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Theorem kercvrlsm 27284
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 15806 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
31, 2syl 15 . . . 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 15824 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  K  e.  U )
81, 7syl 15 . . . 4  |-  ( ph  ->  K  e.  U )
9 kercvrlsm.d . . . 4  |-  ( ph  ->  D  e.  U )
10 kercvrlsm.p . . . . 5  |-  .(+)  =  (
LSSum `  S )
116, 10lsmcl 15852 . . . 4  |-  ( ( S  e.  LMod  /\  K  e.  U  /\  D  e.  U )  ->  ( K  .(+)  D )  e.  U )
123, 8, 9, 11syl3anc 1182 . . 3  |-  ( ph  ->  ( K  .(+)  D )  e.  U )
13 kercvrlsm.b . . . 4  |-  B  =  ( Base `  S
)
1413, 6lssss 15710 . . 3  |-  ( ( K  .(+)  D )  e.  U  ->  ( K 
.(+)  D )  C_  B
)
1512, 14syl 15 . 2  |-  ( ph  ->  ( K  .(+)  D ) 
C_  B )
16 eqid 2296 . . . . . . . . . . 11  |-  ( Base `  T )  =  (
Base `  T )
1713, 16lmhmf 15807 . . . . . . . . . 10  |-  ( F  e.  ( S LMHom  T
)  ->  F : B
--> ( Base `  T
) )
181, 17syl 15 . . . . . . . . 9  |-  ( ph  ->  F : B --> ( Base `  T ) )
19 ffn 5405 . . . . . . . . 9  |-  ( F : B --> ( Base `  T )  ->  F  Fn  B )
2018, 19syl 15 . . . . . . . 8  |-  ( ph  ->  F  Fn  B )
21 fnfvelrn 5678 . . . . . . . 8  |-  ( ( F  Fn  B  /\  a  e.  B )  ->  ( F `  a
)  e.  ran  F
)
2220, 21sylan 457 . . . . . . 7  |-  ( (
ph  /\  a  e.  B )  ->  ( F `  a )  e.  ran  F )
23 kercvrlsm.cv . . . . . . . 8  |-  ( ph  ->  ( F " D
)  =  ran  F
)
2423adantr 451 . . . . . . 7  |-  ( (
ph  /\  a  e.  B )  ->  ( F " D )  =  ran  F )
2522, 24eleqtrrd 2373 . . . . . 6  |-  ( (
ph  /\  a  e.  B )  ->  ( F `  a )  e.  ( F " D
) )
2620adantr 451 . . . . . . 7  |-  ( (
ph  /\  a  e.  B )  ->  F  Fn  B )
2713, 6lssss 15710 . . . . . . . . 9  |-  ( D  e.  U  ->  D  C_  B )
289, 27syl 15 . . . . . . . 8  |-  ( ph  ->  D  C_  B )
2928adantr 451 . . . . . . 7  |-  ( (
ph  /\  a  e.  B )  ->  D  C_  B )
30 fvelimab 5594 . . . . . . 7  |-  ( ( F  Fn  B  /\  D  C_  B )  -> 
( ( F `  a )  e.  ( F " D )  <->  E. b  e.  D  ( F `  b )  =  ( F `  a ) ) )
3126, 29, 30syl2anc 642 . . . . . 6  |-  ( (
ph  /\  a  e.  B )  ->  (
( F `  a
)  e.  ( F
" D )  <->  E. b  e.  D  ( F `  b )  =  ( F `  a ) ) )
3225, 31mpbid 201 . . . . 5  |-  ( (
ph  /\  a  e.  B )  ->  E. b  e.  D  ( F `  b )  =  ( F `  a ) )
33 lmodgrp 15650 . . . . . . . . . . . . 13  |-  ( S  e.  LMod  ->  S  e. 
