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Theorem lkrin 29976
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 3371 . . 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 2296 . . . . . . 7  |-  ( Base `  W )  =  (
Base `  W )
8 lkrin.f . . . . . . 7  |-  F  =  (LFnl `  W )
9 lkrin.k . . . . . . 7  |-  K  =  (LKer `  W )
107, 8, 9lkrcl 29904 . . . . . 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 2296 . . . . . . 7  |-  (Scalar `  W )  =  (Scalar `  W )
13 eqid 2296 . . . . . . 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 29943 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  (
( G  .+  H
) `  v )  =  ( ( G `
 v ) ( +g  `  (Scalar `  W ) ) ( H `  v ) ) )
19 eqid 2296 . . . . . . . . 9  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
2012, 19, 8, 9lkrf0 29905 . . . . . . . 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 29905 . . . . . . . 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 5892 . . . . . 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 15651 . . . . . . . . . 10  |-  ( W  e.  LMod  ->  (Scalar `  W )  e.  Ring )
272, 26syl 15 . . . . . . . . 9  |-  ( ph  ->  (Scalar `  W )  e.  Ring )
28 rnggrp 15362 . . . . . . . . 9  |-  ( (Scalar `  W )  e.  Ring  -> 
(Scalar `  W )  e.  Grp )
2927, 28syl 15 . . . . . . . 8  |-  ( ph  ->  (Scalar `  W )  e.  Grp )
30 eqid 2296 . . . . . . . . . 10  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
3130, 19grpidcl 14526 . . . . . . . . 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 14528 . . . . . . . 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 2332 . . . . 5  |-  ( (
ph  /\  ( v  e.  ( K `  G
)  /\  v  e.  ( K `  H ) ) )  ->  (
( G  .+  H
) `  v )  =  ( 0g `  (Scalar `  W ) ) )
378, 14, 15, 2, 4, 16ldualvaddcl 29942 . . . . . . 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 29901 . . . . . 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 3198 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 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224  Scalarcsca 13227   0gc0g 13416   Grpcgrp 14378   Ringcrg 15353   LModclmod 15643  LFnlclfn 29869  LKerclk 29897  LDualcld 29935
This theorem is referenced by:  lclkrlem2e  32323  lclkrlem2f  32324  lclkrlem2r  32336  lclkrlem2v  32340  lclkrslem2  32350  lcfrlem2  32355
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-plusg 13237  df-sca 13240  df-vsca 13241  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-lmod 15645  df-lfl 29870  df-lkr 29898  df-ldual 29936
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