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Theorem lkrin 29354
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 3358 . . 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 451 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  W  e.  LMod )
4 lkrin.e . . . . . . 7  |-  ( ph  ->  G  e.  F )
54adantr 451 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  G  e.  F )
6 simprl 732 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  v  e.  ( K `  G
) )
7 eqid 2283 . . . . . . 7  |-  ( Base `  W )  =  (
Base `  W )
8 lkrin.f . . . . . . 7  |-  F  =  (LFnl `  W )
9 lkrin.k . . . . . . 7  |-  K  =  (LKer `  W )
107, 8, 9lkrcl 29282 . . . . . 6  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  v  e.  ( K `  G
) )  ->  v  e.  ( Base `  W
) )
113, 5, 6, 10syl3anc 1182 . . . . 5  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  v  e.  ( Base `  W
) )
12 eqid 2283 . . . . . . 7  |-  (Scalar `  W )  =  (Scalar `  W )
13 eqid 2283 . . . . . . 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 451 . . . . . . 7  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  H  e.  F )
187, 12, 13, 8, 14, 15, 3, 5, 17, 11ldualvaddval 29321 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  (
( G  .+  H
) `  v )  =  ( ( G `
 v ) ( +g  `  (Scalar `  W ) ) ( H `  v ) ) )
19 eqid 2283 . . . . . . . . 9  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
2012, 19, 8, 9lkrf0 29283 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  v  e.  ( K `  G
) )  ->  ( G `  v )  =  ( 0g `  (Scalar `  W ) ) )
213, 5, 6, 20syl3anc 1182 . . . . . . 7  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  ( G `  v )  =  ( 0g `  (Scalar `  W ) ) )
22 simprr 733 . . . . . . . 8  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  v  e.  ( K `  H
) )
2312, 19, 8, 9lkrf0 29283 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  H  e.  F  /\  v  e.  ( K `  H
) )  ->  ( H `  v )  =  ( 0g `  (Scalar `  W ) ) )
243, 17, 22, 23syl3anc 1182 . . . . . . 7  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  ( H `  v )  =  ( 0g `  (Scalar `  W ) ) )
2521, 24oveq12d 5876 . . . . . 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 15635 . . . . . . . . . 10  |-  ( W  e.  LMod  ->  (Scalar `  W )  e.  Ring )
272, 26syl 15 . . . . . . . . 9  |-  ( ph  ->  (Scalar `  W )  e.  Ring )
28 rnggrp 15346 . . . . . . . . 9  |-  ( (Scalar `  W )  e.  Ring  -> 
(Scalar `  W )  e.  Grp )
2927, 28syl 15 . . . . . . . 8  |-  ( ph  ->  (Scalar `  W )  e.  Grp )
30 eqid 2283 . . . . . . . . . 10  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
3130, 19grpidcl 14510 . . . . . . . . 9  |-  ( (Scalar `  W )  e.  Grp  ->  ( 0g `  (Scalar `  W ) )  e.  ( Base `  (Scalar `  W ) ) )
3229, 31syl 15 . . . . . . . 8  |-  ( ph  ->  ( 0g `  (Scalar `  W ) )  e.  ( Base `  (Scalar `  W ) ) )
3330, 13, 19grplid 14512 . . . . . . . 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 642 . . . . . . 7  |-  ( ph  ->  ( ( 0g `  (Scalar `  W ) ) ( +g  `  (Scalar `  W ) ) ( 0g `  (Scalar `  W ) ) )  =  ( 0g `  (Scalar `  W ) ) )
3534adantr 451 . . . . . 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 2319 . . . . 5  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  (
( G  .+  H
) `  v )  =  ( 0g `  (Scalar `  W ) ) )
378, 14, 15, 2, 4, 16ldualvaddcl 29320 . . . . . . 7  |-  ( ph  ->  ( G  .+  H
)  e.  F )
3837adantr 451 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  ( G  .+  H )  e.  F )
397, 12, 19, 8, 9ellkr 29279 . . . . . 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 642 . . . . 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 888 . . . 4  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  v  e.  ( K `  ( G  .+  H ) ) )
4241ex 423 . . 3  |-  ( ph  ->  ( ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) )  ->  v  e.  ( K `  ( G 
.+  H ) ) ) )
431, 42syl5bi 208 . 2  |-  ( ph  ->  ( v  e.  ( ( K `  G
)  i^i  ( K `  H ) )  -> 
v  e.  ( K `
 ( G  .+  H ) ) ) )
4443ssrdv 3185 1  |-  ( ph  ->  ( ( K `  G )  i^i  ( K `  H )
)  C_  ( K `  ( G  .+  H
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   ` cfv 5255  (class class class)co 5858   Basecbs 13148   +g cplusg 13208  Scalarcsca 13211   0gc0g 13400   Grpcgrp 14362   Ringcrg 15337   LModclmod 15627  LFnlclfn 29247  LKerclk 29275  LDualcld 29313
This theorem is referenced by:  lclkrlem2e  31701  lclkrlem2f  31702  lclkrlem2r  31714  lclkrlem2v  31718  lclkrslem2  31728  lcfrlem2  31733
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-int 3863  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-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  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-3 9805  df-4 9806  df-5 9807  df-6 9808  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-plusg 13221  df-sca 13224  df-vsca 13225  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-lmod 15629  df-lfl 29248  df-lkr 29276  df-ldual 29314
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