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Theorem lkreqN 35038
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 2459 . . . . . . . 8  |-  ( ph  ->  ( G  =  ( 0g `  D )  <-> 
( A  .x.  H
)  =  ( 0g
`  D ) ) )
3 eqid 2457 . . . . . . . . . 10  |-  ( Base `  D )  =  (
Base `  D )
4 lkreq.t . . . . . . . . . 10  |-  .x.  =  ( .s `  D )
5 eqid 2457 . . . . . . . . . 10  |-  (Scalar `  D )  =  (Scalar `  D )
6 eqid 2457 . . . . . . . . . 10  |-  ( Base `  (Scalar `  D )
)  =  ( Base `  (Scalar `  D )
)
7 eqid 2457 . . . . . . . . . 10  |-  ( 0g
`  (Scalar `  D )
)  =  ( 0g
`  (Scalar `  D )
)
8 eqid 2457 . . . . . . . . . 10  |-  ( 0g
`  D )  =  ( 0g `  D
)
9 lkreq.d . . . . . . . . . . 11  |-  D  =  (LDual `  W )
10 lkreq.w . . . . . . . . . . 11  |-  ( ph  ->  W  e.  LVec )
119, 10lduallvec 35022 . . . . . . . . . 10  |-  ( ph  ->  D  e.  LVec )
12 lkreq.a . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  ( R 
\  {  .0.  }
) )
1312eldifad 3483 . . . . . . . . . . 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 35001 . . . . . . . . . . 11  |-  ( ph  ->  ( Base `  (Scalar `  D ) )  =  R )
1713, 16eleqtrrd 2548 . . . . . . . . . 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 34995 . . . . . . . . . 10  |-  ( ph  ->  H  e.  ( Base `  D ) )
213, 4, 5, 6, 7, 8, 11, 17, 20lvecvs0or 17881 . . . . . . . . 9  |-  ( ph  ->  ( ( A  .x.  H )  =  ( 0g `  D )  <-> 
( A  =  ( 0g `  (Scalar `  D ) )  \/  H  =  ( 0g
`  D ) ) ) )
22 lkreq.o . . . . . . . . . . . . 13  |-  .0.  =  ( 0g `  S )
23 lveclmod 17879 . . . . . . . . . . . . . 14  |-  ( W  e.  LVec  ->  W  e. 
LMod )
2410, 23syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  W  e.  LMod )
2514, 22, 9, 5, 7, 24ldual0 35015 . . . . . . . . . . . 12  |-  ( ph  ->  ( 0g `  (Scalar `  D ) )  =  .0.  )
2625eqeq2d 2471 . . . . . . . . . . 11  |-  ( ph  ->  ( A  =  ( 0g `  (Scalar `  D ) )  <->  A  =  .0.  ) )
27 eldifsni 4158 . . . . . . . . . . . . . 14  |-  ( A  e.  ( R  \  {  .0.  } )  ->  A  =/=  .0.  )
2812, 27syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  A  =/=  .0.  )
2928a1d 25 . . . . . . . . . . . 12  |-  ( ph  ->  ( H  =/=  ( 0g `  D )  ->  A  =/=  .0.  ) )
3029necon4d 2684 . . . . . . . . . . 11  |-  ( ph  ->  ( A  =  .0. 
