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Theorem lmhmlnmsplit 30665
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 17479 . . 3  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
213ad2ant1 1017 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LNoeM  /\  V  e. LNoeM )  ->  S  e.  LMod )
3 eqid 2467 . . . . . 6  |-  ( LSubSp `  S )  =  (
LSubSp `  S )
4 eqid 2467 . . . . . 6  |-  ( Ss  a )  =  ( Ss  a )
53, 4reslmhm 17498 . . . . 5  |-  ( ( F  e.  ( S LMHom 
T )  /\  a  e.  ( LSubSp `  S )
)  ->  ( F  |`  a )  e.  ( ( Ss  a ) LMHom  T
) )
653ad2antl1 1158 . . . 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 5496 . . . . . . . 8  |-  ( `' ( F  |`  a
) " {  .0.  } )  =  ( ( `' F " {  .0.  } )  i^i  a )
8 lmhmfgsplit.k . . . . . . . . . 10  |-  K  =  ( `' F " {  .0.  } )
98eqcomi 2480 . . . . . . . . 9  |-  ( `' F " {  .0.  } )  =  K
109ineq1i 3696 . . . . . . . 8  |-  ( ( `' F " {  .0.  } )  i^i  a )  =  ( K  i^i  a )
11 incom 3691 . . . . . . . 8  |-  ( K  i^i  a )  =  ( a  i^i  K
)
127, 10, 113eqtri 2500 . . . . . . 7  |-  ( `' ( F  |`  a
) " {  .0.  } )  =  ( a  i^i  K )
1312oveq2i 6295 . . . . . 6  |-  ( ( Ss  a )s  ( `' ( F  |`  a ) " {  .0.  } ) )  =  ( ( Ss  a )s  ( a  i^i 
K ) )
14 lmhmfgsplit.u . . . . . . . . 9  |-  U  =  ( Ss  K )
1514oveq1i 6294 . . . . . . . 8  |-  ( Us  ( a  i^i  K ) )  =  ( ( Ss  K )s  ( a  i^i 
K ) )
16 simpl1 999 . . . . . . . . . 10  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  F  e.  ( S LMHom  T ) )
17 cnvexg 6730 . . . . . . . . . . . 12  |-  ( F  e.  ( S LMHom  T
)  ->  `' F  e.  _V )
18 imaexg 6721 . . . . . . . . . . . 12  |-  ( `' F  e.  _V  ->  ( `' F " {  .0.  } )  e.  _V )
1917, 18syl 16 . . . . . . . . . . 11  |-  ( F  e.  ( S LMHom  T
)  ->  ( `' F " {  .0.  }
)  e.  _V )
208, 19syl5eqel 2559 . . . . . . . . . 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 3719 . . . . . . . . 9  |-  ( a  i^i  K )  C_  K
23 ressabs 14553 . . . . . . . . 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 2520 . . . . . . 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 3116 . . . . . . . 8  |-  a  e. 
_V
27 inss1 3718 . . . . . . . 8  |-  ( a  i^i  K )  C_  a
28 ressabs 14553 . . . . . . . 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 2527 . . . . . 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 2520 . . . . 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 1000 . . . . . 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 17497 . . . . . . . . 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 17411 . . . . . . . 8  |-  ( ( S  e.  LMod  /\  a  e.  ( LSubSp `  S )  /\  K  e.  ( LSubSp `
 S ) )  ->  ( a  i^i 
K )  e.  (
LSubSp `  S ) )
3933, 34, 37, 38syl3anc 1228 . . . . . . 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 2467 . . . . . . . . 9  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
4214, 3, 41lsslss 17407 . . . . . . . 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 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 2467 . . . . . . 7  |-  ( Us  ( a  i^i  K ) )  =  ( Us  ( a  i^i  K ) )
4641, 45lnmlssfg 30658 . . . . . 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 2555 . . . 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 6294 . . . . . . . 8  |-  ( Vs  ran  ( F  |`  a
) )  =  ( ( Ts  ran  F )s  ran  ( F  |`  a ) )
51 rnexg 6716 . . . . . . . . 9  |-  ( F  e.  ( S LMHom  T
)  ->  ran  F  e. 
