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Theorem lkreqN 29665
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 2420 . . . . . . . 8  |-  ( ph  ->  ( G  =  ( 0g `  D )  <-> 
( A  .x.  H
)  =  ( 0g
`  D ) ) )
3 eqid 2412 . . . . . . . . . 10  |-  ( Base `  D )  =  (
Base `  D )
4 lkreq.t . . . . . . . . . 10  |-  .x.  =  ( .s `  D )
5 eqid 2412 . . . . . . . . . 10  |-  (Scalar `  D )  =  (Scalar `  D )
6 eqid 2412 . . . . . . . . . 10  |-  ( Base `  (Scalar `  D )
)  =  ( Base `  (Scalar `  D )
)
7 eqid 2412 . . . . . . . . . 10  |-  ( 0g
`  (Scalar `  D )
)  =  ( 0g
`  (Scalar `  D )
)
8 eqid 2412 . . . . . . . . . 10  |-  ( 0g
`  D )  =  ( 0g `  D
)
9 lkreq.d . . . . . . . . . . 11  |-  D  =  (LDual `  W )
10 lkreq.w . . . . . . . . . . 11  |-  ( ph  ->  W  e.  LVec )
119, 10lduallvec 29649 . . . . . . . . . 10  |-  ( ph  ->  D  e.  LVec )
12 lkreq.a . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  ( R 
\  {  .0.  }
) )
1312eldifad 3300 . . . . . . . . . . 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 29628 . . . . . . . . . . 11  |-  ( ph  ->  ( Base `  (Scalar `  D ) )  =  R )
1713, 16eleqtrrd 2489 . . . . . . . . . 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 29622 . . . . . . . . . 10  |-  ( ph  ->  H  e.  ( Base `  D ) )
213, 4, 5, 6, 7, 8, 11, 17, 20lvecvs0or 16143 . . . . . . . . 9  |-  ( ph  ->  ( ( A  .x.  H )  =  ( 0g `  D )  <-> 
( A  =  ( 0g `  (Scalar `  D ) )  \/  H  =  ( 0g
`  D ) ) ) )
22 lkreq.o . . . . . . . . . . . . 13  |-  .0.  =  ( 0g `  S )
23 lveclmod 16141 . . . . . . . . . . . . . 14  |-  ( W  e.  LVec  ->  W  e. 
LMod )
2410, 23syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  W  e.  LMod )
2514, 22, 9, 5, 7, 24ldual0 29642 . . . . . . . . . . . 12  |-  ( ph  ->  ( 0g `  (Scalar `  D ) )  =  .0.  )
2625eqeq2d 2423 . . . . . . . . . . 11  |-  ( ph  ->  ( A  =  ( 0g `  (Scalar `  D ) )  <->  A  =  .0.  ) )
27 eldifsni 3896 . . . . . . . . . . . . . 14  |-  ( A  e.  ( R  \  {  .0.  } )  ->  A  =/=  .0.  )
2812, 27syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  A  =/=  .0.  )
2928a1d 23 . . . . . . . . . . . 12  |-  ( ph  ->  ( H  =/=  ( 0g `  D )  ->  A  =/=  .0.  ) )
3029necon4d 2638 . . . . . . . . . . 11  |-  ( ph  ->  ( A  =  .0. 
