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Theorem lmhmlnmsplit 27185
Description: If the kernel and range of a homomorphism of left modules are Noetherian, then so is the domain. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Revised by Stefan O'Rear, 12-Jun-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
lmhmlnmsplit  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LNoeM  /\  V  e. LNoeM )  ->  S  e. LNoeM )

Proof of Theorem lmhmlnmsplit
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 lmhmlmod1 15790 . . 3  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
213ad2ant1 976 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LNoeM  /\  V  e. LNoeM )  ->  S  e.  LMod )
3 eqid 2283 . . . . . 6  |-  ( LSubSp `  S )  =  (
LSubSp `  S )
4 eqid 2283 . . . . . 6  |-  ( Ss  a )  =  ( Ss  a )
53, 4reslmhm 15809 . . . . 5  |-  ( ( F  e.  ( S LMHom 
T )  /\  a  e.  ( LSubSp `  S )
)  ->  ( F  |`  a )  e.  ( ( Ss  a ) LMHom  T
) )
653ad2antl1 1117 . . . 4  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ( F  |`  a )  e.  ( ( Ss  a ) LMHom  T
) )
7 cnvresima 5162 . . . . . . . 8  |-  ( `' ( F  |`  a
) " {  .0.  } )  =  ( ( `' F " {  .0.  } )  i^i  a )
8 lmhmfgsplit.k . . . . . . . . . 10  |-  K  =  ( `' F " {  .0.  } )
98eqcomi 2287 . . . . . . . . 9  |-  ( `' F " {  .0.  } )  =  K
109ineq1i 3366 . . . . . . . 8  |-  ( ( `' F " {  .0.  } )  i^i  a )  =  ( K  i^i  a )
11 incom 3361 . . . . . . . 8  |-  ( K  i^i  a )  =  ( a  i^i  K
)
127, 10, 113eqtri 2307 . . . . . . 7  |-  ( `' ( F  |`  a
) " {  .0.  } )  =  ( a  i^i  K )
1312oveq2i 5869 . . . . . 6  |-  ( ( Ss  a )s  ( `' ( F  |`  a ) " {  .0.  } ) )  =  ( ( Ss  a )s  ( a  i^i 
K ) )
14 lmhmfgsplit.u . . . . . . . . 9  |-  U  =  ( Ss  K )
1514oveq1i 5868 . . . . . . . 8  |-  ( Us  ( a  i^i  K ) )  =  ( ( Ss  K )s  ( a  i^i 
K ) )
16 simpl1 958 . . . . . . . . . 10  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  F  e.  ( S LMHom  T ) )
17 cnvexg 5208 . . . . . . . . . . . 12  |-  ( F  e.  ( S LMHom  T
)  ->  `' F  e.  _V )
18 imaexg 5026 . . . . . . . . . . . 12  |-  ( `' F  e.  _V  ->  ( `' F " {  .0.  } )  e.  _V )
1917, 18syl 15 . . . . . . . . . . 11  |-  ( F  e.  ( S LMHom  T
)  ->  ( `' F " {  .0.  }
)  e.  _V )
208, 19syl5eqel 2367 . . . . . . . . . 10  |-  ( F  e.  ( S LMHom  T
)  ->  K  e.  _V )
2116, 20syl 15 . . . . . . . . 9  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  K  e.  _V )
22 inss2 3390 . . . . . . . . 9  |-  ( a  i^i  K )  C_  K
23 ressabs 13206 . . . . . . . . 9  |-  ( ( K  e.  _V  /\  ( a  i^i  K
)  C_  K )  ->  ( ( Ss  K )s  ( a  i^i  K ) )  =  ( Ss  ( a  i^i  K ) ) )
2421, 22, 23sylancl 643 . . . . . . . 8  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ( ( Ss  K )s  ( a  i^i 
K ) )  =  ( Ss  ( a  i^i 
K ) ) )
2515, 24syl5eq 2327 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ( Us  (
a  i^i  K )
)  =  ( Ss  ( a  i^i  K ) ) )
26 vex 2791 . . . . . . . 8  |-  a  e. 
