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Theorem lmhmlnmsplit 27162
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 16109 . . 3  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
213ad2ant1 978 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LNoeM  /\  V  e. LNoeM )  ->  S  e.  LMod )
3 eqid 2436 . . . . . 6  |-  ( LSubSp `  S )  =  (
LSubSp `  S )
4 eqid 2436 . . . . . 6  |-  ( Ss  a )  =  ( Ss  a )
53, 4reslmhm 16128 . . . . 5  |-  ( ( F  e.  ( S LMHom 
T )  /\  a  e.  ( LSubSp `  S )
)  ->  ( F  |`  a )  e.  ( ( Ss  a ) LMHom  T
) )
653ad2antl1 1119 . . . 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 5359 . . . . . . . 8  |-  ( `' ( F  |`  a
) " {  .0.  } )  =  ( ( `' F " {  .0.  } )  i^i  a )
8 lmhmfgsplit.k . . . . . . . . . 10  |-  K  =  ( `' F " {  .0.  } )
98eqcomi 2440 . . . . . . . . 9  |-  ( `' F " {  .0.  } )  =  K
109ineq1i 3538 . . . . . . . 8  |-  ( ( `' F " {  .0.  } )  i^i  a )  =  ( K  i^i  a )
11 incom 3533 . . . . . . . 8  |-  ( K  i^i  a )  =  ( a  i^i  K
)
127, 10, 113eqtri 2460 . . . . . . 7  |-  ( `' ( F  |`  a
) " {  .0.  } )  =  ( a  i^i  K )
1312oveq2i 6092 . . . . . 6  |-  ( ( Ss  a )s  ( `' ( F  |`  a ) " {  .0.  } ) )  =  ( ( Ss  a )s  ( a  i^i 
K ) )
14 lmhmfgsplit.u . . . . . . . . 9  |-  U  =  ( Ss  K )
1514oveq1i 6091 . . . . . . . 8  |-  ( Us  ( a  i^i  K ) )  =  ( ( Ss  K )s  ( a  i^i 
K ) )
16 simpl1 960 . . . . . . . . . 10  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  F  e.  ( S LMHom  T ) )
17 cnvexg 5405 . . . . . . . . . . . 12  |-  ( F  e.  ( S LMHom  T
)  ->  `' F  e.  _V )
18 imaexg 5217 . . . . . . . . . . . 12  |-  ( `' F  e.  _V  ->  ( `' F " {  .0.  } )  e.  _V )
1917, 18syl 16 . . . . . . . . . . 11  |-  ( F  e.  ( S LMHom  T
)  ->  ( `' F " {  .0.  }
)  e.  _V )
208, 19syl5eqel 2520 . . . . . . . . . 10  |-  ( F  e.  ( S LMHom  T
)  ->  K  e.  _V )
2116, 20syl 16 . . . . . . . . 9  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  K  e.  _V )
22 inss2 3562 . . . . . . . . 9  |-  ( a  i^i  K )  C_  K
23 ressabs 13527 . . . . . . . . 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 644 . . . . . . . 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 2480 . . . . . . 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 2959 . . . . . . . 8  |-  a  e. 
_V
27 inss1 3561 . . . . . . . 8  |-  ( a  i^i  K )  C_  a
28 ressabs 13527 . . . . . . . 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 654 . . . . . . 7  |-  ( ( Ss  a )s  ( a  i^i 
K ) )  =  ( Ss  ( a  i^i 
K ) )
3025, 29syl6reqr 2487 . . . . . 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 2480 . . . . 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 961 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  U  e. LNoeM )
332adantr 452 . . . . . . . 8  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  S  e.  LMod )
34 simpr 448 . . . . . . . 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 16127 . . . . . . . . 9  |-  ( F  e.  ( S LMHom  T
)  ->  K  e.  ( LSubSp `  S )
)
3716, 36syl 16 . . . . . . . 8  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  K  e.  ( LSubSp `  S )
)
383lssincl 16041 . . . . . . . 8  |-  ( ( S  e.  LMod  /\  a  e.  ( LSubSp `  S )  /\  K  e.  ( LSubSp `
 S ) )  ->  ( a  i^i 
K )  e.  (
LSubSp `  S ) )
3933, 34, 37, 38syl3anc 1184 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ( a  i^i  K )  e.  (
LSubSp `  S ) )
4022a1i 11 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ( a  i^i  K )  C_  K
)
41 eqid 2436 . . . . . . . . 9  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
4214, 3, 41lsslss 16037 . . . . . . . 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 643 . . . . . . 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 889 . . . . . 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 2436 . . . . . . 7  |-  ( Us  ( a  i^i  K ) )  =  ( Us  ( a  i^i  K ) )
4641, 45lnmlssfg 27155 . . . . . 6  |-  ( ( U  e. LNoeM  /\  (
a  i^i  K )  e.  ( LSubSp `  U )
)  ->  ( Us  (
a  i^i  K )
)  e. LFinGen )
4732, 44, 46syl2anc 643 . . . . 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 2510 . . . 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 6091 . . . . . . . 8  |-  ( Vs  ran  ( F  |`  a
) )  =  ( ( Ts  ran  F )s  ran  ( F  |`  a ) )
51 rnexg 5131 . . . . . . . . 9  |-  ( F  e.  ( S LMHom  T
)  ->  ran  F  e. 
