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Theorem lkrin 32531
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 3536 . . 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 462 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  W  e.  LMod )
4 lkrin.e . . . . . . 7  |-  ( ph  ->  G  e.  F )
54adantr 462 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  G  e.  F )
6 simprl 750 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  v  e.  ( K `  G
) )
7 eqid 2441 . . . . . . 7  |-  ( Base `  W )  =  (
Base `  W )
8 lkrin.f . . . . . . 7  |-  F  =  (LFnl `  W )
9 lkrin.k . . . . . . 7  |-  K  =  (LKer `  W )
107, 8, 9lkrcl 32459 . . . . . 6  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  v  e.  ( K `  G
) )  ->  v  e.  ( Base `  W
) )
113, 5, 6, 10syl3anc 1213 . . . . 5  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  v  e.  ( Base `  W
) )
12 eqid 2441 . . . . . . 7  |-  (Scalar `  W )  =  (Scalar `  W )
13 eqid 2441 . . . . . . 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 462 . . . . . . 7  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  H  e.  F )
187, 12, 13, 8, 14, 15, 3, 5, 17, 11ldualvaddval 32498 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  (
( G  .+  H
) `  v )  =  ( ( G `
 v ) ( +g  `  (Scalar `  W ) ) ( H `  v ) ) )
19 eqid 2441 . . . . . . . . 9  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
2012, 19, 8, 9lkrf0 32460 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  v  e.  ( K `  G
) )  ->  ( G `  v )  =  ( 0g `  (Scalar `  W ) ) )
213, 5, 6, 20syl3anc 1213 . . . . . . 7  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  ( G `  v )  =  ( 0g `  (Scalar `  W ) ) )
22 simprr 751 . . . . . . . 8  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  v  e.  ( K `  H
) )
2312, 19, 8, 9lkrf0 32460 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  H  e.  F  /\  v  e.  ( K `  H
) )  ->  ( H `  v )  =  ( 0g `  (Scalar `  W ) ) )
243, 17, 22, 23syl3anc 1213 . . . . . . 7  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  ( H `  v )  =  ( 0g `  (Scalar `  W ) ) )
2521, 24oveq12d 6108 . . . . . 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 ) ) ) )
2612lmodrng 16936 . . . . . . . . . 10  |-  ( W  e.  LMod  ->  (Scalar `  W )  e.  Ring )
272, 26syl 16 . . . . . . . . 9  |-  ( ph  ->  (Scalar `  W )  e.  Ring )
28 rnggrp 16640 . . . . . . . . 9  |-  ( (Scalar `  W )  e.  Ring  -> 
(Scalar `  W )  e.  Grp )
2927, 28syl 16 . . . . . . . 8  |-  ( ph  ->  (Scalar `  W )  e.  Grp )
30 eqid 2441 . . . . . . . . . 10  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
3130, 19grpidcl 15559 . . . . . . . . 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 15561 . . . . . . . 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 656 . . . . . . 7  |-  ( ph  ->  ( ( 0g `  (Scalar `  W ) ) ( +g  `  (Scalar `  W ) ) ( 0g `  (Scalar `  W ) ) )  =  ( 0g `  (Scalar `  W ) ) )
3534adantr 462 . . . . . 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 2477 . . . . 5  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  (
( G  .+  H
) `  v )  =  ( 0g `  (Scalar `  W ) ) )
378, 14, 15, 2, 4, 16ldualvaddcl 32497 . . . . . . 7  |-  ( ph  ->  ( G  .+  H
)  e.  F )
3837adantr 462 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  ( G  .+  H )  e.  F )
397, 12, 19, 8, 9ellkr 32456 . . . . . 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 656 . . . . 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 908 . . . 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 3359 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 1364    e. wcel 1761    i^i cin 3324    C_ wss 3325   ` cfv 5415  (class class class)co 6090   Basecbs 14170   +g cplusg 14234  Scalarcsca 14237   0gc0g 14374   Grpcgrp 15406   Ringcrg 16635   LModclmod 16928  LFnlclfn 32424  LKerclk 32452  LDualcld 32490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-map 7212  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-plusg 14247  df-sca 14250  df-vsca 14251  df-0g 14376  df-mnd 15411  df-grp 15538  df-minusg 15539  df-sbg 15540  df-cmn 16272  df-abl 16273  df-mgp 16582  df-ur 16594  df-rng 16637  df-lmod 16930  df-lfl 32425  df-lkr 32453  df-ldual 32491
This theorem is referenced by:  lclkrlem2e  34878  lclkrlem2f  34879  lclkrlem2r  34891  lclkrlem2v  34895  lclkrslem2  34905  lcfrlem2  34910
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