Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lkrlss Structured version   Unicode version

Theorem lkrlss 34543
Description: The kernel of a linear functional is a subspace. (nlelshi 26848 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 2441 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2441 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
3 eqid 2441 . . . 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 34538 . . 3  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( K `  G )  =  { x  e.  (
Base `  W )  |  ( G `  x )  =  ( 0g `  (Scalar `  W ) ) } )
7 ssrab2 3568 . . 3  |-  { x  e.  ( Base `  W
)  |  ( G `
 x )  =  ( 0g `  (Scalar `  W ) ) } 
C_  ( Base `  W
)
86, 7syl6eqss 3537 . 2  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( K `  G )  C_  ( Base `  W
) )
9 eqid 2441 . . . . . 6  |-  ( 0g
`  W )  =  ( 0g `  W
)
101, 9lmod0vcl 17412 . . . . 5  |-  ( W  e.  LMod  ->  ( 0g
`  W )  e.  ( Base `  W
) )
1110adantr 465 . . . 4  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( 0g `  W )  e.  ( Base `  W
) )
122, 3, 9, 4lfl0 34513 . . . 4  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( G `  ( 0g `  W ) )  =  ( 0g `  (Scalar `  W ) ) )
131, 2, 3, 4, 5ellkr 34537 . . . 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 920 . . 3  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( 0g `  W )  e.  ( K `  G
) )
15 ne0i 3774 . . 3  |-  ( ( 0g `  W )  e.  ( K `  G )  ->  ( K `  G )  =/=  (/) )
1614, 15syl 16 . 2  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( K `  G )  =/=  (/) )
17 simplll 757 . . . . . 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 754 . . . . . . 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 758 . . . . . . . 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 755 . . . . . . . 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 34540 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  x  e.  ( K `  G
) )  ->  x  e.  ( Base `  W
) )
2217, 19, 20, 21syl3anc 1227 . . . . . . 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 2441 . . . . . . . 8  |-  ( .s
`  W )  =  ( .s `  W
)
24 eqid 2441 . . . . . . . 8  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
251, 2, 23, 24lmodvscl 17400 . . . . . . 7  |-  ( ( W  e.  LMod  /\  r  e.  ( Base `  (Scalar `  W ) )  /\  x  e.  ( Base `  W ) )  -> 
( r ( .s
`  W ) x )  e.  ( Base `  W ) )
2617, 18, 22, 25syl3anc 1227 . . . . . 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 756 . . . . . . 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 34540 . . . . . . 7  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  y  e.  ( K `  G
) )  ->  y  e.  ( Base `  W
) )
2917, 19, 27, 28syl3anc 1227 . . . . . 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 2441 . . . . . . 7  |-  ( +g  `  W )  =  ( +g  `  W )
311, 30lmodvacl 17397 . . . . . 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 1227 . . . . 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 2441 . . . . . . . 8  |-  ( +g  `  (Scalar `  W )
)  =  ( +g  `  (Scalar `  W )
)
34 eqid 2441 . . . . . . . 8  |-  ( .r
`  (Scalar `  W )
)  =  ( .r
`  (Scalar `  W )
)
351, 30, 2, 23, 24, 33, 34, 4lfli 34509 . . . . . . 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 1239 . . . . . 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 34541 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  x  e.  ( K `  G
) )  ->  ( G `  x )  =  ( 0g `  (Scalar `  W ) ) )
3817, 19, 20, 37syl3anc 1227 . . . . . . . . 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 6294 . . . . . . . 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 ) ) ) )
402lmodring 17391 . . . . . . . . . 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, 3ringrz 17107 . . . . . . . . 9  |-  ( ( (Scalar `  W )  e.  Ring  /\  r  e.  ( Base `  (Scalar `  W
) ) )  -> 
( r ( .r
`  (Scalar `  W )
) ( 0g `  (Scalar `  W ) ) )  =  ( 0g
`  (Scalar `  W )
) )
4341, 18, 42syl2anc 661 . . . . . . . 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 2482 . . . . . . 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 34541 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  y  e.  ( K `  G
) )  ->  ( G `  y )  =  ( 0g `  (Scalar `  W ) ) )
4617, 19, 27, 45syl3anc 1227 . . . . . . 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 6296 . . . . . 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 17392 . . . . . . . 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 15949 . . . . . . . 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 15951 . . . . . . 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 661 . . . . . 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 2486 . . . . 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 34537 . . . . . 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 725 . . . . 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 920 . . . 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 2862 . . 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 2855 . 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 17452 . 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 1179 1  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( K `  G )  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1381    e. wcel 1802    =/= wne 2636   A.wral 2791   {crab 2795    C_ wss 3459   (/)c0 3768   ` cfv 5575  (class class class)co 6278   Basecbs 14506   +g cplusg 14571   .rcmulr 14572  Scalarcsca 14574   .scvsca 14575   0gc0g 14711   Grpcgrp 15924   Ringcrg 17069   LModclmod 17383   LSubSpclss 17449  LFnlclfn 34505  LKerclk 34533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4545  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-pss 3475  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-tp 4016  df-op 4018  df-uni 4232  df-iun 4314  df-br 4435  df-opab 4493  df-mpt 4494  df-tr 4528  df-eprel 4778  df-id 4782  df-po 4787  df-so 4788  df-fr 4825  df-we 4827  df-ord 4868  df-on 4869  df-lim 4870  df-suc 4871  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-riota 6239  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6683  df-1st 6782  df-2nd 6783  df-recs 7041  df-rdg 7075  df-er 7310  df-map 7421  df-en 7516  df-dom 7517  df-sdom 7518  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9809  df-neg 9810  df-nn 10540  df-2 10597  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-plusg 14584  df-0g 14713  df-mgm 15743  df-sgrp 15782  df-mnd 15792  df-grp 15928  df-minusg 15929  df-sbg 15930  df-mgp 17013  df-ur 17025  df-ring 17071  df-lmod 17385  df-lss 17450  df-lfl 34506  df-lkr 34534
This theorem is referenced by:  lkrssv  34544  lkrlsp  34550  lkrlsp3  34552  lkrshp  34553  lclkrlem2f  36962  lclkrlem2n  36970  lclkrlem2v  36978  lcfrlem25  37017  lcfrlem35  37027
  Copyright terms: Public domain W3C validator