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Theorem lmhmlnmsplit 31272
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 17874 . . 3  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
213ad2ant1 1015 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LNoeM  /\  V  e. LNoeM )  ->  S  e.  LMod )
3 eqid 2454 . . . . . 6  |-  ( LSubSp `  S )  =  (
LSubSp `  S )
4 eqid 2454 . . . . . 6  |-  ( Ss  a )  =  ( Ss  a )
53, 4reslmhm 17893 . . . . 5  |-  ( ( F  e.  ( S LMHom 
T )  /\  a  e.  ( LSubSp `  S )
)  ->  ( F  |`  a )  e.  ( ( Ss  a ) LMHom  T
) )
653ad2antl1 1156 . . . 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 5479 . . . . . . . 8  |-  ( `' ( F  |`  a
) " {  .0.  } )  =  ( ( `' F " {  .0.  } )  i^i  a )
8 lmhmfgsplit.k . . . . . . . . . 10  |-  K  =  ( `' F " {  .0.  } )
98eqcomi 2467 . . . . . . . . 9  |-  ( `' F " {  .0.  } )  =  K
109ineq1i 3682 . . . . . . . 8  |-  ( ( `' F " {  .0.  } )  i^i  a )  =  ( K  i^i  a )
11 incom 3677 . . . . . . . 8  |-  ( K  i^i  a )  =  ( a  i^i  K
)
127, 10, 113eqtri 2487 . . . . . . 7  |-  ( `' ( F  |`  a
) " {  .0.  } )  =  ( a  i^i  K )
1312oveq2i 6281 . . . . . 6  |-  ( ( Ss  a )s  ( `' ( F  |`  a ) " {  .0.  } ) )  =  ( ( Ss  a )s  ( a  i^i 
K ) )
14 lmhmfgsplit.u . . . . . . . . 9  |-  U  =  ( Ss  K )
1514oveq1i 6280 . . . . . . . 8  |-  ( Us  ( a  i^i  K ) )  =  ( ( Ss  K )s  ( a  i^i 
K ) )
16 simpl1 997 . . . . . . . . . 10  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  F  e.  ( S LMHom  T ) )
17 cnvexg 6719 . . . . . . . . . . . 12  |-  ( F  e.  ( S LMHom  T
)  ->  `' F  e.  _V )
18 imaexg 6710 . . . . . . . . . . . 12  |-  ( `' F  e.  _V  ->  ( `' F " {  .0.  } )  e.  _V )
1917, 18syl 16 . . . . . . . . . . 11  |-  ( F  e.  ( S LMHom  T
)  ->  ( `' F " {  .0.  }
)  e.  _V )
208, 19syl5eqel 2546 . . . . . . . . . 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 3705 . . . . . . . . 9  |-  ( a  i^i  K )  C_  K
23 ressabs 14782 . . . . . . . . 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 660 . . . . . . . 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 2507 . . . . . . 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 3109 . . . . . . . 8  |-  a  e. 
_V
27 inss1 3704 . . . . . . . 8  |-  ( a  i^i  K )  C_  a
28 ressabs 14782 . . . . . . . 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 670 . . . . . . 7  |-  ( ( Ss  a )s  ( a  i^i 
K ) )  =  ( Ss  ( a  i^i 
K ) )
3025, 29syl6reqr 2514 . . . . . 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 2507 . . . . 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 998 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  U  e. LNoeM )
332adantr 463 . . . . . . . 8  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  S  e.  LMod )
34 simpr 459 . . . . . . . 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 17892 . . . . . . . . 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 17806 . . . . . . . 8  |-  ( ( S  e.  LMod  /\  a  e.  ( LSubSp `  S )  /\  K  e.  ( LSubSp `
 S ) )  ->  ( a  i^i 
K )  e.  (
LSubSp `  S ) )
3933, 34, 37, 38syl3anc 1226 . . . . . . 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 2454 . . . . . . . . 9  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
4214, 3, 41lsslss 17802 . . . . . . . 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 659 . . . . . . 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 920 . . . . . 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 2454 . . . . . . 7  |-  ( Us  ( a  i^i  K ) )  =  ( Us  ( a  i^i  K ) )
4641, 45lnmlssfg 31265 . . . . . 6  |-  ( ( U  e. LNoeM  /\  (
a  i^i  K )  e.  ( LSubSp `  U )
)  ->  ( Us  (
a  i^i  K )
)  e. LFinGen )
4732, 44, 46syl2anc 659 . . . . 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 2542 . . . 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 6280 . . . . . . . 8  |-  ( Vs  ran  ( F  |`  a
) )  =  ( ( Ts  ran  F )s  ran  ( F  |`  a ) )
51 rnexg 6705 . . . . . . . . 9  |-  ( F  e.  ( S LMHom  T
)  ->  ran  F  e. 