->  H  =  ( 0g `  D ) ) )
3126, 30sylbid 207 . . . . . . . . . 10  |-  ( ph  ->  ( A  =  ( 0g `  (Scalar `  D ) )  ->  H  =  ( 0g `  D ) ) )
32 idd 22 . . . . . . . . . 10  |-  ( ph  ->  ( H  =  ( 0g `  D )  ->  H  =  ( 0g `  D ) ) )
3331, 32jaod 370 . . . . . . . . 9  |-  ( ph  ->  ( ( A  =  ( 0g `  (Scalar `  D ) )  \/  H  =  ( 0g
`  D ) )  ->  H  =  ( 0g `  D ) ) )
3421, 33sylbid 207 . . . . . . . 8  |-  ( ph  ->  ( ( A  .x.  H )  =  ( 0g `  D )  ->  H  =  ( 0g `  D ) ) )
352, 34sylbid 207 . . . . . . 7  |-  ( ph  ->  ( G  =  ( 0g `  D )  ->  H  =  ( 0g `  D ) ) )
36 nne 2579 . . . . . . 7  |-  ( -.  H  =/=  ( 0g
`  D )  <->  H  =  ( 0g `  D ) )
3735, 36syl6ibr 219 . . . . . 6  |-  ( ph  ->  ( G  =  ( 0g `  D )  ->  -.  H  =/=  ( 0g `  D ) ) )
3837con3d 127 . . . . 5  |-  ( ph  ->  ( -.  -.  H  =/=  ( 0g `  D
)  ->  -.  G  =  ( 0g `  D ) ) )
3938orrd 368 . . . 4  |-  ( ph  ->  ( -.  H  =/=  ( 0g `  D
)  \/  -.  G  =  ( 0g `  D ) ) )
40 ianor 475 . . . 4  |-  ( -.  ( H  =/=  ( 0g `  D )  /\  G  =  ( 0g `  D ) )  <->  ( -.  H  =/=  ( 0g `  D )  \/  -.  G  =  ( 0g `  D ) ) )
4139, 40sylibr 204 . . 3  |-  ( ph  ->  -.  ( H  =/=  ( 0g `  D
)  /\  G  =  ( 0g `  D ) ) )
42 df-pss 3304 . . . . . 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 29634 . . . . . . . 8  |-  ( ph  ->  ( A  .x.  H
)  e.  F )
451, 44eqeltrd 2486 . . . . . . 7  |-  ( ph  ->  G  e.  F )
4618, 43, 9, 8, 10, 19, 45lkrpssN 29658 . . . . . 6  |-  ( ph  ->  ( ( K `  H )  C.  ( K `  G )  <->  ( H  =/=  ( 0g
`  D )  /\  G  =  ( 0g `  D ) ) ) )
4742, 46syl5rbbr 252 . . . . 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 29663 . . . . . . 7  |-  ( ph  ->  ( K `  H
)  C_  ( K `  ( A  .x.  H
) ) )
491fveq2d 5699 . . . . . . 7  |-  ( ph  ->  ( K `  G
)  =  ( K `
 ( A  .x.  H ) ) )
5048, 49sseqtr4d 3353 . . . . . 6  |-  ( ph  ->  ( K `  H
)  C_  ( K `  G ) )
5150biantrurd 495 . . . . 5  |-  ( ph  ->  ( ( K `  H )  =/=  ( K `  G )  <->  ( ( K `  H
)  C_  ( K `  G )  /\  ( K `  H )  =/=  ( K `  G
) ) ) )
5247, 51bitr4d 248 . . . 4  |-  ( ph  ->  ( ( H  =/=  ( 0g `  D
)  /\  G  =  ( 0g `  D ) )  <->  ( K `  H )  =/=  ( K `  G )
) )
5352necon2bbid 2633 . . 3  |-  ( ph  ->  ( ( K `  H )  =  ( K `  G )  <->  -.  ( H  =/=  ( 0g `  D )  /\  G  =  ( 0g `  D ) ) ) )
5441, 53mpbird 224 . 2  |-  ( ph  ->  ( K `  H
)  =  ( K `
 G ) )
5554eqcomd 2417 1  |-  ( ph  ->  ( K `  G
)  =  ( K `
 H ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2575    \ cdif 3285    C_ wss 3288    C. wpss 3289   {csn 3782   ` cfv 5421  (class class class)co 6048   Basecbs 13432  Scalarcsca 13495   .scvsca 13496   0gc0g 13686   LModclmod 15913   LVecclvec 16137  LFnlclfn 29552  LKerclk 29580  LDualcld 29618
This theorem is referenced by:  lkrlspeqN  29666  lcdlkreq2N  32118
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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