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Theorem lmhmlnmsplit 31008
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 17553 . . 3  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
213ad2ant1 1018 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LNoeM  /\  V  e. LNoeM )  ->  S  e.  LMod )
3 eqid 2443 . . . . . 6  |-  ( LSubSp `  S )  =  (
LSubSp `  S )
4 eqid 2443 . . . . . 6  |-  ( Ss  a )  =  ( Ss  a )
53, 4reslmhm 17572 . . . . 5  |-  ( ( F  e.  ( S LMHom 
T )  /\  a  e.  ( LSubSp `  S )
)  ->  ( F  |`  a )  e.  ( ( Ss  a ) LMHom  T
) )
653ad2antl1 1159 . . . 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 5486 . . . . . . . 8  |-  ( `' ( F  |`  a
) " {  .0.  } )  =  ( ( `' F " {  .0.  } )  i^i  a )
8 lmhmfgsplit.k . . . . . . . . . 10  |-  K  =  ( `' F " {  .0.  } )
98eqcomi 2456 . . . . . . . . 9  |-  ( `' F " {  .0.  } )  =  K
109ineq1i 3681 . . . . . . . 8  |-  ( ( `' F " {  .0.  } )  i^i  a )  =  ( K  i^i  a )
11 incom 3676 . . . . . . . 8  |-  ( K  i^i  a )  =  ( a  i^i  K
)
127, 10, 113eqtri 2476 . . . . . . 7  |-  ( `' ( F  |`  a
) " {  .0.  } )  =  ( a  i^i  K )
1312oveq2i 6292 . . . . . 6  |-  ( ( Ss  a )s  ( `' ( F  |`  a ) " {  .0.  } ) )  =  ( ( Ss  a )s  ( a  i^i 
K ) )
14 lmhmfgsplit.u . . . . . . . . 9  |-  U  =  ( Ss  K )
1514oveq1i 6291 . . . . . . . 8  |-  ( Us  ( a  i^i  K ) )  =  ( ( Ss  K )s  ( a  i^i 
K ) )
16 simpl1 1000 . . . . . . . . . 10  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  F  e.  ( S LMHom  T ) )
17 cnvexg 6731 . . . . . . . . . . . 12  |-  ( F  e.  ( S LMHom  T
)  ->  `' F  e.  _V )
18 imaexg 6722 . . . . . . . . . . . 12  |-  ( `' F  e.  _V  ->  ( `' F " {  .0.  } )  e.  _V )
1917, 18syl 16 . . . . . . . . . . 11  |-  ( F  e.  ( S LMHom  T
)  ->  ( `' F " {  .0.  }
)  e.  _V )
208, 19syl5eqel 2535 . . . . . . . . . 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 3704 . . . . . . . . 9  |-  ( a  i^i  K )  C_  K
23 ressabs 14572 . . . . . . . . 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 662 . . . . . . . 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 2496 . . . . . . 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 3098 . . . . . . . 8  |-  a  e. 
_V
27 inss1 3703 . . . . . . . 8  |-  ( a  i^i  K )  C_  a
28 ressabs 14572 . . . . . . . 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 672 . . . . . . 7  |-  ( ( Ss  a )s  ( a  i^i 
K ) )  =  ( Ss  ( a  i^i 
K ) )
3025, 29syl6reqr 2503 . . . . . 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 2496 . . . . 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 1001 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  U  e. LNoeM )
332adantr 465 . . . . . . . 8  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  S  e.  LMod )
34 simpr 461 . . . . . . . 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 17571 . . . . . . . . 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 17485 . . . . . . . 8  |-  ( ( S  e.  LMod  /\  a  e.  ( LSubSp `  S )  /\  K  e.  ( LSubSp `
 S ) )  ->  ( a  i^i 
K )  e.  (
LSubSp `  S ) )
3933, 34, 37, 38syl3anc 1229 . . . . . . 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 2443 . . . . . . . . 9  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
4214, 3, 41lsslss 17481 . . . . . . . 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 661 . . . . . . 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 922 . . . . . 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 2443 . . . . . . 7  |-  ( Us  ( a  i^i  K ) )  =  ( Us  ( a  i^i  K ) )
4641, 45lnmlssfg 31001 . . . . . 6  |-  ( ( U  e. LNoeM  /\  (
a  i^i  K )  e.  ( LSubSp `  U )
)  ->  ( Us  (
a  i^i  K )
)  e. LFinGen )
4732, 44, 46syl2anc 661 . . . . 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 2531 . . . 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 6291 . . . . . . . 8  |-  ( Vs  ran  ( F  |`  a
) )  =  ( ( Ts  ran  F )s  ran  ( F  |`  a ) )
51 rnexg 6717 . . . . . . . . 9  |-  ( F  e.  ( S LMHom  T
)  ->  ran  F  e. 
