Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  kercvrlsm Structured version   Unicode version

Theorem kercvrlsm 29576
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 17222 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
31, 2syl 16 . . . 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 17240 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  K  e.  U )
81, 7syl 16 . . . 4  |-  ( ph  ->  K  e.  U )
9 kercvrlsm.d . . . 4  |-  ( ph  ->  D  e.  U )
10 kercvrlsm.p . . . . 5  |-  .(+)  =  (
LSSum `  S )
116, 10lsmcl 17272 . . . 4  |-  ( ( S  e.  LMod  /\  K  e.  U  /\  D  e.  U )  ->  ( K  .(+)  D )  e.  U )
123, 8, 9, 11syl3anc 1219 . . 3  |-  ( ph  ->  ( K  .(+)  D )  e.  U )
13 kercvrlsm.b . . . 4  |-  B  =  ( Base `  S
)
1413, 6lssss 17126 . . 3  |-  ( ( K  .(+)  D )  e.  U  ->  ( K 
.(+)  D )  C_  B
)
1512, 14syl 16 . 2  |-  ( ph  ->  ( K  .(+)  D ) 
C_  B )
16 eqid 2451 . . . . . . . . . . 11  |-  ( Base `  T )  =  (
Base `  T )
1713, 16lmhmf 17223 . . . . . . . . . 10  |-  ( F  e.  ( S LMHom  T
)  ->  F : B
--> ( Base `  T
) )
181, 17syl 16 . . . . . . . . 9  |-  ( ph  ->  F : B --> ( Base `  T ) )
19 ffn 5659 . . . . . . . . 9  |-  ( F : B --> ( Base `  T )  ->  F  Fn  B )
2018, 19syl 16 . . . . . . . 8  |-  ( ph  ->  F  Fn  B )
21 fnfvelrn 5941 . . . . . . . 8  |-  ( ( F  Fn  B  /\  a  e.  B )  ->  ( F `  a
)  e.  ran  F
)
2220, 21sylan 471 . . . . . . 7  |-  ( (
ph  /\  a  e.  B )  ->  ( F `  a )  e.  ran  F )
23 kercvrlsm.cv . . . . . . . 8  |-  ( ph  ->  ( F " D
)  =  ran  F
)
2423adantr 465 . . . . . . 7  |-  ( (
ph  /\  a  e.  B )  ->  ( F " D )  =  ran  F )
2522, 24eleqtrrd 2542 . . . . . 6  |-  ( (
ph  /\  a  e.  B )  ->  ( F `  a )  e.  ( F " D
) )
2620adantr 465 . . . . . . 7  |-  ( (
ph  /\  a  e.  B )  ->  F  Fn  B )
2713, 6lssss 17126 . . . . . . . . 9  |-  ( D  e.  U  ->  D  C_  B )
289, 27syl 16 . . . . . . . 8  |-  ( ph  ->  D  C_  B )
2928adantr 465 . . . . . . 7  |-  ( (
ph  /\  a  e.  B )  ->  D  C_  B )
30 fvelimab 5848 . . . . . . 7  |-  ( ( F  Fn  B  /\  D  C_  B )  -> 
( ( F `  a )  e.  ( F " D )  <->  E. b  e.  D  ( F `  b )  =  ( F `  a ) ) )
3126, 29, 30syl2anc 661 . . . . . 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 17063 . . . . . . . . . . . . 13  |-  ( S  e.  LMod  ->  S  e. 
