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Theorem lkrlss 29578
Description: The kernel of a linear functional is a subspace. (nlelshi 23516 analog.) (Contributed by NM, 16-Apr-2014.)
Hypotheses
Ref Expression
lkrlss.f  |-  F  =  (LFnl `  W )
lkrlss.k  |-  K  =  (LKer `  W )
lkrlss.s  |-  S  =  ( LSubSp `  W )
Assertion
Ref Expression
lkrlss  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( K `  G )  e.  S )

Proof of Theorem lkrlss
Dummy variables  x  r  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2404 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2404 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
3 eqid 2404 . . . 4  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
4 lkrlss.f . . . 4  |-  F  =  (LFnl `  W )
5 lkrlss.k . . . 4  |-  K  =  (LKer `  W )
61, 2, 3, 4, 5lkrval2 29573 . . 3  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( K `  G )  =  { x  e.  (
Base `  W )  |  ( G `  x )  =  ( 0g `  (Scalar `  W ) ) } )
7 ssrab2 3388 . . 3  |-  { x  e.  ( Base `  W
)  |  ( G `
 x )  =  ( 0g `  (Scalar `  W ) ) } 
C_  ( Base `  W
)
86, 7syl6eqss 3358 . 2  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( K `  G )  C_  ( Base `  W
) )
9 eqid 2404 . . . . . 6  |-  ( 0g
`  W )  =  ( 0g `  W
)
101, 9lmod0vcl 15934 . . . . 5  |-  ( W  e.  LMod  ->  ( 0g
`  W )  e.  ( Base `  W
) )
1110adantr 452 . . . 4  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( 0g `  W )  e.  ( Base `  W
) )
122, 3, 9, 4lfl0 29548 . . . 4  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( G `  ( 0g `  W ) )  =  ( 0g `  (Scalar `  W ) ) )
131, 2, 3, 4, 5ellkr 29572 . . . 4  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  (
( 0g `  W
)  e.  ( K `
 G )  <->  ( ( 0g `  W )  e.  ( Base `  W
)  /\  ( G `  ( 0g `  W
) )  =  ( 0g `  (Scalar `  W ) ) ) ) )
1411, 12, 13mpbir2and 889 . . 3  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( 0g `  W )  e.  ( K `  G
) )
15 ne0i 3594 . . 3  |-  ( ( 0g `  W )  e.  ( K `  G )  ->  ( K `  G )  =/=  (/) )
1614, 15syl 16 . 2  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( K `  G )  =/=  (/) )
17 simplll 735 . . . . . 6  |-  ( ( ( ( W  e. 
LMod  /\  G  e.  F
)  /\  r  e.  ( Base `  (Scalar `  W
) ) )  /\  ( x  e.  ( K `  G )  /\  y  e.  ( K `  G )
) )  ->  W  e.  LMod )
18 simplr 732 . . . . . . 7  |-  ( ( ( ( W  e. 
LMod  /\  G  e.  F
)  /\  r  e.  ( Base `  (Scalar `  W
) ) )  /\  ( x  e.  ( K `  G )  /\  y  e.  ( K `  G )
) )  ->  r  e.  ( Base `  (Scalar `  W ) ) )
19 simpllr 736 . . . . . . . 8  |-  ( ( ( ( W  e. 
LMod  /\  G  e.  F
)  /\  r  e.  ( Base `  (Scalar `  W
) ) )  /\  ( x  e.  ( K `  G )  /\  y  e.  ( K `  G )
) )  ->  G  e.  F )
20 simprl 733 . . . . . . . 8  |-  ( ( ( ( W  e. 
LMod  /\  G  e.  F
)  /\  r  e.  ( Base `  (Scalar `  W
) ) )  /\  ( x  e.  ( K `  G )  /\  y  e.  ( K `  G )
) )  ->  x  e.  ( K `  G
) )
211, 4, 5lkrcl 29575 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  x  e.  ( K `  G
) )  ->  x  e.  ( Base `  W
) )
2217, 19, 20, 21syl3anc 1184 . . . . . . 7  |-  ( ( ( ( W  e. 
LMod  /\  G  e.  F
)  /\  r  e.  ( Base `  (Scalar `  W
) ) )  /\  ( x  e.  ( K `  G )  /\  y  e.  ( K `  G )
) )  ->  x  e.  ( Base `  W
) )
23 eqid 2404 . . . . . . . 8  |-  ( .s
`  W )  =  ( .s `  W
)
24 eqid 2404 . . . . . . . 8  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
251, 2, 23, 24lmodvscl 15922 . . . . . . 7  |-  ( ( W  e.  LMod  /\  r  e.  ( Base `  (Scalar `  W ) )  /\  x  e.  ( Base `  W ) )  -> 
( r ( .s
`  W ) x )  e.  ( Base `  W ) )
2617, 18, 22, 25syl3anc 1184 . . . . . 6  |-  ( ( ( ( W  e. 
