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Theorem kercvrlsm 35391
Description: The domain of a linear function is the subspace sum of the kernel and any subspace which covers the range. (Contributed by Stefan O'Rear, 24-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
Hypotheses
Ref Expression
kercvrlsm.u  |-  U  =  ( LSubSp `  S )
kercvrlsm.p  |-  .(+)  =  (
LSSum `  S )
kercvrlsm.z  |-  .0.  =  ( 0g `  T )
kercvrlsm.k  |-  K  =  ( `' F " {  .0.  } )
kercvrlsm.b  |-  B  =  ( Base `  S
)
kercvrlsm.f  |-  ( ph  ->  F  e.  ( S LMHom 
T ) )
kercvrlsm.d  |-  ( ph  ->  D  e.  U )
kercvrlsm.cv  |-  ( ph  ->  ( F " D
)  =  ran  F
)
Assertion
Ref Expression
kercvrlsm  |-  ( ph  ->  ( K  .(+)  D )  =  B )

Proof of Theorem kercvrlsm
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 kercvrlsm.f . . . . 5  |-  ( ph  ->  F  e.  ( S LMHom 
T ) )
2 lmhmlmod1 17999 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
31, 2syl 17 . . . 4  |-  ( ph  ->  S  e.  LMod )
4 kercvrlsm.k . . . . . 6  |-  K  =  ( `' F " {  .0.  } )
5 kercvrlsm.z . . . . . 6  |-  .0.  =  ( 0g `  T )
6 kercvrlsm.u . . . . . 6  |-  U  =  ( LSubSp `  S )
74, 5, 6lmhmkerlss 18017 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  K  e.  U )
81, 7syl 17 . . . 4  |-  ( ph  ->  K  e.  U )
9 kercvrlsm.d . . . 4  |-  ( ph  ->  D  e.  U )
10 kercvrlsm.p . . . . 5  |-  .(+)  =  (
LSSum `  S )
116, 10lsmcl 18049 . . . 4  |-  ( ( S  e.  LMod  /\  K  e.  U  /\  D  e.  U )  ->  ( K  .(+)  D )  e.  U )
123, 8, 9, 11syl3anc 1230 . . 3  |-  ( ph  ->  ( K  .(+)  D )  e.  U )
13 kercvrlsm.b . . . 4  |-  B  =  ( Base `  S
)
1413, 6lssss 17903 . . 3  |-  ( ( K  .(+)  D )  e.  U  ->  ( K 
.(+)  D )  C_  B
)
1512, 14syl 17 . 2  |-  ( ph  ->  ( K  .(+)  D ) 
C_  B )
16 eqid 2402 . . . . . . . . . . 11  |-  ( Base `  T )  =  (
Base `  T )
1713, 16lmhmf 18000 . . . . . . . . . 10  |-  ( F  e.  ( S LMHom  T
)  ->  F : B
--> ( Base `  T
) )
181, 17syl 17 . . . . . . . . 9  |-  ( ph  ->  F : B --> ( Base `  T ) )
19 ffn 5714 . . . . . . . . 9  |-  ( F : B --> ( Base `  T )  ->  F  Fn  B )
2018, 19syl 17 . . . . . . . 8  |-  ( ph  ->  F  Fn  B )
21 fnfvelrn 6006 . . . . . . . 8  |-  ( ( F  Fn  B  /\  a  e.  B )  ->  ( F `  a
)  e.  ran  F
)
2220, 21sylan 469 . . . . . . 7  |-  ( (
ph  /\  a  e.  B )  ->  ( F `  a )  e.  ran  F )
23 kercvrlsm.cv . . . . . . . 8  |-  ( ph  ->  ( F " D
)  =  ran  F
)
2423adantr 463 . . . . . . 7  |-  ( (
ph  /\  a  e.  B )  ->  ( F " D )  =  ran  F )
2522, 24eleqtrrd 2493 . . . . . 6  |-  ( (
ph  /\  a  e.  B )  ->  ( F `  a )  e.  ( F " D
) )
2620adantr 463 . . . . . . 7  |-  ( (
ph  /\  a  e.  B )  ->  F  Fn  B )
2713, 6lssss 17903 . . . . . . . . 9  |-  ( D  e.  U  ->  D  C_  B )
289, 27syl 17 . . . . . . . 8  |-  ( ph  ->  D  C_  B )