Grp )
343, 33syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  S  e.  Grp )
3534adantr 465 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  D ) )  ->  S  e.  Grp )
36 simprl 755 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  D ) )  -> 
a  e.  B )
3728sselda 3456 . . . . . . . . . . . 12  |-  ( (
ph  /\  b  e.  D )  ->  b  e.  B )
3837adantrl 715 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  D ) )  -> 
b  e.  B )
39 eqid 2451 . . . . . . . . . . . 12  |-  ( +g  `  S )  =  ( +g  `  S )
40 eqid 2451 . . . . . . . . . . . 12  |-  ( -g `  S )  =  (
-g `  S )
4113, 39, 40grpnpcan 15721 . . . . . . . . . . 11  |-  ( ( S  e.  Grp  /\  a  e.  B  /\  b  e.  B )  ->  ( ( a (
-g `  S )
b ) ( +g  `  S ) b )  =  a )
4235, 36, 38, 41syl3anc 1219 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  D ) )  -> 
( ( a (
-g `  S )
b ) ( +g  `  S ) b )  =  a )
4342adantr 465 . . . . . . . . 9  |-  ( ( ( ph  /\  (
a  e.  B  /\  b  e.  D )
)  /\  ( F `  b )  =  ( F `  a ) )  ->  ( (
a ( -g `  S
) b ) ( +g  `  S ) b )  =  a )
443ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
a  e.  B  /\  b  e.  D )
)  /\  ( F `  b )  =  ( F `  a ) )  ->  S  e.  LMod )
4513, 6lssss 17126 . . . . . . . . . . . 12  |-  ( K  e.  U  ->  K  C_  B )
468, 45syl 16 . . . . . . . . . . 11  |-  ( ph  ->  K  C_  B )
4746ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
a  e.  B  /\  b  e.  D )
)  /\  ( F `  b )  =  ( F `  a ) )  ->  K  C_  B
)
4828ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
a  e.  B  /\  b  e.  D )
)  /\  ( F `  b )  =  ( F `  a ) )  ->  D  C_  B
)
49 eqcom 2460 . . . . . . . . . . . 12  |-  ( ( F `  b )  =  ( F `  a )  <->  ( F `  a )  =  ( F `  b ) )
50 lmghm 17220 . . . . . . . . . . . . . . 15  |-  ( F  e.  ( S LMHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
511, 50syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  e.  ( S 
GrpHom  T ) )
5251adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  D ) )  ->  F  e.  ( S  GrpHom  T ) )
5313, 5, 4, 40ghmeqker 15877 . . . . . . . . . . . . 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 1219 . . . . . . . . . . . 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 484 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
a  e.  B  /\  b  e.  D )
)  /\  ( F `  b )  =  ( F `  a ) )  ->  ( a
( -g `  S ) b )  e.  K
)
57 simplrr 760 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
a  e.  B  /\  b  e.  D )
)  /\  ( F `  b )  =  ( F `  a ) )  ->  b  e.  D )
5813, 39, 10lsmelvalix 16246 . . . . . . . . . 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 1227 . . . . . . . . 9  |-  ( ( ( ph  /\  (
a  e.  B  /\  b  e.  D )
)  /\  ( F `  b )  =  ( F `  a ) )  ->  ( (
a ( -g `  S
) b ) ( +g  `  S ) b )  e.  ( K  .(+)  D )
)
6043, 59eqeltrrd 2540 . . . . . . . 8  |-  ( ( ( ph  /\  (
a  e.  B  /\  b  e.  D )
)  /\  ( F `  b )  =  ( F `  a ) )  ->  a  e.  ( K  .(+)  D ) )
6160ex 434 . . . . . . 7  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  D ) )  -> 
( ( F `  b )  =  ( F `  a )  ->  a  e.  ( K  .(+)  D )
) )
6261anassrs 648 . . . . . 6  |-  ( ( ( ph  /\  a  e.  B )  /\  b  e.  D )  ->  (
( F `  b
)  =  ( F `
 a )  -> 
a  e.  ( K 
.(+)  D ) ) )
6362rexlimdva 2939 . . . . 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 434 . . 3  |-  ( ph  ->  ( a  e.  B  ->  a  e.  ( K 
.(+)  D ) ) )
6665ssrdv 3462 . 2  |-  ( ph  ->  B  C_  ( K  .(+) 
D ) )
6715, 66eqssd 3473 1  |-  ( ph  ->  ( K  .(+)  D )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   E.wrex 2796    C_ wss 3428   {csn 3977   `'ccnv 4939   ran crn 4941   "cima 4943    Fn wfn 5513   -->wf 5514   ` cfv 5518  (class class class)co 6192   Basecbs 14278   +g cplusg 14342   0gc0g 14482   Grpcgrp 15514   -gcsg 15517    GrpHom cghm 15848   LSSumclsm 16239   LModclmod 17056   LSubSpclss 17121   LMHom clmhm 17208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-1st 6679  df-2nd 6680  df-recs 6934  df-rdg 6968  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-ndx 14281  df-slot 14282  df-base 14283  df-sets 14284  df-ress 14285  df-plusg 14355  df-0g 14484  df-mnd 15519  df-submnd 15569  df-grp 15649  df-minusg 15650  df-sbg 15651  df-subg 15782  df-ghm 15849  df-cntz 15939  df-lsm 16241  df-cmn 16385  df-abl 16386  df-mgp 16699  df-ur 16711  df-rng 16755  df-lmod 17058  df-lss 17122  df-lmhm 17211
This theorem is referenced by:  lmhmfgsplit  29579
  Copyright terms: Public domain W3C validator