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Theorem lkrin 34362
Description: Intersection of the kernels of 2 functionals is included in the kernel of their sum. (Contributed by NM, 7-Jan-2015.)
Hypotheses
Ref Expression
lkrin.f  |-  F  =  (LFnl `  W )
lkrin.k  |-  K  =  (LKer `  W )
lkrin.d  |-  D  =  (LDual `  W )
lkrin.p  |-  .+  =  ( +g  `  D )
lkrin.w  |-  ( ph  ->  W  e.  LMod )
lkrin.e  |-  ( ph  ->  G  e.  F )
lkrin.g  |-  ( ph  ->  H  e.  F )
Assertion
Ref Expression
lkrin  |-  ( ph  ->  ( ( K `  G )  i^i  ( K `  H )
)  C_  ( K `  ( G  .+  H
) ) )

Proof of Theorem lkrin
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 elin 3692 . . 3  |-  ( v  e.  ( ( K `
 G )  i^i  ( K `  H
) )  <->  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )
2 lkrin.w . . . . . . 7  |-  ( ph  ->  W  e.  LMod )
32adantr 465 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  W  e.  LMod )
4 lkrin.e . . . . . . 7  |-  ( ph  ->  G  e.  F )
54adantr 465 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  G  e.  F )
6 simprl 755 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  v  e.  ( K `  G
) )
7 eqid 2467 . . . . . . 7  |-  ( Base `  W )  =  (
Base `  W )
8 lkrin.f . . . . . . 7  |-  F  =  (LFnl `  W )
9 lkrin.k . . . . . . 7  |-  K  =  (LKer `  W )
107, 8, 9lkrcl 34290 . . . . . 6  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  v  e.  ( K `  G
) )  ->  v  e.  ( Base `  W
) )
113, 5, 6, 10syl3anc 1228 . . . . 5  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  v  e.  ( Base `  W
) )
12 eqid 2467 . . . . . . 7  |-  (Scalar `  W )  =  (Scalar `  W )
13 eqid 2467 . . . . . . 7  |-  ( +g  `  (Scalar `  W )
)  =  ( +g  `  (Scalar `  W )
)
14 lkrin.d . . . . . . 7  |-  D  =  (LDual `  W )
15 lkrin.p . . . . . . 7  |-  .+  =  ( +g  `  D )
16 lkrin.g . . . . . . . 8  |-  ( ph  ->  H  e.  F )
1716adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  H  e.  F )
187, 12, 13, 8, 14, 15, 3, 5, 17, 11ldualvaddval 34329 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  (
( G  .+  H
) `  v )  =  ( ( G `
 v ) ( +g  `  (Scalar `  W ) ) ( H `  v ) ) )
19 eqid 2467 . . . . . . . . 9  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
2012, 19, 8, 9lkrf0 34291 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  v  e.  ( K `  G
) )  ->  ( G `  v )  =  ( 0g `  (Scalar `  W ) ) )
213, 5, 6, 20syl3anc 1228 . . . . . . 7  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  ( G `  v )  =  ( 0g `  (Scalar `  W ) ) )
22 simprr 756 . . . . . . . 8  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  v  e.  ( K `  H
) )
2312, 19, 8, 9lkrf0 34291 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  H  e.  F  /\  v  e.  ( K `  H
) )  ->  ( H `  v )  =  ( 0g `  (Scalar `  W ) ) )
243, 17, 22, 23syl3anc 1228 . . . . . . 7  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  ( H `  v )  =  ( 0g `  (Scalar `  W ) ) )
2521, 24oveq12d 6313 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  (
( G `  v
) ( +g  `  (Scalar `  W ) ) ( H `  v ) )  =  ( ( 0g `  (Scalar `  W ) ) ( +g  `  (Scalar `  W ) ) ( 0g `  (Scalar `  W ) ) ) )
2612lmodring 17391 . . . . . . . . . 10  |-  ( W  e.  LMod  ->  (Scalar `  W )  e.  Ring )