LMod  /\  G  e.  F
)  /\  r  e.  ( Base `  (Scalar `  W
) ) )  /\  ( x  e.  ( K `  G )  /\  y  e.  ( K `  G )
) )  ->  (
r ( .s `  W ) x )  e.  ( Base `  W
) )
27 simprr 734 . . . . . . 7  |-  ( ( ( ( W  e. 
LMod  /\  G  e.  F
)  /\  r  e.  ( Base `  (Scalar `  W
) ) )  /\  ( x  e.  ( K `  G )  /\  y  e.  ( K `  G )
) )  ->  y  e.  ( K `  G
) )
281, 4, 5lkrcl 29575 . . . . . . 7  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  y  e.  ( K `  G
) )  ->  y  e.  ( Base `  W
) )
2917, 19, 27, 28syl3anc 1184 . . . . . 6  |-  ( ( ( ( W  e. 
LMod  /\  G  e.  F
)  /\  r  e.  ( Base `  (Scalar `  W
) ) )  /\  ( x  e.  ( K `  G )  /\  y  e.  ( K `  G )
) )  ->  y  e.  ( Base `  W
) )
30 eqid 2404 . . . . . . 7  |-  ( +g  `  W )  =  ( +g  `  W )
311, 30lmodvacl 15919 . . . . . 6  |-  ( ( W  e.  LMod  /\  (
r ( .s `  W ) x )  e.  ( Base `  W
)  /\  y  e.  ( Base `  W )
)  ->  ( (
r ( .s `  W ) x ) ( +g  `  W
) y )  e.  ( Base `  W
) )
3217, 26, 29, 31syl3anc 1184 . . . . 5  |-  ( ( ( ( W  e. 
LMod  /\  G  e.  F
)  /\  r  e.  ( Base `  (Scalar `  W
) ) )  /\  ( x  e.  ( K `  G )  /\  y  e.  ( K `  G )
) )  ->  (
( r ( .s
`  W ) x ) ( +g  `  W
) y )  e.  ( Base `  W
) )
33 eqid 2404 . . . . . . . 8  |-  ( +g  `  (Scalar `  W )
)  =  ( +g  `  (Scalar `  W )
)
34 eqid 2404 . . . . . . . 8  |-  ( .r
`  (Scalar `  W )
)  =  ( .r
`  (Scalar `  W )
)
351, 30, 2, 23, 24, 33, 34, 4lfli 29544 . . . . . . 7  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  (
r  e.  ( Base `  (Scalar `  W )
)  /\  x  e.  ( Base `  W )  /\  y  e.  ( Base `  W ) ) )  ->  ( G `  ( ( r ( .s `  W ) x ) ( +g  `  W ) y ) )  =  ( ( r ( .r `  (Scalar `  W ) ) ( G `  x
) ) ( +g  `  (Scalar `  W )
) ( G `  y ) ) )
3617, 19, 18, 22, 29, 35syl113anc 1196 . . . . . 6  |-  ( ( ( ( W  e. 
LMod  /\  G  e.  F
)  /\  r  e.  ( Base `  (Scalar `  W
) ) )  /\  ( x  e.  ( K `  G )  /\  y  e.  ( K `  G )
) )  ->  ( G `  ( (
r ( .s `  W ) x ) ( +g  `  W
) y ) )  =  ( ( r ( .r `  (Scalar `  W ) ) ( G `  x ) ) ( +g  `  (Scalar `  W ) ) ( G `  y ) ) )
372, 3, 4, 5lkrf0 29576 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  x  e.  ( K `  G
) )  ->  ( G `  x )  =  ( 0g `  (Scalar `  W ) ) )
3817, 19, 20, 37syl3anc 1184 . . . . . . . . 9  |-  ( ( ( ( W  e. 
LMod  /\  G  e.  F
)  /\  r  e.  ( Base `  (Scalar `  W
) ) )  /\  ( x  e.  ( K `  G )  /\  y  e.  ( K `  G )
) )  ->  ( G `  x )  =  ( 0g `  (Scalar `  W ) ) )
3938oveq2d 6056 . . . . . . . 8  |-  ( ( ( ( W  e. 
LMod  /\  G  e.  F
)  /\  r  e.  ( Base `  (Scalar `  W
) ) )  /\  ( x  e.  ( K `  G )  /\  y  e.  ( K `  G )
) )  ->  (
r ( .r `  (Scalar `  W ) ) ( G `  x
) )  =  ( r ( .r `  (Scalar `  W ) ) ( 0g `  (Scalar `  W ) ) ) )
402lmodrng 15913 . . . . . . . . . 10  |-  ( W  e.  LMod  ->  (Scalar `  W )  e.  Ring )
4117, 40syl 16 . . . . . . . . 9  |-  ( ( ( ( W  e. 
