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Theorem lkrlss 35233
Description: The kernel of a linear functional is a subspace. (nlelshi 27095 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 2382 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2382 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
3 eqid 2382 . . . 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 35228 . . 3  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( K `  G )  =  { x  e.  (
Base `  W )  |  ( G `  x )  =  ( 0g `  (Scalar `  W ) ) } )
7 ssrab2 3499 . . 3  |-  { x  e.  ( Base `  W
)  |  ( G `
 x )  =  ( 0g `  (Scalar `  W ) ) } 
C_  ( Base `  W
)
86, 7syl6eqss 3467 . 2  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( K `  G )  C_  ( Base `  W
) )
9 eqid 2382 . . . . . 6  |-  ( 0g
`  W )  =  ( 0g `  W
)
101, 9lmod0vcl 17654 . . . . 5  |-  ( W  e.  LMod  ->  ( 0g
`  W )  e.  ( Base `  W
) )
1110adantr 463 . . . 4  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( 0g `  W )  e.  ( Base `  W
) )
122, 3, 9, 4lfl0 35203 . . . 4  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( G `  ( 0g `  W ) )  =  ( 0g `  (Scalar `  W ) ) )
131, 2, 3, 4, 5ellkr 35227 . . . 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 3717 . . 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 753 . . . . . . 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 754 . . . . . . . 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 35230 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  x  e.  ( K `  G
) )  ->  x  e.  ( Base `  W
) )
2217, 19, 20, 21syl3anc 1226 . . . . . . 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 2382 . . . . . . . 8  |-  ( .s
`  W )  =  ( .s `  W
)
24 eqid 2382 . . . . . . . 8  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
251, 2, 23, 24lmodvscl 17642 . . . . . . 7  |-  ( ( W  e.  LMod  /\  r  e.  ( Base `  (Scalar `  W ) )  /\  x  e.  ( Base `  W ) )  -> 
( r ( .s
`  W ) x )  e.  ( Base `  W ) )
2617, 18, 22, 25syl3anc 1226 . . . . . 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 755 . . . . . . 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 35230 . . . . . . 7  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  y  e.  ( K `  G
) )  ->  y  e.  ( Base `  W
) )
2917, 19, 27, 28syl3anc 1226 . . . . . 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 2382 . . . . . . 7  |-  ( +g  `  W )  =  ( +g  `  W )
311, 30lmodvacl 17639 . . . . . 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 1226 . . . . 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 2382 . . . . . . . 8  |-  ( +g  `  (Scalar `  W )
)  =  ( +g  `  (Scalar `  W )
)
34 eqid 2382 . . . . . . . 8  |-  ( .r
`  (Scalar `  W )
)  =  ( .r
`  (Scalar `  W )
)
351, 30, 2, 23, 24, 33, 34, 4lfli 35199 . . . . . . 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 1238 . . . . . 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 35231 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  x  e.  ( K `  G
) )  ->  ( G `  x )  =  ( 0g `  (Scalar `  W ) ) )
3817, 19, 20, 37syl3anc 1226 . . . . . . . . 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 6212 . . . . . . . 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 17633 . . . . . . . . . 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 17349 . . . . . . . . 9  |-  ( ( (Scalar `  W )  e.  Ring  /\  r  e.  ( Base `  (Scalar `  W
) ) )  -> 
( r ( .r
`  (Scalar `  W )
) ( 0g `  (Scalar `  W ) ) )  =  ( 0g
`  (Scalar `  W )
) )
4341, 18, 42syl2anc 659 . . . . . . . 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 2423 . . . . . . 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 35231 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  y  e.  ( K `  G
) )  ->  ( G `  y )  =  ( 0g `  (Scalar `  W ) ) )
4617, 19, 27, 45syl3anc 1226 . . . . . . 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 6214 . . . . . 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 17634 . . . . . . . 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 16195 . . . . . . . 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 16197 . . . . . . 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 659 . . . . . 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 2427 . . . . 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 35227 . . . . . 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 723 . . . . 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 2803 . . 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 2796 . 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 17694 . 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 1178 1  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( K `  G )  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826    =/= wne 2577   A.wral 2732   {crab 2736    C_ wss 3389   (/)c0 3711   ` cfv 5496  (class class class)co 6196   Basecbs 14634   +g cplusg 14702   .rcmulr 14703  Scalarcsca 14705   .scvsca 14706   0gc0g 14847   Grpcgrp 16170   Ringcrg 17311   LModclmod 17625   LSubSpclss 17691  LFnlclfn 35195  LKerclk 35223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-er 7229  df-map 7340  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-plusg 14715  df-0g 14849  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-grp 16174  df-minusg 16175  df-sbg 16176  df-mgp 17255  df-ur 17267  df-ring 17313  df-lmod 17627  df-lss 17692  df-lfl 35196  df-lkr 35224
This theorem is referenced by:  lkrssv  35234  lkrlsp  35240  lkrlsp3  35242  lkrshp  35243  lclkrlem2f  37652  lclkrlem2n  37660  lclkrlem2v  37668  lcfrlem25  37707  lcfrlem35  37717
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