_V
27 inss1 3389 . . . . . . . 8  |-  ( a  i^i  K )  C_  a
28 ressabs 13206 . . . . . . . 8  |-  ( ( a  e.  _V  /\  ( a  i^i  K
)  C_  a )  ->  ( ( Ss  a )s  ( a  i^i  K ) )  =  ( Ss  ( a  i^i  K ) ) )
2926, 27, 28mp2an 653 . . . . . . 7  |-  ( ( Ss  a )s  ( a  i^i 
K ) )  =  ( Ss  ( a  i^i 
K ) )
3025, 29syl6reqr 2334 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ( ( Ss  a )s  ( a  i^i 
K ) )  =  ( Us  ( a  i^i 
K ) ) )
3113, 30syl5eq 2327 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ( ( Ss  a )s  ( `' ( F  |`  a ) " {  .0.  } ) )  =  ( Us  ( a  i^i  K ) ) )
32 simpl2 959 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  U  e. LNoeM )
332adantr 451 . . . . . . . 8  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  S  e.  LMod )
34 simpr 447 . . . . . . . 8  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  a  e.  ( LSubSp `  S )
)
35 lmhmfgsplit.z . . . . . . . . . 10  |-  .0.  =  ( 0g `  T )
368, 35, 3lmhmkerlss 15808 . . . . . . . . 9  |-  ( F  e.  ( S LMHom  T
)  ->  K  e.  ( LSubSp `  S )
)
3716, 36syl 15 . . . . . . . 8  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  K  e.  ( LSubSp `  S )
)
383lssincl 15722 . . . . . . . 8  |-  ( ( S  e.  LMod  /\  a  e.  ( LSubSp `  S )  /\  K  e.  ( LSubSp `
 S ) )  ->  ( a  i^i 
K )  e.  (
LSubSp `  S ) )
3933, 34, 37, 38syl3anc 1182 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ( a  i^i  K )  e.  (
LSubSp `  S ) )
4022a1i 10 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ( a  i^i  K )  C_  K
)
41 eqid 2283 . . . . . . . . 9  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
4214, 3, 41lsslss 15718 . . . . . . . 8  |-  ( ( S  e.  LMod  /\  K  e.  ( LSubSp `  S )
)  ->  ( (
a  i^i  K )  e.  ( LSubSp `  U )  <->  ( ( a  i^i  K
)  e.  ( LSubSp `  S )  /\  (
a  i^i  K )  C_  K ) ) )
4333, 37, 42syl2anc 642 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ( (
a  i^i  K )  e.  ( LSubSp `  U )  <->  ( ( a  i^i  K
)  e.  ( LSubSp `  S )  /\  (
a  i^i  K )  C_  K ) ) )
4439, 40, 43mpbir2and 888 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ( a  i^i  K )  e.  (
LSubSp `  U ) )
45 eqid 2283 . . . . . . 7  |-  ( Us  ( a  i^i  K ) )  =  ( Us  ( a  i^i  K ) )
4641, 45lnmlssfg 27178 . . . . . 6  |-  ( ( U  e. LNoeM  /\  (
a  i^i  K )  e.  ( LSubSp `  U )
)  ->  ( Us  (
a  i^i  K )
)  e. LFinGen )
4732, 44, 46syl2anc 642 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ( Us  (
a  i^i  K )
)  e. LFinGen )
4831, 47eqeltrd 2357 . . . 4  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ( ( Ss  a )s  ( `' ( F  |`  a ) " {  .0.  } ) )  e. LFinGen )
49 lmhmfgsplit.v . . . . . . . . 9  |-  V  =  ( Ts  ran  F )
5049oveq1i 5868 . . . . . . . 8  |-  ( Vs  ran  ( F  |`  a
) )  =  ( ( Ts  ran  F )s  ran  ( F  |`  a ) )
51 rnexg 4940 . . . . . . . . 9  |-  ( F  e.  ( S LMHom  T
)  ->  ran  F  e. 