LMod  /\  G  e.  F
)  /\  r  e.  ( Base `  (Scalar `  W
) ) )  /\  ( x  e.  ( K `  G )  /\  y  e.  ( K `  G )
) )  ->  (Scalar `  W )  e.  Ring )
4224, 34, 3rngrz 15656 . . . . . . . . 9  |-  ( ( (Scalar `  W )  e.  Ring  /\  r  e.  ( Base `  (Scalar `  W
) ) )  -> 
( r ( .r
`  (Scalar `  W )
) ( 0g `  (Scalar `  W ) ) )  =  ( 0g
`  (Scalar `  W )
) )
4341, 18, 42syl2anc 643 . . . . . . . 8  |-  ( ( ( ( W  e. 
LMod  /\  G  e.  F
)  /\  r  e.  ( Base `  (Scalar `  W
) ) )  /\  ( x  e.  ( K `  G )  /\  y  e.  ( K `  G )
) )  ->  (
r ( .r `  (Scalar `  W ) ) ( 0g `  (Scalar `  W ) ) )  =  ( 0g `  (Scalar `  W ) ) )
4439, 43eqtrd 2436 . . . . . . 7  |-  ( ( ( ( W  e. 
LMod  /\  G  e.  F
)  /\  r  e.  ( Base `  (Scalar `  W
) ) )  /\  ( x  e.  ( K `  G )  /\  y  e.  ( K `  G )
) )  ->  (
r ( .r `  (Scalar `  W ) ) ( G `  x
) )  =  ( 0g `  (Scalar `  W ) ) )
452, 3, 4, 5lkrf0 29576 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  y  e.  ( K `  G
) )  ->  ( G `  y )  =  ( 0g `  (Scalar `  W ) ) )
4617, 19, 27, 45syl3anc 1184 . . . . . . 7  |-  ( ( ( ( W  e. 
LMod  /\  G  e.  F
)  /\  r  e.  ( Base `  (Scalar `  W
) ) )  /\  ( x  e.  ( K `  G )  /\  y  e.  ( K `  G )
) )  ->  ( G `  y )  =  ( 0g `  (Scalar `  W ) ) )
4744, 46oveq12d 6058 . . . . . 6  |-  ( ( ( ( W  e. 
LMod  /\  G  e.  F
)  /\  r  e.  ( Base `  (Scalar `  W
) ) )  /\  ( x  e.  ( K `  G )  /\  y  e.  ( K `  G )
) )  ->  (
( r ( .r
`  (Scalar `  W )
) ( G `  x ) ) ( +g  `  (Scalar `  W ) ) ( G `  y ) )  =  ( ( 0g `  (Scalar `  W ) ) ( +g  `  (Scalar `  W ) ) ( 0g `  (Scalar `  W ) ) ) )
482lmodfgrp 15914 . . . . . . . 8  |-  ( W  e.  LMod  ->  (Scalar `  W )  e.  Grp )
4917, 48syl 16 . . . . . . 7  |-  ( ( ( ( W  e. 
LMod  /\  G  e.  F
)  /\  r  e.  ( Base `  (Scalar `  W
) ) )  /\  ( x  e.  ( K `  G )  /\  y  e.  ( K `  G )
) )  ->  (Scalar `  W )  e.  Grp )
5024, 3grpidcl 14788 . . . . . . . 8  |-  ( (Scalar `  W )  e.  Grp  ->  ( 0g `  (Scalar `  W ) )  e.  ( Base `  (Scalar `  W ) ) )
5149, 50syl 16 . . . . . . 7  |-  ( ( ( ( W  e. 
LMod  /\  G  e.  F
)  /\  r  e.  ( Base `  (Scalar `  W
) ) )  /\  ( x  e.  ( K `  G )  /\  y  e.  ( K `  G )
) )  ->  ( 0g `  (Scalar `  W
) )  e.  (
Base `  (Scalar `  W
) ) )
5224, 33, 3grplid 14790 . . . . . . 7  |-  ( ( (Scalar `  W )  e.  Grp  /\  ( 0g
`  (Scalar `  W )
)  e.  ( Base `  (Scalar `  W )
) )  ->  (
( 0g `  (Scalar `  W ) ) ( +g  `  (Scalar `  W ) ) ( 0g `  (Scalar `  W ) ) )  =  ( 0g `  (Scalar `  W ) ) )
5349, 51, 52syl2anc 643 . . . . . 6  |-  ( ( ( ( W  e. 
