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Theorem lkreqN 33985
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 2469 . . . . . . . 8  |-  ( ph  ->  ( G  =  ( 0g `  D )  <-> 
( A  .x.  H
)  =  ( 0g
`  D ) ) )
3 eqid 2467 . . . . . . . . . 10  |-  ( Base `  D )  =  (
Base `  D )
4 lkreq.t . . . . . . . . . 10  |-  .x.  =  ( .s `  D )
5 eqid 2467 . . . . . . . . . 10  |-  (Scalar `  D )  =  (Scalar `  D )
6 eqid 2467 . . . . . . . . . 10  |-  ( Base `  (Scalar `  D )
)  =  ( Base `  (Scalar `  D )
)
7 eqid 2467 . . . . . . . . . 10  |-  ( 0g
`  (Scalar `  D )
)  =  ( 0g
`  (Scalar `  D )
)
8 eqid 2467 . . . . . . . . . 10  |-  ( 0g
`  D )  =  ( 0g `  D
)
9 lkreq.d . . . . . . . . . . 11  |-  D  =  (LDual `  W )
10 lkreq.w . . . . . . . . . . 11  |-  ( ph  ->  W  e.  LVec )
119, 10lduallvec 33969 . . . . . . . . . 10  |-  ( ph  ->  D  e.  LVec )
12 lkreq.a . . . . . . . . . . . 12  |-  ( ph  ->  A  e.  ( R 
\  {  .0.  }
) )
1312eldifad 3488 . . . . . . . . . . 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 33948 . . . . . . . . . . 11  |-  ( ph  ->  ( Base `  (Scalar `  D ) )  =  R )
1713, 16eleqtrrd 2558 . . . . . . . . . 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 33942 . . . . . . . . . 10  |-  ( ph  ->  H  e.  ( Base `  D ) )
213, 4, 5, 6, 7, 8, 11, 17, 20lvecvs0or 17554 . . . . . . . . 9  |-  ( ph  ->  ( ( A  .x.  H )  =  ( 0g `  D )  <-> 
( A  =  ( 0g `  (Scalar `  D ) )  \/  H  =  ( 0g
`  D ) ) ) )
22 lkreq.o . . . . . . . . . . . . 13  |-  .0.  =  ( 0g `  S )
23 lveclmod 17552 . . . . . . . . . . . . . 14  |-  ( W  e.  LVec  ->  W  e. 
LMod )
2410, 23syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  W  e.  LMod )
2514, 22, 9, 5, 7, 24ldual0 33962 . . . . . . . . . . . 12  |-  ( ph  ->  ( 0g `  (Scalar `  D ) )  =  .0.  )
2625eqeq2d 2481 . . . . . . . . . . 11  |-  ( ph  ->  ( A  =  ( 0g `  (Scalar `  D ) )  <->  A  =  .0.  ) )
27 eldifsni 4153 . . . . . . . . . . . . . 14  |-  ( A  e.  ( R  \  {  .0.  } )  ->  A  =/=  .0.  )
2812, 27syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  A  =/=  .0.  )
2928a1d 25 . . . . . . . . . . . 12  |-  ( ph  ->  ( H  =/=  ( 0g `  D )  ->  A  =/=  .0.  ) )
3029necon4d 2694 . . . . . . . . . . 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 2668 . . . . . . 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 3492 . . . . . 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 33954 . . . . . . . 8  |-  ( ph  ->  ( A  .x.  H
)  e.  F )
451, 44eqeltrd 2555 . . . . . . 7  |-  ( ph  ->  G  e.  F )
4618, 43, 9, 8, 10, 19, 45lkrpssN 33978 . . . . . 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 33983 . . . . . . 7  |-  ( ph  ->  ( K `  H
)  C_  ( K `  ( A  .x.  H
) ) )
491fveq2d 5870 . . . . . . 7  |-  ( ph  ->  ( K `  G
)  =  ( K `
 ( A  .x.  H ) ) )
5048, 49sseqtr4d 3541 . . . . . 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 2723 . . 3  |-  ( ph  ->  ( ( K `  H )  =  ( K `  G )  <->  -.  ( H  =/=  ( 0g `  D )  /\  G  =  ( 0g `  D ) ) ) )
5441, 53mpbird 232 . 2  |-  ( ph  ->  ( K `  H
)  =  ( K `
 G ) )
5554eqcomd 2475 1  |-  ( ph  ->  ( K `  G
)  =  ( K `
 H ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662    \ cdif 3473    C_ wss 3476    C. wpss 3477   {csn 4027   ` cfv 5588  (class class class)co 6284   Basecbs 14490  Scalarcsca 14558   .scvsca 14559   0gc0g 14695   LModclmod 17312   LVecclvec 17548  LFnlclfn 33872  LKerclk 33900  LDualcld 33938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-tpos 6955  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-struct 14492  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-sca 14571  df-vsca 14572  df-0g 14697  df-mnd 15732  df-submnd 15787  df-grp 15867  df-minusg 15868  df-sbg 15869  df-subg 16003  df-cntz 16160  df-lsm 16462  df-cmn 16606  df-abl 16607  df-mgp 16944  df-ur 16956  df-rng 17002  df-oppr 17073  df-dvdsr 17091  df-unit 17092  df-invr 17122  df-drng 17198  df-lmod 17314  df-lss 17379  df-lsp 17418  df-lvec 17549  df-lshyp 33792  df-lfl 33873  df-lkr 33901  df-ldual 33939
This theorem is referenced by:  lkrlspeqN  33986  lcdlkreq2N  36438
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