .0.  )
2928a1d 24 . . . . . . . . . . . 12  |-  ( ph  ->  ( H  =/=  ( 0g `  D )  ->  A  =/=  .0.  ) )
3029necon4d 2673 . . . . . . . . . . 11  |-  ( ph  ->  ( A  =  .0. 
->  H  =  ( 0g `  D ) ) )
3126, 30sylbid 208 . . . . . . . . . 10  |-  ( ph  ->  ( A  =  ( 0g `  (Scalar `  D ) )  ->  H  =  ( 0g `  D ) ) )
32 idd 23 . . . . . . . . . 10  |-  ( ph  ->  ( H  =  ( 0g `  D )  ->  H  =  ( 0g `  D ) ) )
3331, 32jaod 371 . . . . . . . . 9  |-  ( ph  ->  ( ( A  =  ( 0g `  (Scalar `  D ) )  \/  H  =  ( 0g
`  D ) )  ->  H  =  ( 0g `  D ) ) )
3421, 33sylbid 208 . . . . . . . 8  |-  ( ph  ->  ( ( A  .x.  H )  =  ( 0g `  D )  ->  H  =  ( 0g `  D ) ) )
352, 34sylbid 208 . . . . . . 7  |-  ( ph  ->  ( G  =  ( 0g `  D )  ->  H  =  ( 0g `  D ) ) )
36 nne 2611 . . . . . . 7  |-  ( -.  H  =/=  ( 0g
`  D )  <->  H  =  ( 0g `  D ) )
3735, 36syl6ibr 220 . . . . . 6  |-  ( ph  ->  ( G  =  ( 0g `  D )  ->  -.  H  =/=  ( 0g `  D ) ) )
3837con3d 128 . . . . 5  |-  ( ph  ->  ( -.  -.  H  =/=  ( 0g `  D
)  ->  -.  G  =  ( 0g `  D ) ) )
3938orrd 369 . . . 4  |-  ( ph  ->  ( -.  H  =/=  ( 0g `  D
)  \/  -.  G  =  ( 0g `  D ) ) )
40 ianor 476 . . . 4  |-  ( -.  ( H  =/=  ( 0g `  D )  /\  G  =  ( 0g `  D ) )  <->  ( -.  H  =/=  ( 0g `  D )  \/  -.  G  =  ( 0g `  D ) ) )
4139, 40sylibr 205 . . 3  |-  ( ph  ->  -.  ( H  =/=  ( 0g `  D
)  /\  G  =  ( 0g `  D ) ) )
42 df-pss 3322 . . . . . 6  |-  ( ( K `  H ) 
C.  ( K `  G )  <->  ( ( K `  H )  C_  ( K `  G
)  /\  ( K `  H )  =/=  ( K `  G )
) )
43 lkreq.k . . . . . . 7  |-  K  =  (LKer `  W )
4418, 14, 15, 9, 4, 24, 13, 19ldualvscl 30035 . . . . . . . 8  |-  ( ph  ->  ( A  .x.  H
)  e.  F )
451, 44eqeltrd 2516 . . . . . . 7  |-  ( ph  ->  G  e.  F )
4618, 43, 9, 8, 10, 19, 45lkrpssN 30059 . . . . . 6  |-  ( ph  ->  ( ( K `  H )  C.  ( K `  G )  <->  ( H  =/=  ( 0g
`  D )  /\  G  =  ( 0g `  D ) ) ) )
4742, 46syl5rbbr 253 . . . . 5  |-  ( ph  ->  ( ( H  =/=  ( 0g `  D
)  /\  G  =  ( 0g `  D ) )  <->  ( ( K `
 H )  C_  ( K `  G )  /\  ( K `  H )  =/=  ( K `  G )
) ) )
4814, 15, 18, 43, 9, 4, 10, 19, 13lkrss 30064 . . . . . . 7  |-  ( ph  ->  ( K `  H
)  C_  ( K `  ( A  .x.  H
) ) )
491fveq2d 5761 . . . . . . 7  |-  ( ph  ->  ( K `  G
)  =  ( K `
 ( A  .x.  H ) ) )
5048, 49sseqtr4d 3371 . . . . . 6  |-  ( ph  ->  ( K `  H
)  C_  ( K `  G ) )
5150biantrurd 496 . . . . 5  |-  ( ph  ->  ( ( K `  H )  =/=  ( K `  G )  <->  ( ( K `  H
)  C_  ( K `  G )  /\  ( K `  H )  =/=  ( K `  G
) ) ) )
5247, 51bitr4d 249 . . . 4  |-  ( ph  ->  ( ( H  =/=  ( 0g `  D
)  /\  G  =  ( 0g `  D ) )  <->  ( K `  H )  =/=  ( K `  G )
) )
5352necon2bbid 2668 . . 3  |-  ( ph  ->  ( ( K `  H )  =  ( K `  G )  <->  -.  ( H  =/=  ( 0g `  D )  /\  G  =  ( 0g `  D ) ) ) )
5441, 53mpbird 225 . 2  |-  ( ph  ->  ( K `  H
)  =  ( K `
 G ) )
5554eqcomd 2447 1  |-  ( ph  ->  ( K `  G
)  =  ( K `
 H ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1727    =/= wne 2605    \ cdif 3303    C_ wss 3306    C. wpss 3307   {csn 3838   ` cfv 5483  (class class class)co 6110   Basecbs 13500  Scalarcsca 13563   .scvsca 13564   0gc0g 13754   LModclmod 15981   LVecclvec 16205  LFnlclfn 29953  LKerclk 29981  LDualcld 30019
This theorem is referenced by:  lkrlspeqN  30067  lcdlkreq2N  32519
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 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098
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 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-of 6334  df-1st 6378  df-2nd 6379  df-tpos 6508  df-riota 6578  df-recs 6662  df-rdg 6697  df-1o 6753  df-oadd 6757  df-er 6934  df-map 7049  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-nn 10032  df-2 10089  df-3 10090  df-4 10091  df-5 10092  df-6 10093  df-n0 10253  df-z 10314  df-uz 10520  df-fz 11075  df-struct 13502  df-ndx 13503  df-slot 13504  df-base 13505  df-sets 13506  df-ress 13507  df-plusg 13573  df-mulr 13574  df-sca 13576  df-vsca 13577  df-0g 13758  df-mnd 14721  df-submnd 14770  df-grp 14843  df-minusg 14844  df-sbg 14845  df-subg 14972  df-cntz 15147  df-lsm 15301  df-cmn 15445  df-abl 15446  df-mgp 15680  df-rng 15694  df-ur 15696  df-oppr 15759  df-dvdsr 15777  df-unit 15778  df-invr 15808  df-drng 15868  df-lmod 15983  df-lss 16040  df-lsp 16079  df-lvec 16206  df-lshyp 29873  df-lfl 29954  df-lkr 29982  df-ldual 30020
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