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Theorem lkrscss 33049
Description: The kernel of a scalar product of a functional includes the kernel of the functional. (The inclusion is proper for the zero product and equality otherwise.) (Contributed by NM, 9-Oct-2014.)
Hypotheses
Ref Expression
lkrsc.v  |-  V  =  ( Base `  W
)
lkrsc.d  |-  D  =  (Scalar `  W )
lkrsc.k  |-  K  =  ( Base `  D
)
lkrsc.t  |-  .x.  =  ( .r `  D )
lkrsc.f  |-  F  =  (LFnl `  W )
lkrsc.l  |-  L  =  (LKer `  W )
lkrsc.w  |-  ( ph  ->  W  e.  LVec )
lkrsc.g  |-  ( ph  ->  G  e.  F )
lkrsc.r  |-  ( ph  ->  R  e.  K )
Assertion
Ref Expression
lkrscss  |-  ( ph  ->  ( L `  G
)  C_  ( L `  ( G  oF  .x.  ( V  X.  { R } ) ) ) )

Proof of Theorem lkrscss
StepHypRef Expression
1 lkrsc.v . . . . . 6  |-  V  =  ( Base `  W
)
2 lkrsc.f . . . . . 6  |-  F  =  (LFnl `  W )
3 lkrsc.l . . . . . 6  |-  L  =  (LKer `  W )
4 lkrsc.w . . . . . . 7  |-  ( ph  ->  W  e.  LVec )
5 lveclmod 17293 . . . . . . 7  |-  ( W  e.  LVec  ->  W  e. 
LMod )
64, 5syl 16 . . . . . 6  |-  ( ph  ->  W  e.  LMod )
7 lkrsc.g . . . . . 6  |-  ( ph  ->  G  e.  F )
81, 2, 3, 6, 7lkrssv 33047 . . . . 5  |-  ( ph  ->  ( L `  G
)  C_  V )
9 lkrsc.d . . . . . . . 8  |-  D  =  (Scalar `  W )
10 lkrsc.k . . . . . . . 8  |-  K  =  ( Base `  D
)
11 lkrsc.t . . . . . . . 8  |-  .x.  =  ( .r `  D )
12 eqid 2451 . . . . . . . 8  |-  ( 0g
`  D )  =  ( 0g `  D
)
131, 9, 2, 10, 11, 12, 6, 7lfl0sc 33033 . . . . . . 7  |-  ( ph  ->  ( G  oF  .x.  ( V  X.  { ( 0g `  D ) } ) )  =  ( V  X.  { ( 0g
`  D ) } ) )
1413fveq2d 5793 . . . . . 6  |-  ( ph  ->  ( L `  ( G  oF  .x.  ( V  X.  { ( 0g
`  D ) } ) ) )  =  ( L `  ( V  X.  { ( 0g
`  D ) } ) ) )
15 eqid 2451 . . . . . . 7  |-  ( V  X.  { ( 0g
`  D ) } )  =  ( V  X.  { ( 0g
`  D ) } )
169, 12, 1, 2lfl0f 33020 . . . . . . . . 9  |-  ( W  e.  LMod  ->  ( V  X.  { ( 0g
`  D ) } )  e.  F )
176, 16syl 16 . . . . . . . 8  |-  ( ph  ->  ( V  X.  {
( 0g `  D
) } )  e.  F )
189, 12, 1, 2, 3lkr0f 33045 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  ( V  X.  { ( 0g
`  D ) } )  e.  F )  ->  ( ( L `
 ( V  X.  { ( 0g `  D ) } ) )  =  V  <->  ( V  X.  { ( 0g `  D ) } )  =  ( V  X.  { ( 0g `  D ) } ) ) )
196, 17, 18syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( ( L `  ( V  X.  { ( 0g `  D ) } ) )  =  V  <->  ( V  X.  { ( 0g `  D ) } )  =  ( V  X.  { ( 0g `  D ) } ) ) )
2015, 19mpbiri 233 . . . . . 6  |-  ( ph  ->  ( L `  ( V  X.  { ( 0g
`  D ) } ) )  =  V )
2114, 20eqtr2d 2493 . . . . 5  |-  ( ph  ->  V  =  ( L `
 ( G  oF  .x.  ( V  X.  { ( 0g `  D ) } ) ) ) )
228, 21sseqtrd 3490 . . . 4  |-  ( ph  ->  ( L `  G
)  C_  ( L `  ( G  oF  .x.  ( V  X.  { ( 0g `  D ) } ) ) ) )
2322adantr 465 . . 3  |-  ( (
ph  /\  R  =  ( 0g `  D ) )  ->  ( L `  G )  C_  ( L `  ( G  oF  .x.  ( V  X.  { ( 0g