Grp )
343, 33syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  S  e.  Grp )
3534adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  D ) )  ->  S  e.  Grp )
36 simprl 732 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  D ) )  -> 
a  e.  B )
3728sselda 3193 . . . . . . . . . . . 12  |-  ( (
ph  /\  b  e.  D )  ->  b  e.  B )
3837adantrl 696 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  D ) )  -> 
b  e.  B )
39 eqid 2296 . . . . . . . . . . . 12  |-  ( +g  `  S )  =  ( +g  `  S )
40 eqid 2296 . . . . . . . . . . . 12  |-  ( -g `  S )  =  (
-g `  S )
4113, 39, 40grpnpcan 14573 . . . . . . . . . . 11  |-  ( ( S  e.  Grp  /\  a  e.  B  /\  b  e.  B )  ->  ( ( a (
-g `  S )
b ) ( +g  `  S ) b )  =  a )
4235, 36, 38, 41syl3anc 1182 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  D ) )  -> 
( ( a (
-g `  S )
b ) ( +g  `  S ) b )  =  a )
4342adantr 451 . . . . . . . . 9  |-  ( ( ( ph  /\  (
a  e.  B  /\  b  e.  D )
)  /\  ( F `  b )  =  ( F `  a ) )  ->  ( (
a ( -g `  S
) b ) ( +g  `  S ) b )  =  a )
443ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
a  e.  B  /\  b  e.  D )
)  /\  ( F `  b )  =  ( F `  a ) )  ->  S  e.  LMod )
4513, 6lssss 15710 . . . . . . . . . . . 12  |-  ( K  e.  U  ->  K  C_  B )
468, 45syl 15 . . . . . . . . . . 11  |-  ( ph  ->  K  C_  B )
4746ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
a  e.  B  /\  b  e.  D )
)  /\  ( F `  b )  =  ( F `  a ) )  ->  K  C_  B
)
4828ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
a  e.  B  /\  b  e.  D )
)  /\  ( F `  b )  =  ( F `  a ) )  ->  D  C_  B
)
49 eqcom 2298 . . . . . . . . . . . 12  |-  ( ( F `  b )  =  ( F `  a )  <->  ( F `  a )  =  ( F `  b ) )
50 lmghm 15804 . . . . . . . . . . . . . . 15  |-  ( F  e.  ( S LMHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
511, 50syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  e.  ( S 
GrpHom  T ) )
5251adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  D ) )  ->  F  e.  ( S  GrpHom  T ) )
5313, 5, 4, 40ghmeqker 14725 . . . . . . . . . . . . 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 1182 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  D ) )  -> 
( ( F `  a )  =  ( F `  b )  <-> 
( a ( -g `  S ) b )  e.  K ) )
5549, 54syl5bb 248 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  D ) )  -> 
( ( F `  b )  =  ( F `  a )  <-> 
( a ( -g `  S ) b )  e.  K ) )
5655biimpa 470 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
a  e.  B  /\  b  e.  D )
)  /\  ( F `  b )  =  ( F `  a ) )  ->  ( a
( -g `  S ) b )  e.  K
)
57 simplrr 737 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
a  e.  B  /\  b  e.  D )
)  /\  ( F `  b )  =  ( F `  a ) )  ->  b  e.  D )
5813, 39, 10lsmelvalix 14968 . . . . . . . . . 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 1190 . . . . . . . . 9  |-  ( ( ( ph  /\  (
a  e.  B  /\  b  e.  D )
)  /\  ( F `  b )  =  ( F `  a ) )  ->  ( (
a ( -g `  S
) b ) ( +g  `  S ) b )  e.  ( K  .(+)  D )
)
6043, 59eqeltrrd 2371 . . . . . . . 8  |-  ( ( ( ph  /\  (
a  e.  B  /\  b  e.  D )
)  /\  ( F `  b )  =  ( F `  a ) )  ->  a  e.  ( K  .(+)  D ) )
6160ex 423 . . . . . . 7  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  D ) )  -> 
( ( F `  b )  =  ( F `  a )  ->  a  e.  ( K  .(+)  D )
) )
6261anassrs 629 . . . . . 6  |-  ( ( ( ph  /\  a  e.  B )  /\  b  e.  D )  ->  (
( F `  b
)  =  ( F `
 a )  -> 
a  e.  ( K 
.(+)  D ) ) )
6362rexlimdva 2680 . . . . 5  |-  ( (
ph  /\  a  e.  B )  ->  ( E. b  e.  D  ( F `  b )  =  ( F `  a )  ->  a  e.  ( K  .(+)  D ) ) )
6432, 63mpd 14 . . . 4  |-  ( (
ph  /\  a  e.  B )  ->  a  e.  ( K  .(+)  D ) )
6564ex 423 . . 3  |-  ( ph  ->  ( a  e.  B  ->  a  e.  ( K 
.(+)  D ) ) )
6665ssrdv 3198 . 2  |-  ( ph  ->  B  C_  ( K  .(+) 
D ) )
6715, 66eqssd 3209 1  |-  ( ph  ->  ( K  .(+)  D )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557    C_ wss 3165   {csn 3653   `'ccnv 4704   ran crn 4706   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   0gc0g 13416   Grpcgrp 14378   -gcsg 14381    GrpHom cghm 14696   LSSumclsm 14961   LModclmod 15643   LSubSpclss 15705   LMHom clmhm 15792
This theorem is referenced by:  lmhmfgsplit  27287
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-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-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  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-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-ghm 14697  df-cntz 14809  df-lsm 14963  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-lmod 15645  df-lss 15706  df-lmhm 15795
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