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Theorem lmhmfgsplit 27052
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 959 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  V  e. LFinGen )
2 lmhmlmod2 16063 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
323ad2ant1 978 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  T  e.  LMod )
4 lmhmrnlss 16081 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  ran  F  e.  ( LSubSp `  T )
)
543ad2ant1 978 . . . 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 2404 . . . . 5  |-  ( LSubSp `  T )  =  (
LSubSp `  T )
8 eqid 2404 . . . . 5  |-  ( LSpan `  T )  =  (
LSpan `  T )
96, 7, 8islssfg 27036 . . . 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 643 . . 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 202 . 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 960 . . . . 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 2404 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
14 eqid 2404 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
1513, 14lmhmf 16065 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
16 ffn 5550 . . . . 5  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  F  Fn  ( Base `  S )
)
1712, 15, 163syl 19 . . . 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 3767 . . . . 5  |-  ( a  e.  ~P ran  F  ->  a  C_  ran  F )
1918ad2antrl 709 . . . 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 741 . . . 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 7370 . . . 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 1184 . . 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 2404 . . . . . . 7  |-  ( LSubSp `  S )  =  (
LSubSp `  S )
24 eqid 2404 . . . . . . 7  |-  ( LSSum `  S )  =  (
LSSum `  S )
25 lmhmfgsplit.z . . . . . . 7  |-  .0.  =  ( 0g `  T )
26 lmhmfgsplit.k . . . . . . 7  |-  K  =  ( `' F " {  .0.  } )
27 simpll1 996 . . . . . . 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 16064 . . . . . . . . . 10  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
29283ad2ant1 978 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  S  e.  LMod )
3029ad2antrr 707 . . . . . . . 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 3521 . . . . . . . . . . 11  |-  ( ~P ( Base `  S
)  i^i  Fin )  C_ 
~P ( Base `  S
)
3231sseli 3304 . . . . . . . . . 10  |-  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  ->  b  e.  ~P ( Base `  S
) )
33 elpwi 3767 . . . . . . . . . 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 709 . . . . . . . 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 2404 . . . . . . . . 9  |-  ( LSpan `  S )  =  (
LSpan `  S )
3713, 23, 36lspcl 16007 . . . . . . . 8  |-  ( ( S  e.  LMod  /\  b  C_  ( Base `  S
) )  ->  (
( LSpan `  S ) `  b )  e.  (
LSubSp `  S ) )
3830, 35, 37syl2anc 643 . . . . . . 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 16080 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  b  C_  ( Base `  S
) )  ->  ( F " ( ( LSpan `  S ) `  b
) )  =  ( ( LSpan `  T ) `  ( F " b
) ) )
4027, 35, 39syl2anc 643 . . . . . . . 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 5687 . . . . . . . . 9  |-  ( ( F " b )  =  a  ->  (
( LSpan `  T ) `  ( F " b
) )  =  ( ( LSpan `  T ) `  a ) )
4241ad2antll 710 . . . . . . . 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 1027 . . . . . . . . 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 1153 . . . . . . . 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 2440 . . . . . . 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 27049 . . . . . 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 6056 . . . . 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 13479 . . . . . . 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 707 . . . . 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 2437 . . . 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 2404 . . . . 5  |-  ( Ss  ( ( LSpan `  S ) `  b ) )  =  ( Ss  ( ( LSpan `  S ) `  b
) )
54 eqid 2404 . . . . 5  |-  ( Ss  ( K ( LSSum `  S
) ( ( LSpan `  S ) `  b
) ) )  =  ( Ss  ( K (
LSSum `  S ) ( ( LSpan `  S ) `  b ) ) )
5526, 25, 23lmhmkerlss 16082 . . . . . . 7  |-  ( F  e.  ( S LMHom  T
)  ->  K  e.  ( LSubSp `  S )
)
56553ad2ant1 978 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  K  e.  ( LSubSp `  S ) )
5756ad2antrr 707 . . . . 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 997 . . . . 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 3522 . . . . . . . 8  |-  ( ~P ( Base `  S
)  i^i  Fin )  C_ 
Fin
6059sseli 3304 . . . . . . 7  |-  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  ->  b  e.  Fin )
6160ad2antrl 709 . . . . . 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 27038 . . . . . 6  |-  ( ( S  e.  LMod  /\  b  C_  ( Base `  S
)  /\  b  e.  Fin )  ->  ( Ss  ( ( LSpan `  S ) `  b ) )  e. LFinGen )
6330, 35, 61, 62syl3anc 1184 . . . . 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 27040 . . . 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 2478 . . 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 2794 . 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 2794 1  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  S  e. LFinGen )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   E.wrex 2667    i^i cin 3279    C_ wss 3280   ~Pcpw 3759   {csn 3774   `'ccnv 4836   ran crn 4838   "cima 4840    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   Fincfn 7068   Basecbs 13424   ↾s cress 13425   0gc0g 13678   LSSumclsm 15223   LModclmod 15905   LSubSpclss 15963   LSpanclspn 16002   LMHom clmhm 16050  LFinGenclfig 27033
This theorem is referenced by:  lmhmlnmsplit  27053
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-sca 13500  df-vsca 13501  df-0g 13682  df-mnd 14645  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-subg 14896  df-ghm 14959  df-cntz 15071  df-lsm 15225  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-ur 15620  df-lmod 15907  df-lss 15964  df-lsp 16003  df-lmhm 16053  df-lfig 27034
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