`  D ) } ) ) ) )
24 sneq 3985 . . . . . . 7  |-  ( R  =  ( 0g `  D )  ->  { R }  =  { ( 0g `  D ) } )
2524xpeq2d 4962 . . . . . 6  |-  ( R  =  ( 0g `  D )  ->  ( V  X.  { R }
)  =  ( V  X.  { ( 0g
`  D ) } ) )
2625oveq2d 6206 . . . . 5  |-  ( R  =  ( 0g `  D )  ->  ( G  oF  .x.  ( V  X.  { R }
) )  =  ( G  oF  .x.  ( V  X.  { ( 0g `  D ) } ) ) )
2726fveq2d 5793 . . . 4  |-  ( R  =  ( 0g `  D )  ->  ( L `  ( G  oF  .x.  ( V  X.  { R }
) ) )  =  ( L `  ( G  oF  .x.  ( V  X.  { ( 0g
`  D ) } ) ) ) )
2827adantl 466 . . 3  |-  ( (
ph  /\  R  =  ( 0g `  D ) )  ->  ( L `  ( G  oF  .x.  ( V  X.  { R } ) ) )  =  ( L `
 ( G  oF  .x.  ( V  X.  { ( 0g `  D ) } ) ) ) )
2923, 28sseqtr4d 3491 . 2  |-  ( (
ph  /\  R  =  ( 0g `  D ) )  ->  ( L `  G )  C_  ( L `  ( G  oF  .x.  ( V  X.  { R }
) ) ) )
304adantr 465 . . . 4  |-  ( (
ph  /\  R  =/=  ( 0g `  D ) )  ->  W  e.  LVec )
317adantr 465 . . . 4  |-  ( (
ph  /\  R  =/=  ( 0g `  D ) )  ->  G  e.  F )
32 lkrsc.r . . . . 5  |-  ( ph  ->  R  e.  K )
3332adantr 465 . . . 4  |-  ( (
ph  /\  R  =/=  ( 0g `  D ) )  ->  R  e.  K )
34 simpr 461 . . . 4  |-  ( (
ph  /\  R  =/=  ( 0g `  D ) )  ->  R  =/=  ( 0g `  D ) )
351, 9, 10, 11, 2, 3, 30, 31, 33, 12, 34lkrsc 33048 . . 3  |-  ( (
ph  /\  R  =/=  ( 0g `  D ) )  ->  ( L `  ( G  oF  .x.  ( V  X.  { R } ) ) )  =  ( L `
 G ) )
36 eqimss2 3507 . . 3  |-  ( ( L `  ( G  oF  .x.  ( V  X.  { R }
) ) )  =  ( L `  G
)  ->  ( L `  G )  C_  ( L `  ( G  oF  .x.  ( V  X.  { R }
) ) ) )
3735, 36syl 16 . 2  |-  ( (
ph  /\  R  =/=  ( 0g `  D ) )  ->  ( L `  G )  C_  ( L `  ( G  oF  .x.  ( V  X.  { R }
) ) ) )
3829, 37pm2.61dane 2766 1  |-  ( ph  ->  ( L `  G
)  C_  ( L `  ( G  oF  .x.  ( V  X.  { R } ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644    C_ wss 3426   {csn 3975    X. cxp 4936   ` cfv 5516  (class class class)co 6190    oFcof 6418   Basecbs 14276   .rcmulr 14341  Scalarcsca 14343   0gc0g 14480   LModclmod 17054   LVecclvec 17289  LFnlclfn 33008  LKerclk 33036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-of 6420  df-om 6577  df-1st 6677  df-2nd 6678  df-tpos 6845  df-recs 6932  df-rdg 6966  df-er 7201  df-map 7316  df-en 7411  df-dom 7412  df-sdom 7413  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-2 10481  df-3 10482  df-ndx 14279  df-slot 14280  df-base 14281  df-sets 14282  df-ress 14283  df-plusg 14353  df-mulr 14354  df-0g 14482  df-mnd 15517  df-grp 15647  df-minusg 15648  df-sbg 15649  df-mgp 16697  df-ur 16709  df-rng 16753  df-oppr 16821  df-dvdsr 16839  df-unit 16840  df-invr 16870  df-drng 16940  df-lmod 17056  df-lss 17120  df-lvec 17290  df-lfl 33009  df-lkr 33037
This theorem is referenced by:  lfl1dim  33072  lfl1dim2N  33073  lkrss  33119
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