272, 26syl 16 . . . . . . . . 9  |-  ( ph  ->  (Scalar `  W )  e.  Ring )
28 ringgrp 17075 . . . . . . . . 9  |-  ( (Scalar `  W )  e.  Ring  -> 
(Scalar `  W )  e.  Grp )
2927, 28syl 16 . . . . . . . 8  |-  ( ph  ->  (Scalar `  W )  e.  Grp )
30 eqid 2467 . . . . . . . . . 10  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
3130, 19grpidcl 15950 . . . . . . . . 9  |-  ( (Scalar `  W )  e.  Grp  ->  ( 0g `  (Scalar `  W ) )  e.  ( Base `  (Scalar `  W ) ) )
3229, 31syl 16 . . . . . . . 8  |-  ( ph  ->  ( 0g `  (Scalar `  W ) )  e.  ( Base `  (Scalar `  W ) ) )
3330, 13, 19grplid 15952 . . . . . . . 8  |-  ( ( (Scalar `  W )  e.  Grp  /\  ( 0g
`  (Scalar `  W )
)  e.  ( Base `  (Scalar `  W )
) )  ->  (
( 0g `  (Scalar `  W ) ) ( +g  `  (Scalar `  W ) ) ( 0g `  (Scalar `  W ) ) )  =  ( 0g `  (Scalar `  W ) ) )
3429, 32, 33syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( ( 0g `  (Scalar `  W ) ) ( +g  `  (Scalar `  W ) ) ( 0g `  (Scalar `  W ) ) )  =  ( 0g `  (Scalar `  W ) ) )
3534adantr 465 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  (
( 0g `  (Scalar `  W ) ) ( +g  `  (Scalar `  W ) ) ( 0g `  (Scalar `  W ) ) )  =  ( 0g `  (Scalar `  W ) ) )
3618, 25, 353eqtrd 2512 . . . . 5  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  (
( G  .+  H
) `  v )  =  ( 0g `  (Scalar `  W ) ) )
378, 14, 15, 2, 4, 16ldualvaddcl 34328 . . . . . . 7  |-  ( ph  ->  ( G  .+  H
)  e.  F )
3837adantr 465 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  ( G  .+  H )  e.  F )
397, 12, 19, 8, 9ellkr 34287 . . . . . 6  |-  ( ( W  e.  LMod  /\  ( G  .+  H )  e.  F )  ->  (
v  e.  ( K `
 ( G  .+  H ) )  <->  ( v  e.  ( Base `  W
)  /\  ( ( G  .+  H ) `  v )  =  ( 0g `  (Scalar `  W ) ) ) ) )
403, 38, 39syl2anc 661 . . . . 5  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  (
v  e.  ( K `
 ( G  .+  H ) )  <->  ( v  e.  ( Base `  W
)  /\  ( ( G  .+  H ) `  v )  =  ( 0g `  (Scalar `  W ) ) ) ) )
4111, 36, 40mpbir2and 920 . . . 4  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  v  e.  ( K `  ( G  .+  H ) ) )
4241ex 434 . . 3  |-  ( ph  ->  ( ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) )  ->  v  e.  ( K `  ( G 
.+  H ) ) ) )
431, 42syl5bi 217 . 2  |-  ( ph  ->  ( v  e.  ( ( K `  G
)  i^i  ( K `  H ) )  -> 
v  e.  ( K `
 ( G  .+  H ) ) ) )
4443ssrdv 3515 1  |-  ( ph  ->  ( ( K `  G )  i^i  ( K `  H )
)  C_  ( K `  ( G  .+  H
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    i^i cin 3480    C_ wss 3481   ` cfv 5594  (class class class)co 6295   Basecbs 14507   +g cplusg 14572  Scalarcsca 14575   0gc0g 14712   Grpcgrp 15925   Ringcrg 17070   LModclmod 17383  LFnlclfn 34255  LKerclk 34283  LDualcld 34321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-plusg 14585  df-sca 14588  df-vsca 14589  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-sbg 15931  df-cmn 16673  df-abl 16674  df-mgp 17014  df-ur 17026  df-ring 17072  df-lmod 17385  df-lfl 34256  df-lkr 34284  df-ldual 34322
This theorem is referenced by:  lclkrlem2e  36709  lclkrlem2f  36710  lclkrlem2r  36722  lclkrlem2v  36726  lclkrslem2  36736  lcfrlem2  36741
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