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Theorem lkrin 29962
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 3530 . . 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 452 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  W  e.  LMod )
4 lkrin.e . . . . . . 7  |-  ( ph  ->  G  e.  F )
54adantr 452 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  G  e.  F )
6 simprl 733 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  v  e.  ( K `  G
) )
7 eqid 2436 . . . . . . 7  |-  ( Base `  W )  =  (
Base `  W )
8 lkrin.f . . . . . . 7  |-  F  =  (LFnl `  W )
9 lkrin.k . . . . . . 7  |-  K  =  (LKer `  W )
107, 8, 9lkrcl 29890 . . . . . 6  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  v  e.  ( K `  G
) )  ->  v  e.  ( Base `  W
) )
113, 5, 6, 10syl3anc 1184 . . . . 5  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  v  e.  ( Base `  W
) )
12 eqid 2436 . . . . . . 7  |-  (Scalar `  W )  =  (Scalar `  W )
13 eqid 2436 . . . . . . 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 452 . . . . . . 7  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  H  e.  F )
187, 12, 13, 8, 14, 15, 3, 5, 17, 11ldualvaddval 29929 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  (
( G  .+  H
) `  v )  =  ( ( G `
 v ) ( +g  `  (Scalar `  W ) ) ( H `  v ) ) )
19 eqid 2436 . . . . . . . . 9  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
2012, 19, 8, 9lkrf0 29891 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  G  e.  F  /\  v  e.  ( K `  G
) )  ->  ( G `  v )  =  ( 0g `  (Scalar `  W ) ) )
213, 5, 6, 20syl3anc 1184 . . . . . . 7  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  ( G `  v )  =  ( 0g `  (Scalar `  W ) ) )
22 simprr 734 . . . . . . . 8  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  v  e.  ( K `  H
) )
2312, 19, 8, 9lkrf0 29891 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  H  e.  F  /\  v  e.  ( K `  H
) )  ->  ( H `  v )  =  ( 0g `  (Scalar `  W ) ) )
243, 17, 22, 23syl3anc 1184 . . . . . . 7  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  ( H `  v )  =  ( 0g `  (Scalar `  W ) ) )
2521, 24oveq12d 6099 . . . . . 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 15958 . . . . . . . . . 10  |-  ( W  e.  LMod  ->  (Scalar `  W )  e.  Ring )
272, 26syl 16 . . . . . . . . 9  |-  ( ph  ->  (Scalar `  W )  e.  Ring )
28 rnggrp 15669 . . . . . . . . 9  |-  ( (Scalar `  W )  e.  Ring  -> 
(Scalar `  W )  e.  Grp )
2927, 28syl 16 . . . . . . . 8  |-  ( ph  ->  (Scalar `  W )  e.  Grp )
30 eqid 2436 . . . . . . . . . 10  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
3130, 19grpidcl 14833 . . . . . . . . 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 14835 . . . . . . . 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 643 . . . . . . 7  |-  ( ph  ->  ( ( 0g `  (Scalar `  W ) ) ( +g  `  (Scalar `  W ) ) ( 0g `  (Scalar `  W ) ) )  =  ( 0g `  (Scalar `  W ) ) )
3534adantr 452 . . . . . 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 2472 . . . . 5  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  (
( G  .+  H
) `  v )  =  ( 0g `  (Scalar `  W ) ) )
378, 14, 15, 2, 4, 16ldualvaddcl 29928 . . . . . . 7  |-  ( ph  ->  ( G  .+  H
)  e.  F )
3837adantr 452 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  ( G  .+  H )  e.  F )
397, 12, 19, 8, 9ellkr 29887 . . . . . 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 643 . . . . 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 889 . . . 4  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  v  e.  ( K `  ( G  .+  H ) ) )
4241ex 424 . . 3  |-  ( ph  ->  ( ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) )  ->  v  e.  ( K `  ( G 
.+  H ) ) ) )
431, 42syl5bi 209 . 2  |-  ( ph  ->  ( v  e.  ( ( K `  G
)  i^i  ( K `  H ) )  -> 
v  e.  ( K `
 ( G  .+  H ) ) ) )
4443ssrdv 3354 1  |-  ( ph  ->  ( ( K `  G )  i^i  ( K `  H )
)  C_  ( K `  ( G  .+  H
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    i^i cin 3319    C_ wss 3320   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529  Scalarcsca 13532   0gc0g 13723   Grpcgrp 14685   Ringcrg 15660   LModclmod 15950  LFnlclfn 29855  LKerclk 29883  LDualcld 29921
This theorem is referenced by:  lclkrlem2e  32309  lclkrlem2f  32310  lclkrlem2r  32322  lclkrlem2v  32326  lclkrslem2  32336  lcfrlem2  32341
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-plusg 13542  df-sca 13545  df-vsca 13546  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-sbg 14814  df-cmn 15414  df-abl 15415  df-mgp 15649  df-rng 15663  df-ur 15665  df-lmod 15952  df-lfl 29856  df-lkr 29884  df-ldual 29922
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