_V )
52 resexg 5316 . . . . . . . . . 10  |-  ( F  e.  ( S LMHom  T
)  ->  ( F  |`  a )  e.  _V )
53 rnexg 6716 . . . . . . . . . 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 14552 . . . . . . . . 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 2520 . . . . . . 7  |-  ( F  e.  ( S LMHom  T
)  ->  ( Vs  ran  ( F  |`  a ) )  =  ( Ts  ( ran  F  i^i  ran  ( F  |`  a ) ) ) )
58 incom 3691 . . . . . . . . 9  |-  ( ran 
F  i^i  ran  ( F  |`  a ) )  =  ( ran  ( F  |`  a )  i^i  ran  F )
59 resss 5297 . . . . . . . . . . 11  |-  ( F  |`  a )  C_  F
60 rnss 5231 . . . . . . . . . . 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 3490 . . . . . . . . . 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 2497 . . . . . . . 8  |-  ran  ( F  |`  a )  =  ( ran  F  i^i  ran  ( F  |`  a
) )
6564oveq2i 6295 . . . . . . 7  |-  ( Ts  ran  ( F  |`  a
) )  =  ( Ts  ( ran  F  i^i  ran  ( F  |`  a
) ) )
6657, 65syl6reqr 2527 . . . . . 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 1001 . . . . . 6  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  V  e. LNoeM )
69 lmhmrnlss 17496 . . . . . . . 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 17478 . . . . . . . . 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 17496 . . . . . . . . 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 2467 . . . . . . . . 9  |-  ( LSubSp `  T )  =  (
LSubSp `  T )
77 eqid 2467 . . . . . . . . 9  |-  ( LSubSp `  V )  =  (
LSubSp `  V )
7849, 76, 77lsslss 17407 . . . . . . . 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 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 2467 . . . . . . 7  |-  ( Vs  ran  ( F  |`  a
) )  =  ( Vs 
ran  ( F  |`  a ) )
8277, 81lnmlssfg 30658 . . . . . 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 2555 . . . 4  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ( Ts  ran  ( F  |`  a ) )  e. LFinGen )
85 eqid 2467 . . . . 5  |-  ( `' ( F  |`  a
) " {  .0.  } )  =  ( `' ( F  |`  a
) " {  .0.  } )
86 eqid 2467 . . . . 5  |-  ( ( Ss  a )s  ( `' ( F  |`  a ) " {  .0.  } ) )  =  ( ( Ss  a )s  ( `' ( F  |`  a ) " {  .0.  } ) )
87 eqid 2467 . . . . 5  |-  ( Ts  ran  ( F  |`  a
) )  =  ( Ts 
ran  ( F  |`  a ) )
8835, 85, 86, 87lmhmfgsplit 30664 . . . 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 1228 . . 3  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LNoeM  /\  V  e. LNoeM
)  /\  a  e.  ( LSubSp `  S )
)  ->  ( Ss  a
)  e. LFinGen )
9089ralrimiva 2878 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LNoeM  /\  V  e. LNoeM )  ->  A. a  e.  (
LSubSp `  S ) ( Ss  a )  e. LFinGen )
913islnm 30655 . 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 973    = wceq 1379    e. wcel 1767   A.wral 2814   _Vcvv 3113    i^i cin 3475    C_ wss 3476   {csn 4027   `'ccnv 4998   ran crn 5000    |` cres 5001   "cima 5002   ` cfv 5588  (class class class)co 6284   ↾s cress 14491   0gc0g 14695   LModclmod 17312   LSubSpclss 17378   LMHom clmhm 17465  LFinGenclfig 30645  LNoeMclnm 30653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-4 10596  df-5 10597  df-6 10598  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-sca 14571  df-vsca 14572  df-0g 14697  df-mnd 15732  df-submnd 15787  df-grp 15867  df-minusg 15868  df-sbg 15869  df-subg 16003  df-ghm 16070  df-cntz 16160  df-lsm 16462  df-cmn 16606  df-abl 16607  df-mgp 16944  df-ur 16956  df-rng 17002  df-lmod 17314  df-lss 17379  df-lsp 17418  df-lmhm 17468  df-lfig 30646  df-lnm 30654
This theorem is referenced by:  pwslnmlem2  30671
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