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Theorem lkrlss 29285
Description: The kernel of a linear functional is a subspace. (nlelshi 22640 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 2283 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2283 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
3 eqid 2283 . . . 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 29280 . . 3  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( K `  G )  =  { x  e.  (
Base `  W )  |  ( G `  x )  =  ( 0g `  (Scalar `  W ) ) } )
7 ssrab2 3258 . . . 4  |-  { x  e.  ( Base `  W
)  |  ( G `
 x )  =  ( 0g `  (Scalar `  W ) ) } 
C_  ( Base `  W
)
87a1i 10 . . 3  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  { x  e.  ( Base `  W
)  |  ( G `
 x )  =  ( 0g `  (Scalar `  W ) ) } 
C_  ( Base `  W
) )
96, 8eqsstrd 3212 . 2  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( K `  G )  C_  ( Base `  W
) )
10 eqid 2283 . . . . . 6  |-  ( 0g
`  W )  =  ( 0g `  W
)
111, 10lmod0vcl 15659 . . . . 5  |-  ( W  e.  LMod  ->  ( 0g
`  W )  e.  ( Base `  W
) )
1211adantr 451 . . . 4  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( 0g `  W )  e.  ( Base `  W
) )
132, 3, 10, 4lfl0 29255 . . . 4  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( G `  ( 0g `  W ) )  =  ( 0g `  (Scalar `  W ) ) )
141, 2, 3, 4, 5ellkr 29279 . . . 4  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  (
( 0g `  W
)  e.  ( K `
 G )  <->  ( ( 0g `  W )  e.  ( Base `  W
)  /\  ( G `  ( 0g `  W
) )  =  ( 0g `  (Scalar `  W ) ) ) ) )
1512, 13, 14mpbir2and 888 . . 3  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( 0g `  W )  e.  ( K `  G
) )
16 ne0i 3461 . . 3  |-  ( ( 0g `  W )  e.  ( K `  G )  ->  ( K `  G )  =/=  (/) )
1715, 16syl 15 . 2  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( K `  G )  =/=  (/) )
18 simplll 734 . . . . . 6  |-  ( ( ( ( W  e. 
LMod  /\  G  e.  F
)  /\  r  e.  ( Base `  (Scalar `  W
) ) )  /\  ( x  e.  ( K `  G )  /\  y  e.  ( K `  G )
) )  ->  W  e.  LMod )
19 simplr 731 . . . . . . 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 ) ) )
20 simpllr 735 . . . . . . . 8  |-  ( ( ( ( W  e. 
LMod  /\  G  e.  F
)  /\  r  e.  ( Base `  (Scalar `  W
) ) )  /\  ( x  e.  ( K `  G )  /\  y  e.  ( K `  G )
) )  ->  G  e.  F )
21 simprl 732 . . . . . . . 8  |-  ( ( ( ( W  e. 
LMod  /\  G  e.  F
)  /\  r  e.  ( Base `  (Scalar `  W
) ) )  /\  ( x  e.  ( K `  G )  /\  y  e.  ( K `  G )
) )  ->  x  e.  ( K `  G
) )
221, 4, 5lkrcl 29282 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  x  e.  ( K `  G
) )  ->  x  e.  ( Base `  W
) )
2318, 20, 21, 22syl3anc 1182 . . . . . . 7  |-  ( ( ( ( W  e. 
LMod  /\  G  e.  F
)  /\  r  e.  ( Base `  (Scalar `  W
) ) )  /\  ( x  e.  ( K `  G )  /\  y  e.  ( K `  G )
) )  ->  x  e.  ( Base `  W
) )
24 eqid 2283 . . . . . . . 8  |-  ( .s
`  W )  =  ( .s `  W
)
25 eqid 2283 . . . . . . . 8  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
261, 2, 24, 25lmodvscl 15644 . . . . . . 7  |-  ( ( W  e.  LMod  /\  r  e.  ( Base `  (Scalar `  W ) )  /\  x  e.  ( Base `  W ) )  -> 
( r ( .s
`  W ) x )  e.  ( Base `  W ) )
2718, 19, 23, 26syl3anc 1182 . . . . . 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
) )
28 simprr 733 . . . . . . 7  |-  ( ( ( ( W  e. 
LMod  /\  G  e.  F
)  /\  r  e.  ( Base `  (Scalar `  W
) ) )  /\  ( x  e.  ( K `  G )  /\  y  e.  ( K `  G )
) )  ->  y  e.  ( K `  G
) )
291, 4, 5lkrcl 29282 . . . . . . 7  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  y  e.  ( K `  G
) )  ->  y  e.  ( Base `  W
) )
3018, 20, 28, 29syl3anc 1182 . . . . . 6  |-  ( ( ( ( W  e. 
