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Theorem lkrscss 29288
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 15859 . . . . . . 7  |-  ( W  e.  LVec  ->  W  e. 
LMod )
64, 5syl 15 . . . . . 6  |-  ( ph  ->  W  e.  LMod )
7 lkrsc.g . . . . . 6  |-  ( ph  ->  G  e.  F )
81, 2, 3, 6, 7lkrssv 29286 . . . . 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 2283 . . . . . . . 8  |-  ( 0g
`  D )  =  ( 0g `  D
)
131, 9, 2, 10, 11, 12, 6, 7lfl0sc 29272 . . . . . . 7  |-  ( ph  ->  ( G  o F 
.x.  ( V  X.  { ( 0g `  D ) } ) )  =  ( V  X.  { ( 0g
`  D ) } ) )
1413fveq2d 5529 . . . . . 6  |-  ( ph  ->  ( L `  ( G  o F  .x.  ( V  X.  { ( 0g
`  D ) } ) ) )  =  ( L `  ( V  X.  { ( 0g
`  D ) } ) ) )
15 eqid 2283 . . . . . . 7  |-  ( V  X.  { ( 0g
`  D ) } )  =  ( V  X.  { ( 0g
`  D ) } )
169, 12, 1, 2lfl0f 29259 . . . . . . . . 9  |-  ( W  e.  LMod  ->  ( V  X.  { ( 0g
`  D ) } )  e.  F )
176, 16syl 15 . . . . . . . 8  |-  ( ph  ->  ( V  X.  {
( 0g `  D
) } )  e.  F )
189, 12, 1, 2, 3lkr0f 29284 . . . . . . . 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 642 . . . . . . 7  |-  ( ph  ->  ( ( L `  ( V  X.  { ( 0g `  D ) } ) )  =  V  <->  ( V  X.  { ( 0g `  D ) } )  =  ( V  X.  { ( 0g `  D ) } ) ) )
2015, 19mpbiri 224 . . . . . 6  |-  ( ph  ->  ( L `  ( V  X.  { ( 0g
`  D ) } ) )  =  V )
2114, 20eqtr2d 2316 . . . . 5  |-  ( ph  ->  V  =  ( L `
 ( G  o F  .x.  ( V  X.  { ( 0g `  D ) } ) ) ) )
228, 21sseqtrd 3214 . . . 4  |-  ( ph  ->  ( L `  G
)  C_  ( L `  ( G  o F 
.x.  ( V  X.  { ( 0g `  D ) } ) ) ) )
2322adantr 451 . . 3  |-  ( (
ph  /\  R  =  ( 0g `  D ) )  ->  ( L `  G )  C_  ( L `  ( G  o F  .x.  ( V  X.  { ( 0g
`  D ) } ) ) ) )
24 sneq 3651 . . . . . . 7  |-  ( R  =  ( 0g `  D )  ->  { R }  =  { ( 0g `  D ) } )
2524xpeq2d 4713 . . . . . 6  |-  ( R  =  ( 0g `  D )  ->  ( V  X.  { R }
)  =  ( V  X.  { ( 0g
`  D ) } ) )
2625oveq2d 5874 . . . . 5  |-  ( R  =  ( 0g `  D )  ->  ( G  o F  .x.  ( V  X.  { R }
) )  =  ( G  o F  .x.  ( V  X.  { ( 0g `  D ) } ) ) )
2726fveq2d 5529 . . . 4  |-  ( R  =  ( 0g `  D )  ->  ( L `  ( G  o F  .x.  ( V  X.  { R }
) ) )  =  ( L `  ( G  o F  .x.  ( V  X.  { ( 0g
`  D ) } ) ) ) )
2827adantl 452 . . 3  |-  ( (
ph  /\  R  =  ( 0g `  D ) )  ->  ( L `  ( G  o F 
.x.  ( V  X.  { R } ) ) )  =  ( L `
 ( G  o F  .x.  ( V  X.  { ( 0g `  D ) } ) ) ) )
2923, 28sseqtr4d 3215 . 2  |-  ( (
ph  /\  R  =  ( 0g `  D ) )  ->  ( L `  G )  C_  ( L `  ( G  o F  .x.  ( V  X.  { R }
) ) ) )
304adantr 451 . . . 4  |-  ( (
ph  /\  R  =/=  ( 0g `  D ) )  ->  W  e.  LVec )
317adantr 451 . . . 4  |-  ( (
ph  /\  R  =/=  ( 0g `  D ) )  ->  G  e.  F )
32 lkrsc.r . . . . 5  |-  ( ph  ->  R  e.  K )
3332adantr 451 . . . 4  |-  ( (
ph  /\  R  =/=  ( 0g `  D ) )  ->  R  e.  K )
34 simpr 447 . . . 4  |-  ( (
ph  /\  R  =/=  ( 0g `  D ) )  ->  R  =/=  ( 0g `  D ) )
351, 9, 10, 11, 2, 3, 30, 31, 33, 12, 34lkrsc 29287 . . 3  |-  ( (
ph  /\  R  =/=  ( 0g `  D ) )  ->  ( L `  ( G  o F 
.x.  ( V  X.  { R } ) ) )  =  ( L `
 G ) )
36 eqimss2 3231 . . 3  |-  ( ( L `  ( G  o F  .x.  ( V  X.  { R }
) ) )  =  ( L `  G
)  ->  ( L `  G )  C_  ( L `  ( G  o F  .x.  ( V  X.  { R }
) ) ) )
3735, 36syl 15 . 2  |-  ( (
ph  /\  R  =/=  ( 0g `  D ) )  ->  ( L `  G )  C_  ( L `  ( G  o F  .x.  ( V  X.  { R }
) ) ) )
3829, 37pm2.61dane 2524 1  |-  ( ph  ->  ( L `  G
)  C_  ( L `  ( G  o F 
.x.  ( V  X.  { R } ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    C_ wss 3152   {csn 3640    X. cxp 4687   ` cfv 5255  (class class class)co 5858    o Fcof 6076   Basecbs 13148   .rcmulr 13209  Scalarcsca 13211   0gc0g 13400   LModclmod 15627   LVecclvec 15855  LFnlclfn 29247  LKerclk 29275
This theorem is referenced by:  lfl1dim  29311  lfl1dim2N  29312  lkrss  29358
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-drng 15514  df-lmod 15629  df-lss 15690  df-lvec 15856  df-lfl 29248  df-lkr 29276
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