_V )
52 resexg 5185 . . . . . . . . . 10  |-  ( F  e.  ( S LMHom  T
)  ->  ( F  |`  a )  e.  _V )
53 rnexg 5131 . . . . . . . . . 10  |-  ( ( F  |`  a )  e.  _V  ->  ran  ( F  |`  a )  e.  _V )
5452, 53syl 16 . . . . . . . . 9  |-  ( F  e.  ( S LMHom  T
)  ->  ran  ( F  |`  a )  e.  _V )
55 ressress 13526 . . . . . . . . 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 643 . . . . . . . 8  |-  ( F  e.  ( S LMHom  T
)  ->  ( ( Ts  ran  F )s  ran  ( F  |`  a ) )  =  ( Ts  ( ran  F  i^i  ran  ( F  |`  a ) ) ) )
5750, 56syl5eq 2480 . . . . . . 7  |-  ( F  e.  ( S LMHom  T
)  ->  ( Vs  ran  ( F  |`  a ) )  =  ( Ts  ( ran  F  i^i  ran  ( F  |`  a ) ) ) )
58 incom 3533 . . . . . . . . 9  |-  ( ran 
F  i^i  ran  ( F  |`  a ) )  =  ( ran  ( F  |`  a )  i^i  ran  F )
59 resss 5170 . . . . . . . . . . 11  |-  ( F  |`  a )  C_  F
60 rnss 5098 . . . . . . . . . . 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 3334 . . . . . . . . . 10  |-  ( ran  ( F  |`  a
)  C_  ran  F  <->  ( ran  ( F  |`  a )  i^i  ran  F )  =  ran  ( F  |`  a ) )
6361, 62mpbi 200 . . . . . . . . 9  |-  ( ran  ( F  |`  a
)  i^i  ran  F )  =  ran  ( F  |`  a )
6458, 63eqtr2i 2457 . . . . . . . 8  |-  ran  ( F  |`  a )  =  ( ran  F  i^i  ran  ( F  |`  a
) )
6564oveq2i 6092 . . . . . . 7  |-  ( Ts  ran  ( F  |`  a
) )  =  ( Ts  ( ran  F  i^i  ran  ( F  |`  a
) ) )
6657, 65syl6reqr 2487 . . . . . 6  |-  ( F  e.  ( S LMHom  T
)  ->  ( Ts  ran  ( F  |`  a ) )  =  ( Vs  ran  ( F  |`  a
) ) )
6716, 66syl 16 . . . . 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 962 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  V  e. LNoeM )
69 lmhmrnlss 16126 . . . . . . . 8  |-  ( ( F  |`  a )  e.  ( ( Ss  a ) LMHom 
T )  ->  ran  ( F  |`  a )  e.  ( LSubSp `  T
) )
706, 69syl 16 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ran  ( F  |`  a )  e.  (
LSubSp `  T ) )
7161a1i 11 . . . . . . 7  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ran  ( F  |`  a )  C_  ran  F )
72 lmhmlmod2 16108 . . . . . . . . 9  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
7316, 72syl 16 . . . . . . . 8  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  T  e.  LMod )
74 lmhmrnlss 16126 . . . . . . . . 9  |-  ( F  e.  ( S LMHom  T
)  ->  ran  F  e.  ( LSubSp `  T )
)
7516, 74syl 16 . . . . . . . 8  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ran  F  e.  ( LSubSp `  T )
)
76 eqid 2436 . . . . . . . . 9  |-  ( LSubSp `  T )  =  (
LSubSp `  T )
77 eqid 2436 . . . . . . . . 9  |-  ( LSubSp `  V )  =  (
LSubSp `  V )
7849, 76, 77lsslss 16037 . . . . . . . 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 643 . . . . . . 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 889 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ran  ( F  |`  a )  e.  (
LSubSp `  V ) )
81 eqid 2436 . . . . . . 7  |-  ( Vs  ran  ( F  |`  a
) )  =  ( Vs 
ran  ( F  |`  a ) )
8277, 81lnmlssfg 27155 . . . . . 6  |-  ( ( V  e. LNoeM  /\  ran  ( F  |`  a )  e.  ( LSubSp `  V )
)  ->  ( Vs  ran  ( F  |`  a ) )  e. LFinGen )
8368, 80, 82syl2anc 643 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ( Vs  ran  ( F  |`  a ) )  e. LFinGen )
8467, 83eqeltrd 2510 . . . 4  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ( Ts  ran  ( F  |`  a ) )  e. LFinGen )
85 eqid 2436 . . . . 5  |-  ( `' ( F  |`  a
) " {  .0.  } )  =  ( `' ( F  |`  a
) " {  .0.  } )
86 eqid 2436 . . . . 5  |-  ( ( Ss  a )s  ( `' ( F  |`  a ) " {  .0.  } ) )  =  ( ( Ss  a )s  ( `' ( F  |`  a ) " {  .0.  } ) )
87 eqid 2436 . . . . 5  |-  ( Ts  ran  ( F  |`  a
) )  =  ( Ts 
ran  ( F  |`  a ) )
8835, 85, 86, 87lmhmfgsplit 27161 . . . 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 1184 . . 3  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ( Ss  a
)  e. LFinGen )
9089ralrimiva 2789 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LNoeM  /\  V  e. LNoeM )  ->  A. a  e.  (
LSubSp `  S ) ( Ss  a )  e. LFinGen )
913islnm 27152 . 2  |-  ( S  e. LNoeM 
<->  ( S  e.  LMod  /\ 
A. a  e.  (
LSubSp `  S ) ( Ss  a )  e. LFinGen )
)
922, 90, 91sylanbrc 646 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 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705   _Vcvv 2956    i^i cin 3319    C_ wss 3320   {csn 3814   `'ccnv 4877   ran crn 4879    |` cres 4880   "cima 4881   ` cfv 5454  (class class class)co 6081   ↾s cress 13470   0gc0g 13723   LModclmod 15950   LSubSpclss 16008   LMHom clmhm 16095  LFinGenclfig 27142  LNoeMclnm 27150
This theorem is referenced by:  pwslnmlem2  27172
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  df-lnm 27151
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