LMod  /\  G  e.  F
)  /\  r  e.  ( Base `  (Scalar `  W
) ) )  /\  ( x  e.  ( K `  G )  /\  y  e.  ( K `  G )
) )  ->  (
( 0g `  (Scalar `  W ) ) ( +g  `  (Scalar `  W ) ) ( 0g `  (Scalar `  W ) ) )  =  ( 0g `  (Scalar `  W ) ) )
5436, 47, 533eqtrd 2440 . . . . 5  |-  ( ( ( ( W  e. 
LMod  /\  G  e.  F
)  /\  r  e.  ( Base `  (Scalar `  W
) ) )  /\  ( x  e.  ( K `  G )  /\  y  e.  ( K `  G )
) )  ->  ( G `  ( (
r ( .s `  W ) x ) ( +g  `  W
) y ) )  =  ( 0g `  (Scalar `  W ) ) )
551, 2, 3, 4, 5ellkr 29572 . . . . . 6  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  (
( ( r ( .s `  W ) x ) ( +g  `  W ) y )  e.  ( K `  G )  <->  ( (
( r ( .s
`  W ) x ) ( +g  `  W
) y )  e.  ( Base `  W
)  /\  ( G `  ( ( r ( .s `  W ) x ) ( +g  `  W ) y ) )  =  ( 0g
`  (Scalar `  W )
) ) ) )
5655ad2antrr 707 . . . . 5  |-  ( ( ( ( W  e. 
LMod  /\  G  e.  F
)  /\  r  e.  ( Base `  (Scalar `  W
) ) )  /\  ( x  e.  ( K `  G )  /\  y  e.  ( K `  G )
) )  ->  (
( ( r ( .s `  W ) x ) ( +g  `  W ) y )  e.  ( K `  G )  <->  ( (
( r ( .s
`  W ) x ) ( +g  `  W
) y )  e.  ( Base `  W
)  /\  ( G `  ( ( r ( .s `  W ) x ) ( +g  `  W ) y ) )  =  ( 0g
`  (Scalar `  W )
) ) ) )
5732, 54, 56mpbir2and 889 . . . 4  |-  ( ( ( ( W  e. 
LMod  /\  G  e.  F
)  /\  r  e.  ( Base `  (Scalar `  W
) ) )  /\  ( x  e.  ( K `  G )  /\  y  e.  ( K `  G )
) )  ->  (
( r ( .s
`  W ) x ) ( +g  `  W
) y )  e.  ( K `  G
) )
5857ralrimivva 2758 . . 3  |-  ( ( ( W  e.  LMod  /\  G  e.  F )  /\  r  e.  (
Base `  (Scalar `  W
) ) )  ->  A. x  e.  ( K `  G ) A. y  e.  ( K `  G )
( ( r ( .s `  W ) x ) ( +g  `  W ) y )  e.  ( K `  G ) )
5958ralrimiva 2749 . 2  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  A. r  e.  ( Base `  (Scalar `  W ) ) A. x  e.  ( K `  G ) A. y  e.  ( K `  G
) ( ( r ( .s `  W
) x ) ( +g  `  W ) y )  e.  ( K `  G ) )
60 lkrlss.s . . 3  |-  S  =  ( LSubSp `  W )
612, 24, 1, 30, 23, 60islss 15966 . 2  |-  ( ( K `  G )  e.  S  <->  ( ( K `  G )  C_  ( Base `  W
)  /\  ( K `  G )  =/=  (/)  /\  A. r  e.  ( Base `  (Scalar `  W )
) A. x  e.  ( K `  G
) A. y  e.  ( K `  G
) ( ( r ( .s `  W
) x ) ( +g  `  W ) y )  e.  ( K `  G ) ) )
628, 16, 59, 61syl3anbrc 1138 1  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( K `  G )  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   {crab 2670    C_ wss 3280   (/)c0 3588   ` cfv 5413  (class class class)co 6040   Basecbs 13424   +g cplusg 13484   .rcmulr 13485  Scalarcsca 13487   .scvsca 13488   0gc0g 13678   Grpcgrp 14640   Ringcrg 15615   LModclmod 15905   LSubSpclss 15963  LFnlclfn 29540  LKerclk 29568
This theorem is referenced by:  lkrssv  29579  lkrlsp  29585  lkrlsp3  29587  lkrshp  29588  lclkrlem2f  31995  lclkrlem2n  32003  lclkrlem2v  32011  lcfrlem25  32050  lcfrlem35  32060
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-plusg 13497  df-0g 13682  df-mnd 14645  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mgp 15604  df-rng 15618  df-ur 15620  df-lmod 15907  df-lss 15964  df-lfl 29541  df-lkr 29569
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