2928adantr 463 . . . . . . 7  |-  ( (
ph  /\  a  e.  B )  ->  D  C_  B )
30 fvelimab 5905 . . . . . . 7  |-  ( ( F  Fn  B  /\  D  C_  B )  -> 
( ( F `  a )  e.  ( F " D )  <->  E. b  e.  D  ( F `  b )  =  ( F `  a ) ) )
3126, 29, 30syl2anc 659 . . . . . 6  |-  ( (
ph  /\  a  e.  B )  ->  (
( F `  a
)  e.  ( F
" D )  <->  E. b  e.  D  ( F `  b )  =  ( F `  a ) ) )
3225, 31mpbid 210 . . . . 5  |-  ( (
ph  /\  a  e.  B )  ->  E. b  e.  D  ( F `  b )  =  ( F `  a ) )
33 lmodgrp 17839 . . . . . . . . . . . . 13  |-  ( S  e.  LMod  ->  S  e. 
Grp )
343, 33syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  S  e.  Grp )
3534adantr 463 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  D ) )  ->  S  e.  Grp )
36 simprl 756 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  D ) )  -> 
a  e.  B )
3728sselda 3442 . . . . . . . . . . . 12  |-  ( (
ph  /\  b  e.  D )  ->  b  e.  B )
3837adantrl 714 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  D ) )  -> 
b  e.  B )
39 eqid 2402 . . . . . . . . . . . 12  |-  ( +g  `  S )  =  ( +g  `  S )
40 eqid 2402 . . . . . . . . . . . 12  |-  ( -g `  S )  =  (
-g `  S )
4113, 39, 40grpnpcan 16454 . . . . . . . . . . 11  |-  ( ( S  e.  Grp  /\  a  e.  B  /\  b  e.  B )  ->  ( ( a (
-g `  S )
b ) ( +g  `  S ) b )  =  a )
4235, 36, 38, 41syl3anc 1230 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  D ) )  -> 
( ( a (
-g `  S )
b ) ( +g  `  S ) b )  =  a )
4342adantr 463 . . . . . . . . 9  |-  ( ( ( ph  /\  (
a  e.  B  /\  b  e.  D )
)  /\  ( F `  b )  =  ( F `  a ) )  ->  ( (
a ( -g `  S
) b ) ( +g  `  S ) b )  =  a )
443ad2antrr 724 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
a  e.  B  /\  b  e.  D )
)  /\  ( F `  b )  =  ( F `  a ) )  ->  S  e.  LMod )
4513, 6lssss 17903 . . . . . . . . . . . 12  |-  ( K  e.  U  ->  K  C_  B )
468, 45syl 17 . . . . . . . . . . 11  |-  ( ph  ->  K  C_  B )
4746ad2antrr 724 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
a  e.  B  /\  b  e.  D )
)  /\  ( F `  b )  =  ( F `  a ) )  ->  K  C_  B
)
4828ad2antrr 724 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
a  e.  B  /\  b  e.  D )
)  /\  ( F `  b )  =  ( F `  a ) )  ->  D  C_  B
)
49 eqcom 2411 . . . . . . . . . . . 12  |-  ( ( F `  b )  =  ( F `  a )  <->  ( F `  a )  =  ( F `  b ) )
50 lmghm 17997 . . . . . . . . . . . . . . 15  |-  ( F  e.  ( S LMHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
511, 50syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  e.  ( S 
GrpHom  T ) )
5251adantr 463 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  D ) )  ->  F  e.  ( S  GrpHom  T ) )
5313, 5, 4, 40ghmeqker 16617 . . . . . . . . . . . . 13  |-  ( ( F  e.  ( S 
GrpHom  T )  /\  a  e.  B  /\  b  e.  B )  ->  (
( F `  a
)  =  ( F `