LMod  /\  G  e.  F
)  /\  r  e.  ( Base `  (Scalar `  W
) ) )  /\  ( x  e.  ( K `  G )  /\  y  e.  ( K `  G )
) )  ->  y  e.  ( Base `  W
) )
31 eqid 2283 . . . . . . 7  |-  ( +g  `  W )  =  ( +g  `  W )
321, 31lmodvacl 15641 . . . . . 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
) )
3318, 27, 30, 32syl3anc 1182 . . . . 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
) )
34 eqid 2283 . . . . . . . 8  |-  ( +g  `  (Scalar `  W )
)  =  ( +g  `  (Scalar `  W )
)
35 eqid 2283 . . . . . . . 8  |-  ( .r
`  (Scalar `  W )
)  =  ( .r
`  (Scalar `  W )
)
361, 31, 2, 24, 25, 34, 35, 4lfli 29251 . . . . . . 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 ) ) )
3718, 20, 19, 23, 30, 36syl113anc 1194 . . . . . 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 ) ) )
382, 3, 4, 5lkrf0 29283 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  x  e.  ( K `  G
) )  ->  ( G `  x )  =  ( 0g `  (Scalar `  W ) ) )
3918, 20, 21, 38syl3anc 1182 . . . . . . . . 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 ) ) )
4039oveq2d 5874 . . . . . . . 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 ) ) ) )
412lmodrng 15635 . . . . . . . . . 10  |-  ( W  e.  LMod  ->  (Scalar `  W )  e.  Ring )
4218, 41syl 15 . . . . . . . . 9  |-  ( ( ( ( W  e. 
LMod  /\  G  e.  F
)  /\  r  e.  ( Base `  (Scalar `  W
) ) )  /\  ( x  e.  ( K `  G )  /\  y  e.  ( K `  G )
) )  ->  (Scalar `  W )  e.  Ring )
4325, 35, 3rngrz 15378 . . . . . . . . 9  |-  ( ( (Scalar `  W )  e.  Ring  /\  r  e.  ( Base `  (Scalar `  W
) ) )  -> 
( r ( .r
`  (Scalar `  W )
) ( 0g `  (Scalar `  W ) ) )  =  ( 0g
`  (Scalar `  W )
) )
4442, 19, 43syl2anc 642 . . . . . . . 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 ) ) )
4540, 44eqtrd 2315 . . . . . . 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 ) ) )
462, 3, 4, 5lkrf0 29283 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  y  e.  ( K `  G
) )  ->  ( G `  y )  =  ( 0g `  (Scalar `  W ) ) )
4718, 20, 28, 46syl3anc 1182 . . . . . . 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 ) ) )
4845, 47oveq12d 5876 . . . . . 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 ) ) ) )
492lmodfgrp 15636 . . . . . . . 8  |-  ( W  e.  LMod  ->  (Scalar `  W )  e.  Grp )
5018, 49syl 15 . . . . . . 7  |-  ( ( ( ( W  e. 
LMod  /\  G  e.  F
)  /\  r  e.  ( Base `  (Scalar `  W
) ) )  /\  ( x  e.  ( K `  G )  /\  y  e.  ( K `  G )
) )  ->  (Scalar `  W )  e.  Grp )
5125, 3grpidcl 14510 . . . . . . . 8  |-  ( (Scalar `  W )  e.  Grp  ->  ( 0g `  (Scalar `  W ) )  e.  ( Base `  (Scalar `  W ) ) )
5250, 51syl 15 . . . . . . 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
) ) )
5325, 34, 3grplid 14512 . . . . . . 7  |-  ( ( (Scalar `  W )  e.  Grp  /\  ( 0g
`  (Scalar `  W )
)  e.  ( Base `  (Scalar `  W )
) )  ->  (
( 0g `  (Scalar `  W ) ) ( +g  `  (Scalar `  W ) ) ( 0g `  (Scalar `  W ) ) )  =  ( 0g `  (Scalar `  W ) ) )
5450, 52, 53syl2anc 642 . . . . . 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 ) ) )
5537, 48, 543eqtrd 2319 . . . . 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 ) ) )
561, 2, 3, 4, 5ellkr 29279 . . . . . 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 )
) ) ) )
5756ad2antrr 706 . . . . 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 )
) ) ) )
5833, 55, 57mpbir2and 888 . . . 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
) )
5958ralrimivva 2635 . . 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 ) )
6059ralrimiva 2626 . 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 ) )
61 lkrlss.s . . 3  |-  S  =  ( LSubSp `  W )
622, 25, 1, 31, 24, 61islss 15692 . 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 ) ) )
639, 17, 60, 62syl3anbrc 1136 1  |-  ( ( W  e.  LMod  /\  G  e.  F )  ->  ( K `  G )  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   {crab 2547    C_ wss 3152   (/)c0 3455   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212   0gc0g 13400   Grpcgrp 14362   Ringcrg 15337   LModclmod 15627   LSubSpclss 15689  LFnlclfn 29247  LKerclk 29275
This theorem is referenced by:  lkrssv  29286  lkrlsp  29292  lkrlsp3  29294  lkrshp  29295  lclkrlem2f  31702  lclkrlem2n  31710  lclkrlem2v  31718  lcfrlem25  31757  lcfrlem35  31767
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-1st 6122  df-2nd 6123  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-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  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-lmod 15629  df-lss 15690  df-lfl 29248  df-lkr 29276
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