_V )
52 resexg 5306 . . . . . . . . . 10  |-  ( F  e.  ( S LMHom  T
)  ->  ( F  |`  a )  e.  _V )
53 rnexg 6717 . . . . . . . . . 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 14571 . . . . . . . . 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 661 . . . . . . . 8  |-  ( F  e.  ( S LMHom  T
)  ->  ( ( Ts  ran  F )s  ran  ( F  |`  a ) )  =  ( Ts  ( ran  F  i^i  ran  ( F  |`  a ) ) ) )
5750, 56syl5eq 2496 . . . . . . 7  |-  ( F  e.  ( S LMHom  T
)  ->  ( Vs  ran  ( F  |`  a ) )  =  ( Ts  ( ran  F  i^i  ran  ( F  |`  a ) ) ) )
58 incom 3676 . . . . . . . . 9  |-  ( ran 
F  i^i  ran  ( F  |`  a ) )  =  ( ran  ( F  |`  a )  i^i  ran  F )
59 resss 5287 . . . . . . . . . . 11  |-  ( F  |`  a )  C_  F
60 rnss 5221 . . . . . . . . . . 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 2473 . . . . . . . 8  |-  ran  ( F  |`  a )  =  ( ran  F  i^i  ran  ( F  |`  a
) )
6564oveq2i 6292 . . . . . . 7  |-  ( Ts  ran  ( F  |`  a
) )  =  ( Ts  ( ran  F  i^i  ran  ( F  |`  a
) ) )
6657, 65syl6reqr 2503 . . . . . 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 1002 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  V  e. LNoeM )
69 lmhmrnlss 17570 . . . . . . . 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 17552 . . . . . . . . 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 17570 . . . . . . . . 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 2443 . . . . . . . . 9  |-  ( LSubSp `  T )  =  (
LSubSp `  T )
77 eqid 2443 . . . . . . . . 9  |-  ( LSubSp `  V )  =  (
LSubSp `  V )
7849, 76, 77lsslss 17481 . . . . . . . 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 661 . . . . . . 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 922 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ran  ( F  |`  a )  e.  (
LSubSp `  V ) )
81 eqid 2443 . . . . . . 7  |-  ( Vs  ran  ( F  |`  a
) )  =  ( Vs 
ran  ( F  |`  a ) )
8277, 81lnmlssfg 31001 . . . . . 6  |-  ( ( V  e. LNoeM  /\  ran  ( F  |`  a )  e.  ( LSubSp `  V )
)  ->  ( Vs  ran  ( F  |`  a ) )  e. LFinGen )
8368, 80, 82syl2anc 661 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ( Vs  ran  ( F  |`  a ) )  e. LFinGen )
8467, 83eqeltrd 2531 . . . 4  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ( Ts  ran  ( F  |`  a ) )  e. LFinGen )
85 eqid 2443 . . . . 5  |-  ( `' ( F  |`  a
) " {  .0.  } )  =  ( `' ( F  |`  a
) " {  .0.  } )
86 eqid 2443 . . . . 5  |-  ( ( Ss  a )s  ( `' ( F  |`  a ) " {  .0.  } ) )  =  ( ( Ss  a )s  ( `' ( F  |`  a ) " {  .0.  } ) )
87 eqid 2443 . . . . 5  |-  ( Ts  ran  ( F  |`  a
) )  =  ( Ts 
ran  ( F  |`  a ) )
8835, 85, 86, 87lmhmfgsplit 31007 . . . 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 1229 . . 3  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ( Ss  a
)  e. LFinGen )
9089ralrimiva 2857 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LNoeM  /\  V  e. LNoeM )  ->  A. a  e.  (
LSubSp `  S ) ( Ss  a )  e. LFinGen )
913islnm 30998 . 2  |-  ( S  e. LNoeM 
<->  ( S  e.  LMod  /\ 
A. a  e.  (
LSubSp `  S ) ( Ss  a )  e. LFinGen )
)
922, 90, 91sylanbrc 664 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 369    /\ w3a 974    = wceq 1383    e. wcel 1804   A.wral 2793   _Vcvv 3095    i^i cin 3460    C_ wss 3461   {csn 4014   `'ccnv 4988   ran crn 4990    |` cres 4991   "cima 4992   ` cfv 5578  (class class class)co 6281   ↾s cress 14510   0gc0g 14714   LModclmod 17386   LSubSpclss 17452   LMHom clmhm 17539  LFinGenclfig 30988  LNoeMclnm 30996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-sca 14590  df-vsca 14591  df-0g 14716  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15841  df-grp 15931  df-minusg 15932  df-sbg 15933  df-subg 16072  df-ghm 16139  df-cntz 16229  df-lsm 16530  df-cmn 16674  df-abl 16675  df-mgp 17016  df-ur 17028  df-ring 17074  df-lmod 17388  df-lss 17453  df-lsp 17492  df-lmhm 17542  df-lfig 30989  df-lnm 30997
This theorem is referenced by:  pwslnmlem2  31014
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