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Theorem kercvrlsm 27158
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 16109 . . . . 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 16127 . . . . 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 16155 . . . 4  |-  ( ( S  e.  LMod  /\  K  e.  U  /\  D  e.  U )  ->  ( K  .(+)  D )  e.  U )
123, 8, 9, 11syl3anc 1184 . . 3  |-  ( ph  ->  ( K  .(+)  D )  e.  U )
13 kercvrlsm.b . . . 4  |-  B  =  ( Base `  S
)
1413, 6lssss 16013 . . 3  |-  ( ( K  .(+)  D )  e.  U  ->  ( K 
.(+)  D )  C_  B
)
1512, 14syl 16 . 2  |-  ( ph  ->  ( K  .(+)  D ) 
C_  B )
16 eqid 2436 . . . . . . . . . . 11  |-  ( Base `  T )  =  (
Base `  T )
1713, 16lmhmf 16110 . . . . . . . . . 10  |-  ( F  e.  ( S LMHom  T
)  ->  F : B
--> ( Base `  T
) )
181, 17syl 16 . . . . . . . . 9  |-  ( ph  ->  F : B --> ( Base `  T ) )
19 ffn 5591 . . . . . . . . 9  |-  ( F : B --> ( Base `  T )  ->  F  Fn  B )
2018, 19syl 16 . . . . . . . 8  |-  ( ph  ->  F  Fn  B )
21 fnfvelrn 5867 . . . . . . . 8  |-  ( ( F  Fn  B  /\  a  e.  B )  ->  ( F `  a
)  e.  ran  F
)
2220, 21sylan 458 . . . . . . 7  |-  ( (
ph  /\  a  e.  B )  ->  ( F `  a )  e.  ran  F )
23 kercvrlsm.cv . . . . . . . 8  |-  ( ph  ->  ( F " D
)  =  ran  F
)
2423adantr 452 . . . . . . 7  |-  ( (
ph  /\  a  e.  B )  ->  ( F " D )  =  ran  F )
2522, 24eleqtrrd 2513 . . . . . 6  |-  ( (
ph  /\  a  e.  B )  ->  ( F `  a )  e.  ( F " D
) )
2620adantr 452 . . . . . . 7  |-  ( (
ph  /\  a  e.  B )  ->  F  Fn  B )
2713, 6lssss 16013 . . . . . . . . 9  |-  ( D  e.  U  ->  D  C_  B )
289, 27syl 16 . . . . . . . 8  |-  ( ph  ->  D  C_  B )
2928adantr 452 . . . . . . 7  |-  ( (
ph  /\  a  e.  B )  ->  D  C_  B )
30 fvelimab 5782 . . . . . . 7  |-  ( ( F  Fn  B  /\  D  C_  B )  -> 
( ( F `  a )  e.  ( F " D )  <->  E. b  e.  D  ( F `  b )  =  ( F `  a ) ) )
3126, 29, 30syl2anc 643 . . . . . 6  |-  ( (
ph  /\  a  e.  B )  ->  (
( F `  a
)  e.  ( F
" D )  <->  E. b  e.  D  ( F `  b )  =  ( F `  a ) ) )
3225, 31mpbid 202 . . . . 5  |-  ( (
ph  /\  a  e.  B )  ->  E. b  e.  D  ( F `  b )  =  ( F `  a ) )
33 lmodgrp 15957 . . . . . . . . . . . . 13  |-  ( S  e.  LMod  ->  S  e. 