_V )
52 resexg 5304 . . . . . . . . . 10  |-  ( F  e.  ( S LMHom  T
)  ->  ( F  |`  a )  e.  _V )
53 rnexg 6705 . . . . . . . . . 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 14781 . . . . . . . . 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 659 . . . . . . . 8  |-  ( F  e.  ( S LMHom  T
)  ->  ( ( Ts  ran  F )s  ran  ( F  |`  a ) )  =  ( Ts  ( ran  F  i^i  ran  ( F  |`  a ) ) ) )
5750, 56syl5eq 2507 . . . . . . 7  |-  ( F  e.  ( S LMHom  T
)  ->  ( Vs  ran  ( F  |`  a ) )  =  ( Ts  ( ran  F  i^i  ran  ( F  |`  a ) ) ) )
58 incom 3677 . . . . . . . . 9  |-  ( ran 
F  i^i  ran  ( F  |`  a ) )  =  ( ran  ( F  |`  a )  i^i  ran  F )
59 resss 5285 . . . . . . . . . . 11  |-  ( F  |`  a )  C_  F
60 rnss 5220 . . . . . . . . . . 11  |-  ( ( F  |`  a )  C_  F  ->  ran  ( F  |`  a )  C_  ran  F )
6159, 60ax-mp 5 . . . . . . . . . 10  |-  ran  ( F  |`  a )  C_  ran  F
62 df-ss 3475 . . . . . . . . . 10  |-  ( ran  ( F  |`  a
)  C_  ran  F  <->  ( ran  ( F  |`  a )  i^i  ran  F )  =  ran  ( F  |`  a ) )
6361, 62mpbi 208 . . . . . . . . 9  |-  ( ran  ( F  |`  a
)  i^i  ran  F )  =  ran  ( F  |`  a )
6458, 63eqtr2i 2484 . . . . . . . 8  |-  ran  ( F  |`  a )  =  ( ran  F  i^i  ran  ( F  |`  a
) )
6564oveq2i 6281 . . . . . . 7  |-  ( Ts  ran  ( F  |`  a
) )  =  ( Ts  ( ran  F  i^i  ran  ( F  |`  a
) ) )
6657, 65syl6reqr 2514 . . . . . 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 999 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  V  e. LNoeM )
69 lmhmrnlss 17891 . . . . . . . 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 17873 . . . . . . . . 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 17891 . . . . . . . . 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 2454 . . . . . . . . 9  |-  ( LSubSp `  T )  =  (
LSubSp `  T )
77 eqid 2454 . . . . . . . . 9  |-  ( LSubSp `  V )  =  (
LSubSp `  V )
7849, 76, 77lsslss 17802 . . . . . . . 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 659 . . . . . . 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 920 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ran  ( F  |`  a )  e.  (
LSubSp `  V ) )
81 eqid 2454 . . . . . . 7  |-  ( Vs  ran  ( F  |`  a
) )  =  ( Vs 
ran  ( F  |`  a ) )
8277, 81lnmlssfg 31265 . . . . . 6  |-  ( ( V  e. LNoeM  /\  ran  ( F  |`  a )  e.  ( LSubSp `  V )
)  ->  ( Vs  ran  ( F  |`  a ) )  e. LFinGen )
8368, 80, 82syl2anc 659 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ( Vs  ran  ( F  |`  a ) )  e. LFinGen )
8467, 83eqeltrd 2542 . . . 4  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ( Ts  ran  ( F  |`  a ) )  e. LFinGen )
85 eqid 2454 . . . . 5  |-  ( `' ( F  |`  a
) " {  .0.  } )  =  ( `' ( F  |`  a
) " {  .0.  } )
86 eqid 2454 . . . . 5  |-  ( ( Ss  a )s  ( `' ( F  |`  a ) " {  .0.  } ) )  =  ( ( Ss  a )s  ( `' ( F  |`  a ) " {  .0.  } ) )
87 eqid 2454 . . . . 5  |-  ( Ts  ran  ( F  |`  a
) )  =  ( Ts 
ran  ( F  |`  a ) )
8835, 85, 86, 87lmhmfgsplit 31271 . . . 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 1226 . . 3  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ( Ss  a
)  e. LFinGen )
9089ralrimiva 2868 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LNoeM  /\  V  e. LNoeM )  ->  A. a  e.  (
LSubSp `  S ) ( Ss  a )  e. LFinGen )
913islnm 31262 . 2  |-  ( S  e. LNoeM 
<->  ( S  e.  LMod  /\ 
A. a  e.  (
LSubSp `  S ) ( Ss  a )  e. LFinGen )
)
922, 90, 91sylanbrc 662 1  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LNoeM  /\  V  e. LNoeM )  ->  S  e. LNoeM )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   _Vcvv 3106    i^i cin 3460    C_ wss 3461   {csn 4016   `'ccnv 4987   ran crn 4989    |` cres 4990   "cima 4991   ` cfv 5570  (class class class)co 6270   ↾s cress 14717   0gc0g 14929   LModclmod 17707   LSubSpclss 17773   LMHom clmhm 17860  LFinGenclfig 31252  LNoeMclnm 31260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-sca 14800  df-vsca 14801  df-0g 14931  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-grp 16256  df-minusg 16257  df-sbg 16258  df-subg 16397  df-ghm 16464  df-cntz 16554  df-lsm 16855  df-cmn 16999  df-abl 17000  df-mgp 17337  df-ur 17349  df-ring 17395  df-lmod 17709  df-lss 17774  df-lsp 17813  df-lmhm 17863  df-lfig 31253  df-lnm 31261
This theorem is referenced by:  pwslnmlem2  31278
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