_V )
52 resexg 4994 . . . . . . . . . 10  |-  ( F  e.  ( S LMHom  T
)  ->  ( F  |`  a )  e.  _V )
53 rnexg 4940 . . . . . . . . . 10  |-  ( ( F  |`  a )  e.  _V  ->  ran  ( F  |`  a )  e.  _V )
5452, 53syl 15 . . . . . . . . 9  |-  ( F  e.  ( S LMHom  T
)  ->  ran  ( F  |`  a )  e.  _V )
55 ressress 13205 . . . . . . . . 9  |-  ( ( ran  F  e.  _V  /\ 
ran  ( F  |`  a )  e.  _V )  ->  ( ( Ts  ran 
F )s  ran  ( F  |`  a ) )  =  ( Ts  ( ran  F  i^i  ran  ( F  |`  a ) ) ) )
5651, 54, 55syl2anc 642 . . . . . . . 8  |-  ( F  e.  ( S LMHom  T
)  ->  ( ( Ts  ran  F )s  ran  ( F  |`  a ) )  =  ( Ts  ( ran  F  i^i  ran  ( F  |`  a ) ) ) )
5750, 56syl5eq 2327 . . . . . . 7  |-  ( F  e.  ( S LMHom  T
)  ->  ( Vs  ran  ( F  |`  a ) )  =  ( Ts  ( ran  F  i^i  ran  ( F  |`  a ) ) ) )
58 incom 3361 . . . . . . . . 9  |-  ( ran 
F  i^i  ran  ( F  |`  a ) )  =  ( ran  ( F  |`  a )  i^i  ran  F )
59 resss 4979 . . . . . . . . . . 11  |-  ( F  |`  a )  C_  F
60 rnss 4907 . . . . . . . . . . 11  |-  ( ( F  |`  a )  C_  F  ->  ran  ( F  |`  a )  C_  ran  F )
6159, 60ax-mp 8 . . . . . . . . . 10  |-  ran  ( F  |`  a )  C_  ran  F
62 df-ss 3166 . . . . . . . . . 10  |-  ( ran  ( F  |`  a
)  C_  ran  F  <->  ( ran  ( F  |`  a )  i^i  ran  F )  =  ran  ( F  |`  a ) )
6361, 62mpbi 199 . . . . . . . . 9  |-  ( ran  ( F  |`  a
)  i^i  ran  F )  =  ran  ( F  |`  a )
6458, 63eqtr2i 2304 . . . . . . . 8  |-  ran  ( F  |`  a )  =  ( ran  F  i^i  ran  ( F  |`  a
) )
6564oveq2i 5869 . . . . . . 7  |-  ( Ts  ran  ( F  |`  a
) )  =  ( Ts  ( ran  F  i^i  ran  ( F  |`  a
) ) )
6657, 65syl6reqr 2334 . . . . . 6  |-  ( F  e.  ( S LMHom  T
)  ->  ( Ts  ran  ( F  |`  a ) )  =  ( Vs  ran  ( F  |`  a
) ) )
6716, 66syl 15 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ( Ts  ran  ( F  |`  a ) )  =  ( Vs  ran  ( F  |`  a
) ) )
68 simpl3 960 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  V  e. LNoeM )
69 lmhmrnlss 15807 . . . . . . . 8  |-  ( ( F  |`  a )  e.  ( ( Ss  a ) LMHom 
T )  ->  ran  ( F  |`  a )  e.  ( LSubSp `  T
) )
706, 69syl 15 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ran  ( F  |`  a )  e.  (
LSubSp `  T ) )
7161a1i 10 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ran  ( F  |`  a )  C_  ran  F )
72 lmhmlmod2 15789 . . . . . . . . 9  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
7316, 72syl 15 . . . . . . . 8  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  T  e.  LMod )
74 lmhmrnlss 15807 . . . . . . . . 9  |-  ( F  e.  ( S LMHom  T
)  ->  ran  F  e.  ( LSubSp `  T )
)
7516, 74syl 15 . . . . . . . 8  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ran  F  e.  ( LSubSp `  T )
)
76 eqid 2283 . . . . . . . . 9  |-  ( LSubSp `  T )  =  (
LSubSp `  T )
77 eqid 2283 . . . . . . . . 9  |-  ( LSubSp `  V )  =  (
LSubSp `  V )
