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Theorem lkrscss 35239
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 17950 . . . . . . 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 35237 . . . . 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 2454 . . . . . . . 8  |-  ( 0g
`  D )  =  ( 0g `  D
)
131, 9, 2, 10, 11, 12, 6, 7lfl0sc 35223 . . . . . . 7  |-  ( ph  ->  ( G  oF  .x.  ( V  X.  { ( 0g `  D ) } ) )  =  ( V  X.  { ( 0g
`  D ) } ) )
1413fveq2d 5852 . . . . . 6  |-  ( ph  ->  ( L `  ( G  oF  .x.  ( V  X.  { ( 0g
`  D ) } ) ) )  =  ( L `  ( V  X.  { ( 0g
`  D ) } ) ) )
15 eqid 2454 . . . . . . 7  |-  ( V  X.  { ( 0g
`  D ) } )  =  ( V  X.  { ( 0g
`  D ) } )
169, 12, 1, 2lfl0f 35210 . . . . . . . . 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 35235 . . . . . . . 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 659 . . . . . . 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 2496 . . . . 5  |-  ( ph  ->  V  =  ( L `
 ( G  oF  .x.  ( V  X.  { ( 0g `  D ) } ) ) ) )
228, 21sseqtrd 3525 . . . 4  |-  ( ph  ->  ( L `  G
)  C_  ( L `  ( G  oF  .x.  ( V  X.  { ( 0g `  D ) } ) ) ) )
2322adantr 463 . . 3  |-  ( (
ph  /\  R  =  ( 0g `  D ) )  ->  ( L `  G )  C_  ( L `  ( G  oF  .x.  ( V  X.  { ( 0g
`  D ) } ) ) ) )
24 sneq 4026 . . . . . . 7  |-  ( R  =  ( 0g `  D )  ->  { R }  =  { ( 0g `  D ) } )
2524xpeq2d 5012 . . . . . 6  |-  ( R  =  ( 0g `  D )  ->  ( V  X.  { R }
)  =  ( V  X.  { ( 0g
`  D ) } ) )
2625oveq2d 6286 . . . . 5  |-  ( R  =  ( 0g `  D )  ->  ( G  oF  .x.  ( V  X.  { R }
) )  =  ( G  oF  .x.  ( V  X.  { ( 0g `  D ) } ) ) )
2726fveq2d 5852 . . . 4  |-  ( R  =  ( 0g `  D )  ->  ( L `  ( G  oF  .x.  ( V  X.  { R }
) ) )  =  ( L `  ( G  oF  .x.  ( V  X.  { ( 0g
`  D ) } ) ) ) )
2827adantl 464 . . 3  |-  ( (
ph  /\  R  =  ( 0g `  D ) )  ->  ( L `  ( G  oF  .x.  ( V  X.  { R } ) ) )  =  ( L `
 ( G  oF  .x.  ( V  X.  { ( 0g `  D ) } ) ) ) )
2923, 28sseqtr4d 3526 . 2  |-  ( (
ph  /\  R  =  ( 0g `  D ) )  ->  ( L `  G )  C_  ( L `  ( G  oF  .x.  ( V  X.  { R }
) ) ) )
304adantr 463 . . . 4  |-  ( (
ph  /\  R  =/=  ( 0g `  D ) )  ->  W  e.  LVec )
317adantr 463 . . . 4  |-  ( (
ph  /\  R  =/=  ( 0g `  D ) )  ->  G  e.  F )
32 lkrsc.r . . . . 5  |-  ( ph  ->  R  e.  K )
3332adantr 463 . . . 4  |-  ( (
ph  /\  R  =/=  ( 0g `  D ) )  ->  R  e.  K )
34 simpr 459 . . . 4  |-  ( (
ph  /\  R  =/=  ( 0g `  D ) )  ->  R  =/=  ( 0g `  D ) )
351, 9, 10, 11, 2, 3, 30, 31, 33, 12, 34lkrsc 35238 . . 3  |-  ( (
ph  /\  R  =/=  ( 0g `  D ) )  ->  ( L `  ( G  oF  .x.  ( V  X.  { R } ) ) )  =  ( L `
 G ) )
36 eqimss2 3542 . . 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 2772 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 367    = wceq 1398    e. wcel 1823    =/= wne 2649    C_ wss 3461   {csn 4016    X. cxp 4986   ` cfv 5570  (class class class)co 6270    oFcof 6511   Basecbs 14719   .rcmulr 14788  Scalarcsca 14790   0gc0g 14932   LModclmod 17710   LVecclvec 17946  LFnlclfn 35198  LKerclk 35226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-tpos 6947  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-mulr 14801  df-0g 14934  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-grp 16259  df-minusg 16260  df-sbg 16261  df-mgp 17340  df-ur 17352  df-ring 17398  df-oppr 17470  df-dvdsr 17488  df-unit 17489  df-invr 17519  df-drng 17596  df-lmod 17712  df-lss 17777  df-lvec 17947  df-lfl 35199  df-lkr 35227
This theorem is referenced by:  lfl1dim  35262  lfl1dim2N  35263  lkrss  35309
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