->  H  =  ( 0g `  D ) ) )
3126, 30sylbid 215 . . . . . . . . . 10  |-  ( ph  ->  ( A  =  ( 0g `  (Scalar `  D ) )  ->  H  =  ( 0g `  D ) ) )
32 idd 24 . . . . . . . . . 10  |-  ( ph  ->  ( H  =  ( 0g `  D )  ->  H  =  ( 0g `  D ) ) )
3331, 32jaod 380 . . . . . . . . 9  |-  ( ph  ->  ( ( A  =  ( 0g `  (Scalar `  D ) )  \/  H  =  ( 0g
`  D ) )  ->  H  =  ( 0g `  D ) ) )
3421, 33sylbid 215 . . . . . . . 8  |-  ( ph  ->  ( ( A  .x.  H )  =  ( 0g `  D )  ->  H  =  ( 0g `  D ) ) )
352, 34sylbid 215 . . . . . . 7  |-  ( ph  ->  ( G  =  ( 0g `  D )  ->  H  =  ( 0g `  D ) ) )
36 nne 2658 . . . . . . 7  |-  ( -.  H  =/=  ( 0g
`  D )  <->  H  =  ( 0g `  D ) )
3735, 36syl6ibr 227 . . . . . 6  |-  ( ph  ->  ( G  =  ( 0g `  D )  ->  -.  H  =/=  ( 0g `  D ) ) )
3837con3d 133 . . . . 5  |-  ( ph  ->  ( -.  -.  H  =/=  ( 0g `  D
)  ->  -.  G  =  ( 0g `  D ) ) )
3938orrd 378 . . . 4  |-  ( ph  ->  ( -.  H  =/=  ( 0g `  D
)  \/  -.  G  =  ( 0g `  D ) ) )
40 ianor 488 . . . 4  |-  ( -.  ( H  =/=  ( 0g `  D )  /\  G  =  ( 0g `  D ) )  <->  ( -.  H  =/=  ( 0g `  D )  \/  -.  G  =  ( 0g `  D ) ) )
4139, 40sylibr 212 . . 3  |-  ( ph  ->  -.  ( H  =/=  ( 0g `  D
)  /\  G  =  ( 0g `  D ) ) )
42 df-pss 3487 . . . . . 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 35007 . . . . . . . 8  |-  ( ph  ->  ( A  .x.  H
)  e.  F )
451, 44eqeltrd 2545 . . . . . . 7  |-  ( ph  ->  G  e.  F )
4618, 43, 9, 8, 10, 19, 45lkrpssN 35031 . . . . . 6  |-  ( ph  ->  ( ( K `  H )  C.  ( K `  G )  <->  ( H  =/=  ( 0g
`  D )  /\  G  =  ( 0g `  D ) ) ) )
4742, 46syl5rbbr 260 . . . . 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 35036 . . . . . . 7  |-  ( ph  ->  ( K `  H
)  C_  ( K `  ( A  .x.  H
) ) )
491fveq2d 5876 . . . . . . 7  |-  ( ph  ->  ( K `  G
)  =  ( K `
 ( A  .x.  H ) ) )
5048, 49sseqtr4d 3536 . . . . . 6  |-  ( ph  ->  ( K `  H
)  C_  ( K `  G ) )
5150biantrurd 508 . . . . 5  |-  ( ph  ->  ( ( K `  H )  =/=  ( K `  G )  <->  ( ( K `  H
)  C_  ( K `  G )  /\  ( K `  H )  =/=  ( K `  G
) ) ) )
5247, 51bitr4d 256 . . . 4  |-  ( ph  ->  ( ( H  =/=  ( 0g `  D
)  /\  G  =  ( 0g `  D ) )  <->  ( K `  H )  =/=  ( K `  G )
) )
5352necon2bbid 2713 . . 3  |-  ( ph  ->  ( ( K `  H )  =  ( K `  G )  <->  -.  ( H  =/=  ( 0g `  D )  /\  G  =  ( 0g `  D ) ) ) )
5441, 53mpbird 232 . 2  |-  ( ph  ->  ( K `  H
)  =  ( K `
 G ) )
5554eqcomd 2465 1  |-  ( ph  ->  ( K `  G
)  =  ( K `
 H ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1395    e. wcel 1819    =/= wne 2652    \ cdif 3468    C_ wss 3471    C. wpss 3472   {csn 4032   ` cfv 5594  (class class class)co 6296   Basecbs 14644  Scalarcsca 14715   .scvsca 14716   0gc0g 14857   LModclmod 17639   LVecclvec 17875  LFnlclfn 34925  LKerclk 34953  LDualcld 34991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-tpos 6973  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-struct 14646  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-sca 14728  df-vsca 14729  df-0g 14859  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-submnd 16094  df-grp 16184  df-minusg 16185  df-sbg 16186  df-subg 16325  df-cntz 16482  df-lsm 16783  df-cmn 16927  df-abl 16928  df-mgp 17269  df-ur 17281  df-ring 17327  df-oppr 17399  df-dvdsr 17417  df-unit 17418  df-invr 17448  df-drng 17525  df-lmod 17641  df-lss 17706  df-lsp 17745  df-lvec 17876  df-lshyp 34845  df-lfl 34926  df-lkr 34954  df-ldual 34992
This theorem is referenced by:  lkrlspeqN  35039  lcdlkreq2N  37493
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