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Theorem lkrscss 29823
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  o F 
.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 16170 . . . . . . 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 29821 . . . . 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 2435 . . . . . . . 8  |-  ( 0g
`  D )  =  ( 0g `  D
)
131, 9, 2, 10, 11, 12, 6, 7lfl0sc 29807 . . . . . . 7  |-  ( ph  ->  ( G  o F 
.x.  ( V  X.  { ( 0g `  D ) } ) )  =  ( V  X.  { ( 0g
`  D ) } ) )
1413fveq2d 5724 . . . . . 6  |-  ( ph  ->  ( L `  ( G  o F  .x.  ( V  X.  { ( 0g
`  D ) } ) ) )  =  ( L `  ( V  X.  { ( 0g
`  D ) } ) ) )
15 eqid 2435 . . . . . . 7  |-  ( V  X.  { ( 0g
`  D ) } )  =  ( V  X.  { ( 0g
`  D ) } )
169, 12, 1, 2lfl0f 29794 . . . . . . . . 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 29819 . . . . . . . 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 643 . . . . . . 7  |-  ( ph  ->  ( ( L `  ( V  X.  { ( 0g `  D ) } ) )  =  V  <->  ( V  X.  { ( 0g `  D ) } )  =  ( V  X.  { ( 0g `  D ) } ) ) )
2015, 19mpbiri 225 . . . . . 6  |-  ( ph  ->  ( L `  ( V  X.  { ( 0g
`  D ) } ) )  =  V )
2114, 20eqtr2d 2468 . . . . 5  |-  ( ph  ->  V  =  ( L `
 ( G  o F  .x.  ( V  X.  { ( 0g `  D ) } ) ) ) )
228, 21sseqtrd 3376 . . . 4  |-  ( ph  ->  ( L `  G
)  C_  ( L `  ( G  o F 
.x.  ( V  X.  { ( 0g `  D ) } ) ) ) )
2322adantr 452 . . 3  |-  ( (
ph  /\  R  =  ( 0g `  D ) )  ->  ( L `  G )  C_  ( L `  ( G  o F  .x.  ( V  X.  { ( 0g
`  D ) } ) ) ) )
24 sneq 3817 . . . . . . 7  |-  ( R  =  ( 0g `  D )  ->  { R }  =  { ( 0g `  D ) } )
2524xpeq2d 4894 . . . . . 6  |-  ( R  =  ( 0g `  D )  ->  ( V  X.  { R }
)  =  ( V  X.  { ( 0g
`  D ) } ) )
2625oveq2d 6089 . . . . 5  |-  ( R  =  ( 0g `  D )  ->  ( G  o F  .x.  ( V  X.  { R }
) )  =  ( G  o F  .x.  ( V  X.  { ( 0g `  D ) } ) ) )
2726fveq2d 5724 . . . 4  |-  ( R  =  ( 0g `  D )  ->  ( L `  ( G  o F  .x.  ( V  X.  { R }
) ) )  =  ( L `  ( G  o F  .x.  ( V  X.  { ( 0g
`  D ) } ) ) ) )
2827adantl 453 . . 3  |-  ( (
ph  /\  R  =  ( 0g `  D ) )  ->  ( L `  ( G  o F 
.x.  ( V  X.  { R } ) ) )  =  ( L `
 ( G  o F  .x.  ( V  X.  { ( 0g `  D ) } ) ) ) )
2923, 28sseqtr4d 3377 . 2  |-  ( (
ph  /\  R  =  ( 0g `  D ) )  ->  ( L `  G )  C_  ( L `  ( G  o F  .x.  ( V  X.  { R }
) ) ) )
304adantr 452 . . . 4  |-  ( (
ph  /\  R  =/=  ( 0g `  D ) )  ->  W  e.  LVec )
317adantr 452 . . . 4  |-  ( (
ph  /\  R  =/=  ( 0g `  D ) )  ->  G  e.  F )
32 lkrsc.r . . . . 5  |-  ( ph  ->  R  e.  K )
3332adantr 452 . . . 4  |-  ( (
ph  /\  R  =/=  ( 0g `  D ) )  ->  R  e.  K )
34 simpr 448 . . . 4  |-  ( (
ph  /\  R  =/=  ( 0g `  D ) )  ->  R  =/=  ( 0g `  D ) )
351, 9, 10, 11, 2, 3, 30, 31, 33, 12, 34lkrsc 29822 . . 3  |-  ( (
ph  /\  R  =/=  ( 0g `  D ) )  ->  ( L `  ( G  o F 
.x.  ( V  X.  { R } ) ) )  =  ( L `
 G ) )
36 eqimss2 3393 . . 3  |-  ( ( L `  ( G  o F  .x.  ( V  X.  { R }
) ) )  =  ( L `  G
)  ->  ( L `  G )  C_  ( L `  ( G  o F  .x.  ( V  X.  { R }
) ) ) )
3735, 36syl 16 . 2  |-  ( (
ph  /\  R  =/=  ( 0g `  D ) )  ->  ( L `  G )  C_  ( L `  ( G  o F  .x.  ( V  X.  { R }
) ) ) )
3829, 37pm2.61dane 2676 1  |-  ( ph  ->  ( L `  G
)  C_  ( L `  ( G  o F 
.x.  ( V  X.  { R } ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598    C_ wss 3312   {csn 3806    X. cxp 4868   ` cfv 5446  (class class class)co 6073    o Fcof 6295   Basecbs 13461   .rcmulr 13522  Scalarcsca 13524   0gc0g 13715   LModclmod 15942   LVecclvec 16166  LFnlclfn 29782  LKerclk 29810
This theorem is referenced by:  lfl1dim  29846  lfl1dim2N  29847  lkrss  29893
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-0g 13719  df-mnd 14682  df-grp 14804  df-minusg 14805  df-sbg 14806  df-mgp 15641  df-rng 15655  df-ur 15657  df-oppr 15720  df-dvdsr 15738  df-unit 15739  df-invr 15769  df-drng 15829  df-lmod 15944  df-lss 16001  df-lvec 16167  df-lfl 29783  df-lkr 29811
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