Grp )
343, 33syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  S  e.  Grp )
3534adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  D ) )  ->  S  e.  Grp )
36 simprl 733 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  D ) )  -> 
a  e.  B )
3728sselda 3348 . . . . . . . . . . . 12  |-  ( (
ph  /\  b  e.  D )  ->  b  e.  B )
3837adantrl 697 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  D ) )  -> 
b  e.  B )
39 eqid 2436 . . . . . . . . . . . 12  |-  ( +g  `  S )  =  ( +g  `  S )
40 eqid 2436 . . . . . . . . . . . 12  |-  ( -g `  S )  =  (
-g `  S )
4113, 39, 40grpnpcan 14880 . . . . . . . . . . 11  |-  ( ( S  e.  Grp  /\  a  e.  B  /\  b  e.  B )  ->  ( ( a (
-g `  S )
b ) ( +g  `  S ) b )  =  a )
4235, 36, 38, 41syl3anc 1184 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  D ) )  -> 
( ( a (
-g `  S )
b ) ( +g  `  S ) b )  =  a )
4342adantr 452 . . . . . . . . 9  |-  ( ( ( ph  /\  (
a  e.  B  /\  b  e.  D )
)  /\  ( F `  b )  =  ( F `  a ) )  ->  ( (
a ( -g `  S
) b ) ( +g  `  S ) b )  =  a )
443ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
a  e.  B  /\  b  e.  D )
)  /\  ( F `  b )  =  ( F `  a ) )  ->  S  e.  LMod )
4513, 6lssss 16013 . . . . . . . . . . . 12  |-  ( K  e.  U  ->  K  C_  B )
468, 45syl 16 . . . . . . . . . . 11  |-  ( ph  ->  K  C_  B )
4746ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
a  e.  B  /\  b  e.  D )
)  /\  ( F `  b )  =  ( F `  a ) )  ->  K  C_  B
)
4828ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
a  e.  B  /\  b  e.  D )
)  /\  ( F `  b )  =  ( F `  a ) )  ->  D  C_  B
)
49 eqcom 2438 . . . . . . . . . . . 12  |-  ( ( F `  b )  =  ( F `  a )  <->  ( F `  a )  =  ( F `  b ) )
50 lmghm 16107 . . . . . . . . . . . . . . 15  |-  ( F  e.  ( S LMHom  T
)  ->  F  e.  ( S  GrpHom  T ) )
511, 50syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  e.  ( S 
GrpHom  T ) )
5251adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  D ) )  ->  F  e.  ( S  GrpHom  T ) )
5313, 5, 4, 40ghmeqker 15032 . . . . . . . . . . . . 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 1184 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  D ) )  -> 
( ( F `  a )  =  ( F `  b )  <-> 
( a ( -g `  S ) b )  e.  K ) )
5549, 54syl5bb 249 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  D ) )  -> 
( ( F `  b )  =  ( F `  a )  <-> 
( a ( -g `  S ) b )  e.  K ) )
5655biimpa 471 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
a  e.  B  /\  b  e.  D )
)  /\  ( F `  b )  =  ( F `  a ) )  ->  ( a
( -g `  S ) b )  e.  K
)
57 simplrr 738 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
a  e.  B  /\  b  e.  D )
)  /\  ( F `  b )  =  ( F `  a ) )  ->  b  e.  D )
5813, 39, 10lsmelvalix 15275 . . . . . . . . . 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 1192 . . . . . . . . 9  |-  ( ( ( ph  /\  (
a  e.  B  /\  b  e.  D )
)  /\  ( F `  b )  =  ( F `  a ) )  ->  ( (
a ( -g `  S
) b ) ( +g  `  S ) b )  e.  ( K  .(+)  D )
)
6043, 59eqeltrrd 2511 . . . . . . . 8  |-  ( ( ( ph  /\  (
a  e.  B  /\  b  e.  D )
)  /\  ( F `  b )  =  ( F `  a ) )  ->  a  e.  ( K  .(+)  D ) )
6160ex 424 . . . . . . 7  |-  ( (
ph  /\  ( a  e.  B  /\  b  e.  D ) )  -> 
( ( F `  b )  =  ( F `  a )  ->  a  e.  ( K  .(+)  D )
) )
6261anassrs 630 . . . . . 6  |-  ( ( ( ph  /\  a  e.  B )  /\  b  e.  D )  ->  (
( F `  b
)  =  ( F `
 a )  -> 
a  e.  ( K 
.(+)  D ) ) )
6362rexlimdva 2830 . . . . 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 424 . . 3  |-  ( ph  ->  ( a  e.  B  ->  a  e.  ( K 
.(+)  D ) ) )
6665ssrdv 3354 . 2  |-  ( ph  ->  B  C_  ( K  .(+) 
D ) )
6715, 66eqssd 3365 1  |-  ( ph  ->  ( K  .(+)  D )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2706    C_ wss 3320   {csn 3814   `'ccnv 4877   ran crn 4879   "cima 4881    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   Basecbs 13469   +g cplusg 13529   0gc0g 13723   Grpcgrp 14685   -gcsg 14688    GrpHom cghm 15003   LSSumclsm 15268   LModclmod 15950   LSubSpclss 16008   LMHom clmhm 16095
This theorem is referenced by:  lmhmfgsplit  27161
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-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-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  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-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-0g 13727  df-mnd 14690  df-submnd 14739  df-grp 14812  df-minusg 14813  df-sbg 14814  df-subg 14941  df-ghm 15004  df-cntz 15116  df-lsm 15270  df-cmn 15414  df-abl 15415  df-mgp 15649  df-rng 15663  df-ur 15665  df-lmod 15952  df-lss 16009  df-lmhm 16098
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