 b )  <->  ( a
( -g `  S ) b )  e.  K
) )
5452, 36, 38, 53syl3anc 1230 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  D ) )  -> 
( ( F `  a )  =  ( F `  b )  <-> 
( a ( -g `  S ) b )  e.  K ) )
5549, 54syl5bb 257 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  D ) )  -> 
( ( F `  b )  =  ( F `  a )  <-> 
( a ( -g `  S ) b )  e.  K ) )
5655biimpa 482 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
a  e.  B  /\  b  e.  D )
)  /\  ( F `  b )  =  ( F `  a ) )  ->  ( a
( -g `  S ) b )  e.  K
)
57 simplrr 763 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
a  e.  B  /\  b  e.  D )
)  /\  ( F `  b )  =  ( F `  a ) )  ->  b  e.  D )
5813, 39, 10lsmelvalix 16985 . . . . . . . . . 10  |-  ( ( ( S  e.  LMod  /\  K  C_  B  /\  D  C_  B )  /\  ( ( a (
-g `  S )
b )  e.  K  /\  b  e.  D
) )  ->  (
( a ( -g `  S ) b ) ( +g  `  S
) b )  e.  ( K  .(+)  D ) )
5944, 47, 48, 56, 57, 58syl32anc 1238 . . . . . . . . 9  |-  ( ( ( ph  /\  (
a  e.  B  /\  b  e.  D )
)  /\  ( F `  b )  =  ( F `  a ) )  ->  ( (
a ( -g `  S
) b ) ( +g  `  S ) b )  e.  ( K  .(+)  D )
)
6043, 59eqeltrrd 2491 . . . . . . . 8  |-  ( ( ( ph  /\  (
a  e.  B  /\  b  e.  D )
)  /\  ( F `  b )  =  ( F `  a ) )  ->  a  e.  ( K  .(+)  D ) )
6160ex 432 . . . . . . 7  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  D ) )  -> 
( ( F `  b )  =  ( F `  a )  ->  a  e.  ( K  .(+)  D )
) )
6261anassrs 646 . . . . . 6  |-  ( ( ( ph  /\  a  e.  B )  /\  b  e.  D )  ->  (
( F `  b
)  =  ( F `
 a )  -> 
a  e.  ( K 
.(+)  D ) ) )
6362rexlimdva 2896 . . . . 5  |-  ( (
ph  /\  a  e.  B )  ->  ( E. b  e.  D  ( F `  b )  =  ( F `  a )  ->  a  e.  ( K  .(+)  D ) ) )
6432, 63mpd 15 . . . 4  |-  ( (
ph  /\  a  e.  B )  ->  a  e.  ( K  .(+)  D ) )
6564ex 432 . . 3  |-  ( ph  ->  ( a  e.  B  ->  a  e.  ( K 
.(+)  D ) ) )
6665ssrdv 3448 . 2  |-  ( ph  ->  B  C_  ( K  .(+) 
D ) )
6715, 66eqssd 3459 1  |-  ( ph  ->  ( K  .(+)  D )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   E.wrex 2755    C_ wss 3414   {csn 3972   `'ccnv 4822   ran crn 4824   "cima 4826    Fn wfn 5564   -->wf 5565   ` cfv 5569  (class class class)co 6278   Basecbs 14841   +g cplusg 14909   0gc0g 15054   Grpcgrp 16377   -gcsg 16379    GrpHom cghm 16588   LSSumclsm 16978   LModclmod 17832   LSubSpclss 17898   LMHom clmhm 17985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-0g 15056  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-submnd 16291  df-grp 16381  df-minusg 16382  df-sbg 16383  df-subg 16522  df-ghm 16589  df-cntz 16679  df-lsm 16980  df-cmn 17124  df-abl 17125  df-mgp 17462  df-ur 17474  df-ring 17520  df-lmod 17834  df-lss 17899  df-lmhm 17988
This theorem is referenced by:  lmhmfgsplit  35394
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