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Theorem lmhmfgsplit 30664
Description: If the kernel and range of a homomorphism of left modules are finitely generated, then so is the domain. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
Hypotheses
Ref Expression
lmhmfgsplit.z  |-  .0.  =  ( 0g `  T )
lmhmfgsplit.k  |-  K  =  ( `' F " {  .0.  } )
lmhmfgsplit.u  |-  U  =  ( Ss  K )
lmhmfgsplit.v  |-  V  =  ( Ts  ran  F )
Assertion
Ref Expression
lmhmfgsplit  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  S  e. LFinGen )

Proof of Theorem lmhmfgsplit
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp3 998 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  V  e. LFinGen )
2 lmhmlmod2 17478 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
323ad2ant1 1017 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  T  e.  LMod )
4 lmhmrnlss 17496 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  ran  F  e.  ( LSubSp `  T )
)
543ad2ant1 1017 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  ran  F  e.  (
LSubSp `  T ) )
6 lmhmfgsplit.v . . . . 5  |-  V  =  ( Ts  ran  F )
7 eqid 2467 . . . . 5  |-  ( LSubSp `  T )  =  (
LSubSp `  T )
8 eqid 2467 . . . . 5  |-  ( LSpan `  T )  =  (
LSpan `  T )
96, 7, 8islssfg 30648 . . . 4  |-  ( ( T  e.  LMod  /\  ran  F  e.  ( LSubSp `  T
) )  ->  ( V  e. LFinGen  <->  E. a  e.  ~P  ran  F ( a  e. 
Fin  /\  ( ( LSpan `  T ) `  a )  =  ran  F ) ) )
103, 5, 9syl2anc 661 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  ( V  e. LFinGen  <->  E. a  e.  ~P  ran  F ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )
111, 10mpbid 210 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  E. a  e.  ~P  ran  F ( a  e. 
Fin  /\  ( ( LSpan `  T ) `  a )  =  ran  F ) )
12 simpl1 999 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e. 
~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  ->  F  e.  ( S LMHom  T ) )
13 eqid 2467 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
14 eqid 2467 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
1513, 14lmhmf 17480 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
16 ffn 5731 . . . . 5  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  F  Fn  ( Base `  S )
)
1712, 15, 163syl 20 . . . 4  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e. 
~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  ->  F  Fn  ( Base `  S )
)
18 elpwi 4019 . . . . 5  |-  ( a  e.  ~P ran  F  ->  a  C_  ran  F )
1918ad2antrl 727 . . . 4  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e. 
~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  ->  a  C_  ran  F )
20 simprrl 763 . . . 4  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e. 
~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  ->  a  e.  Fin )
21 fipreima 7826 . . . 4  |-  ( ( F  Fn  ( Base `  S )  /\  a  C_ 
ran  F  /\  a  e.  Fin )  ->  E. b  e.  ( ~P ( Base `  S )  i^i  Fin ) ( F "
b )  =  a )
2217, 19, 20, 21syl3anc 1228 . . 3  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e. 
~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  ->  E. b  e.  ( ~P ( Base `  S )  i^i  Fin ) ( F "
b )  =  a )
23 eqid 2467 . . . . . . 7  |-  ( LSubSp `  S )  =  (
LSubSp `  S )
24 eqid 2467 . . . . . . 7  |-  ( LSSum `  S )  =  (
LSSum `  S )
25 lmhmfgsplit.z . . . . . . 7  |-  .0.  =  ( 0g `  T )
26 lmhmfgsplit.k . . . . . . 7  |-  K  =  ( `' F " {  .0.  } )
27 simpll1 1035 . . . . . . 7  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  ->  F  e.  ( S LMHom  T ) )
28 lmhmlmod1 17479 . . . . . . . . . 10  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
29283ad2ant1 1017 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  S  e.  LMod )
3029ad2antrr 725 . . . . . . . 8  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  ->  S  e.  LMod )
31 inss1 3718 . . . . . . . . . . 11  |-  ( ~P ( Base `  S
)  i^i  Fin )  C_ 
~P ( Base `  S
)
3231sseli 3500 . . . . . . . . . 10  |-  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  ->  b  e.  ~P ( Base `  S
) )
33 elpwi 4019 . . . . . . . . . 10  |-  ( b  e.  ~P ( Base `  S )  ->  b  C_  ( Base `  S
) )
3432, 33syl 16 . . . . . . . . 9  |-  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  ->  b  C_  ( Base `  S
) )
3534ad2antrl 727 . . . . . . . 8  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
b  C_  ( Base `  S ) )
36 eqid 2467 . . . . . . . . 9  |-  ( LSpan `  S )  =  (
LSpan `  S )
3713, 23, 36lspcl 17422 . . . . . . . 8  |-  ( ( S  e.  LMod  /\  b  C_  ( Base `  S
) )  ->  (
( LSpan `  S ) `  b )  e.  (
LSubSp `  S ) )
3830, 35, 37syl2anc 661 . . . . . . 7  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
( ( LSpan `  S
) `  b )  e.  ( LSubSp `  S )
)
3913, 36, 8lmhmlsp 17495 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  b  C_  ( Base `  S
) )  ->  ( F " ( ( LSpan `  S ) `  b
) )  =  ( ( LSpan `  T ) `  ( F " b
) ) )
4027, 35, 39syl2anc 661 . . . . . . . 8  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
( F " (
( LSpan `  S ) `  b ) )  =  ( ( LSpan `  T
) `  ( F " b ) ) )
41 fveq2 5866 . . . . . . . . 9  |-  ( ( F " b )  =  a  ->  (
( LSpan `  T ) `  ( F " b
) )  =  ( ( LSpan `  T ) `  a ) )
4241ad2antll 728 . . . . . . . 8  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
( ( LSpan `  T
) `  ( F " b ) )  =  ( ( LSpan `  T
) `  a )
)
43 simp2rr 1066 . . . . . . . . 9  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e. 
