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Theorem lmhmfgsplit 27161
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 16108 . . . . 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 16126 . . . . 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 2436 . . . . 5  |-  ( LSubSp `  T )  =  (
LSubSp `  T )
8 eqid 2436 . . . . 5  |-  ( LSpan `  T )  =  (
LSpan `  T )
96, 7, 8islssfg 27145 . . . 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 2436 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
14 eqid 2436 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
1513, 14lmhmf 16110 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
16 ffn 5591 . . . . 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 3807 . . . . 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 7412 . . . 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 2436 . . . . . . 7  |-  ( LSubSp `  S )  =  (
LSubSp `  S )
24 eqid 2436 . . . . . . 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 16109 . . . . . . . . . 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 3561 . . . . . . . . . . 11  |-  ( ~P ( Base `  S
)  i^i  Fin )  C_ 
~P ( Base `  S
)
3231sseli 3344 . . . . . . . . . 10  |-  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  ->  b  e.  ~P ( Base `  S
) )
33 elpwi 3807 . . . . . . . . . 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 2436 . . . . . . . . 9  |-  ( LSpan `  S )  =  (
LSpan `  S )
3713, 23, 36lspcl 16052 . . . . . . . 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 16125 . . . . . . . . 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 5728 . . . . . . . . 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 2472 . . . . . . 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 27158 . . . . . 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 6097 . . . . 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 13524 . . . . . . 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 2469 . . . 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 2436 . . . . 5  |-  ( Ss  ( ( LSpan `  S ) `  b ) )  =  ( Ss  ( ( LSpan `  S ) `  b
) )
54 eqid 2436 . . . . 5  |-  ( Ss  ( K ( LSSum `  S
) ( ( LSpan `  S ) `  b
) ) )  =  ( Ss  ( K (
LSSum `  S ) ( ( LSpan `  S ) `  b ) ) )
5526, 25, 23lmhmkerlss 16127 . . . . . . 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 3562 . . . . . . . 8  |-  ( ~P ( Base `  S
)  i^i  Fin )  C_ 
Fin
6059sseli 3344 . . . . . . 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 27147 . . . . . 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 27149 . . . 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 2510 . . 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 2834 . 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 2834 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 1652    e. wcel 1725   E.wrex 2706    i^i cin 3319    C_ wss 3320   ~Pcpw 3799   {csn 3814   `'ccnv 4877   ran crn 4879   "cima 4881    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   Fincfn 7109   Basecbs 13469   ↾s cress 13470   0gc0g 13723   LSSumclsm 15268   LModclmod 15950   LSubSpclss 16008   LSpanclspn 16047   LMHom clmhm 16095  LFinGenclfig 27142
This theorem is referenced by:  lmhmlnmsplit  27162
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-int 4051  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-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  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-3 10059  df-4 10060  df-5 10061  df-6 10062  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-sca 13545  df-vsca 13546  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-lsp 16048  df-lmhm 16098  df-lfig 27143
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