7849, 76, 77lsslss 15718 . . . . . . . 8  |-  ( ( T  e.  LMod  /\  ran  F  e.  ( LSubSp `  T
) )  ->  ( ran  ( F  |`  a
)  e.  ( LSubSp `  V )  <->  ( ran  ( F  |`  a )  e.  ( LSubSp `  T
)  /\  ran  ( F  |`  a )  C_  ran  F ) ) )
7973, 75, 78syl2anc 642 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ( ran  ( F  |`  a )  e.  ( LSubSp `  V
)  <->  ( ran  ( F  |`  a )  e.  ( LSubSp `  T )  /\  ran  ( F  |`  a )  C_  ran  F ) ) )
8070, 71, 79mpbir2and 888 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ran  ( F  |`  a )  e.  (
LSubSp `  V ) )
81 eqid 2283 . . . . . . 7  |-  ( Vs  ran  ( F  |`  a
) )  =  ( Vs 
ran  ( F  |`  a ) )
8277, 81lnmlssfg 27178 . . . . . 6  |-  ( ( V  e. LNoeM  /\  ran  ( F  |`  a )  e.  ( LSubSp `  V )
)  ->  ( Vs  ran  ( F  |`  a ) )  e. LFinGen )
8368, 80, 82syl2anc 642 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ( Vs  ran  ( F  |`  a ) )  e. LFinGen )
8467, 83eqeltrd 2357 . . . 4  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ( Ts  ran  ( F  |`  a ) )  e. LFinGen )
85 eqid 2283 . . . . 5  |-  ( `' ( F  |`  a
) " {  .0.  } )  =  ( `' ( F  |`  a
) " {  .0.  } )
86 eqid 2283 . . . . 5  |-  ( ( Ss  a )s  ( `' ( F  |`  a ) " {  .0.  } ) )  =  ( ( Ss  a )s  ( `' ( F  |`  a ) " {  .0.  } ) )
87 eqid 2283 . . . . 5  |-  ( Ts  ran  ( F  |`  a
) )  =  ( Ts 
ran  ( F  |`  a ) )
8835, 85, 86, 87lmhmfgsplit 27184 . . . 4  |-  ( ( ( F  |`  a
)  e.  ( ( Ss  a ) LMHom  T )  /\  ( ( Ss  a )s  ( `' ( F  |`  a ) " {  .0.  } ) )  e. LFinGen  /\  ( Ts  ran  ( F  |`  a ) )  e. LFinGen )  ->  ( Ss  a )  e. LFinGen )
896, 48, 84, 88syl3anc 1182 . . 3  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ( Ss  a
)  e. LFinGen )
9089ralrimiva 2626 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LNoeM  /\  V  e. LNoeM )  ->  A. a  e.  (
LSubSp `  S ) ( Ss  a )  e. LFinGen )
913islnm 27175 . 2  |-  ( S  e. LNoeM 
<->  ( S  e.  LMod  /\ 
A. a  e.  (
LSubSp `  S ) ( Ss  a )  e. LFinGen )
)
922, 90, 91sylanbrc 645 1  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LNoeM  /\  V  e. LNoeM )  ->  S  e. LNoeM )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    i^i cin 3151    C_ wss 3152   {csn 3640   `'ccnv 4688   ran crn 4690    |` cres 4691   "cima 4692   ` cfv 5255  (class class class)co 5858   ↾s cress 13149   0gc0g 13400   LModclmod 15627   LSubSpclss 15689   LMHom clmhm 15776  LFinGenclfig 27165  LNoeMclnm 27173
This theorem is referenced by:  pwslnmlem2  27195
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-sca 13224  df-vsca 13225  df-0g 13404  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-ghm 14681  df-cntz 14793  df-lsm 14947  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lmhm 15779  df-lfig 27166  df-lnm 27174
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