~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i  Fin )  /\  ( F "
b )  =  a ) )  ->  (
( LSpan `  T ) `  a )  =  ran  F )
44433expa 1196 . . . . . . . 8  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
( ( LSpan `  T
) `  a )  =  ran  F )
4540, 42, 443eqtrd 2512 . . . . . . 7  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
( F " (
( LSpan `  S ) `  b ) )  =  ran  F )
4623, 24, 25, 26, 13, 27, 38, 45kercvrlsm 30661 . . . . . 6  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
( K ( LSSum `  S ) ( (
LSpan `  S ) `  b ) )  =  ( Base `  S
) )
4746oveq2d 6300 . . . . 5  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
( Ss  ( K (
LSSum `  S ) ( ( LSpan `  S ) `  b ) ) )  =  ( Ss  ( Base `  S ) ) )
4813ressid 14550 . . . . . . 7  |-  ( S  e.  LMod  ->  ( Ss  (
Base `  S )
)  =  S )
4929, 48syl 16 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  ( Ss  ( Base `  S
) )  =  S )
5049ad2antrr 725 . . . . 5  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
( Ss  ( Base `  S
) )  =  S )
5147, 50eqtr2d 2509 . . . 4  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  ->  S  =  ( Ss  ( K ( LSSum `  S
) ( ( LSpan `  S ) `  b
) ) ) )
52 lmhmfgsplit.u . . . . 5  |-  U  =  ( Ss  K )
53 eqid 2467 . . . . 5  |-  ( Ss  ( ( LSpan `  S ) `  b ) )  =  ( Ss  ( ( LSpan `  S ) `  b
) )
54 eqid 2467 . . . . 5  |-  ( Ss  ( K ( LSSum `  S
) ( ( LSpan `  S ) `  b
) ) )  =  ( Ss  ( K (
LSSum `  S ) ( ( LSpan `  S ) `  b ) ) )
5526, 25, 23lmhmkerlss 17497 . . . . . . 7  |-  ( F  e.  ( S LMHom  T
)  ->  K  e.  ( LSubSp `  S )
)
56553ad2ant1 1017 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  K  e.  ( LSubSp `  S ) )
5756ad2antrr 725 . . . . 5  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  ->  K  e.  ( LSubSp `  S ) )
58 simpll2 1036 . . . . 5  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  ->  U  e. LFinGen )
59 inss2 3719 . . . . . . . 8  |-  ( ~P ( Base `  S
)  i^i  Fin )  C_ 
Fin
6059sseli 3500 . . . . . . 7  |-  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  ->  b  e.  Fin )
6160ad2antrl 727 . . . . . 6  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
b  e.  Fin )
6236, 13, 53islssfgi 30650 . . . . . 6  |-  ( ( S  e.  LMod  /\  b  C_  ( Base `  S
)  /\  b  e.  Fin )  ->  ( Ss  ( ( LSpan `  S ) `  b ) )  e. LFinGen )
6330, 35, 61, 62syl3anc 1228 . . . . 5  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
( Ss  ( ( LSpan `  S ) `  b
) )  e. LFinGen )
6423, 24, 52, 53, 54, 30, 57, 38, 58, 63lsmfgcl 30652 . . . 4  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
( Ss  ( K (
LSSum `  S ) ( ( LSpan `  S ) `  b ) ) )  e. LFinGen )
6551, 64eqeltrd 2555 . . 3  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  ->  S  e. LFinGen )
6622, 65rexlimddv 2959 . 2  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e. 
~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  ->  S  e. LFinGen )
6711, 66rexlimddv 2959 1  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  S  e. LFinGen )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2815    i^i cin 3475    C_ wss 3476   ~Pcpw 4010   {csn 4027   `'ccnv 4998   ran crn 5000   "cima 5002    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6284   Fincfn 7516   Basecbs 14490   ↾s cress 14491   0gc0g 14695   LSSumclsm 16460   LModclmod 17312   LSubSpclss 17378   LSpanclspn 17417   LMHom clmhm 17465  LFinGenclfig 30645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-sca 14571  df-vsca 14572  df-0g 14697  df-mnd 15732  df-submnd 15787  df-grp 15867  df-minusg 15868  df-sbg 15869  df-subg 16003  df-ghm 16070  df-cntz 16160  df-lsm 16462  df-cmn 16606  df-abl 16607  df-mgp 16944  df-ur 16956  df-rng 17002  df-lmod 17314  df-lss 17379  df-lsp 17418  df-lmhm 17468  df-lfig 30646
This theorem is referenced by:  lmhmlnmsplit  30665
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