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Theorem lkrlss 29955
Description: The kernel of a linear functional is a subspace. (nlelshi 23565 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 2438 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2438 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
3 eqid 2438 . . . 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 29950 . . 3  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( K `  G )  =  { x  e.  (
Base `  W )  |  ( G `  x )  =  ( 0g `  (Scalar `  W ) ) } )
7 ssrab2 3430 . . 3  |-  { x  e.  ( Base `  W
)  |  ( G `
 x )  =  ( 0g `  (Scalar `  W ) ) } 
C_  ( Base `  W
)
86, 7syl6eqss 3400 . 2  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( K `  G )  C_  ( Base `  W
) )
9 eqid 2438 . . . . . 6  |-  ( 0g
`  W )  =  ( 0g `  W
)
101, 9lmod0vcl 15981 . . . . 5  |-  ( W  e.  LMod  ->  ( 0g
`  W )  e.  ( Base `  W
) )
1110adantr 453 . . . 4  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( 0g `  W )  e.  ( Base `  W
) )
122, 3, 9, 4lfl0 29925 . . . 4  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( G `  ( 0g `  W ) )  =  ( 0g `  (Scalar `  W ) ) )
131, 2, 3, 4, 5ellkr 29949 . . . 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 890 . . 3  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( 0g `  W )  e.  ( K `  G
) )
15 ne0i 3636 . . 3  |-  ( ( 0g `  W )  e.  ( K `  G )  ->  ( K `  G )  =/=  (/) )
1614, 15syl 16 . 2  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( K `  G )  =/=  (/) )
17 simplll 736 . . . . . 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 733 . . . . . . 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 737 . . . . . . . 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 734 . . . . . . . 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 29952 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  x  e.  ( K `  G
) )  ->  x  e.  ( Base `  W
) )
2217, 19, 20, 21syl3anc 1185 . . . . . . 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 2438 . . . . . . . 8  |-  ( .s
`  W )  =  ( .s `  W
)
24 eqid 2438 . . . . . . . 8  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
251, 2, 23, 24lmodvscl 15969 . . . . . . 7  |-  ( ( W  e.  LMod  /\  r  e.  ( Base `  (Scalar `  W ) )  /\  x  e.  ( Base `  W ) )  -> 
( r ( .s
`  W ) x )  e.  ( Base `  W ) )
2617, 18, 22, 25syl3anc 1185 . . . . . 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 735 . . . . . . 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 29952 . . . . . . 7  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  y  e.  ( K `  G
) )  ->  y  e.  ( Base `  W
) )
2917, 19, 27, 28syl3anc 1185 . . . . . 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 2438 . . . . . . 7  |-  ( +g  `  W )  =  ( +g  `  W )
311, 30lmodvacl 15966 . . . . . 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 1185 . . . . 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 2438 . . . . . . . 8  |-  ( +g  `  (Scalar `  W )
)  =  ( +g  `  (Scalar `  W )
)
34 eqid 2438 . . . . . . . 8  |-  ( .r
`  (Scalar `  W )
)  =  ( .r
`  (Scalar `  W )
)
351, 30, 2, 23, 24, 33, 34, 4lfli 29921 . . . . . . 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 1197 . . . . . 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 29953 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  x  e.  ( K `  G
) )  ->  ( G `  x )  =  ( 0g `  (Scalar `  W ) ) )
3817, 19, 20, 37syl3anc 1185 . . . . . . . . 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 6099 . . . . . . . 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 15960 . . . . . . . . . 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 15703 . . . . . . . . 9  |-  ( ( (Scalar `  W )  e.  Ring  /\  r  e.  ( Base `  (Scalar `  W
) ) )  -> 
( r ( .r
`  (Scalar `  W )
) ( 0g `  (Scalar `  W ) ) )  =  ( 0g
`  (Scalar `  W )
) )
4341, 18, 42syl2anc 644 . . . . . . . 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 2470 . . . . . . 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 29953 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  y  e.  ( K `  G
) )  ->  ( G `  y )  =  ( 0g `  (Scalar `  W ) ) )
4617, 19, 27, 45syl3anc 1185 . . . . . . 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 6101 . . . . . 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 15961 . . . . . . . 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 14835 . . . . . . . 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 14837 . . . . . . 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 644 . . . . . 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 2474 . . . . 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 29949 . . . . . 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 708 . . . . 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 890 . . . 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 2800 . . 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 2791 . 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 16013 . 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 1139 1  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( K `  G )  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   {crab 2711    C_ wss 3322   (/)c0 3630   ` cfv 5456  (class class class)co 6083   Basecbs 13471   +g cplusg 13531   .rcmulr 13532  Scalarcsca 13534   .scvsca 13535   0gc0g 13725   Grpcgrp 14687   Ringcrg 15662   LModclmod 15952   LSubSpclss 16010  LFnlclfn 29917  LKerclk 29945
This theorem is referenced by:  lkrssv  29956  lkrlsp  29962  lkrlsp3  29964  lkrshp  29965  lclkrlem2f  32372  lclkrlem2n  32380  lclkrlem2v  32388  lcfrlem25  32427  lcfrlem35  32437
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-plusg 13544  df-0g 13729  df-mnd 14692  df-grp 14814  df-minusg 14815  df-sbg 14816  df-mgp 15651  df-rng 15665  df-ur 15667  df-lmod 15954  df-lss 16011  df-lfl 29918  df-lkr 29946
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