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Theorem lmhmfgsplit 29465
Description: If the kernel and range of a homomorphism of left modules are finitely generated, then so is the domain. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Revised by Stefan O'Rear, 6-May-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
lmhmfgsplit  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  S  e. LFinGen )

Proof of Theorem lmhmfgsplit
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp3 990 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  V  e. LFinGen )
2 lmhmlmod2 17135 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  T  e.  LMod )
323ad2ant1 1009 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  T  e.  LMod )
4 lmhmrnlss 17153 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  ran  F  e.  ( LSubSp `  T )
)
543ad2ant1 1009 . . . 4  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  ran  F  e.  (
LSubSp `  T ) )
6 lmhmfgsplit.v . . . . 5  |-  V  =  ( Ts  ran  F )
7 eqid 2443 . . . . 5  |-  ( LSubSp `  T )  =  (
LSubSp `  T )
8 eqid 2443 . . . . 5  |-  ( LSpan `  T )  =  (
LSpan `  T )
96, 7, 8islssfg 29449 . . . 4  |-  ( ( T  e.  LMod  /\  ran  F  e.  ( LSubSp `  T
) )  ->  ( V  e. LFinGen  <->  E. a  e.  ~P  ran  F ( a  e. 
Fin  /\  ( ( LSpan `  T ) `  a )  =  ran  F ) ) )
103, 5, 9syl2anc 661 . . 3  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  ( V  e. LFinGen  <->  E. a  e.  ~P  ran  F ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )
111, 10mpbid 210 . 2  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  E. a  e.  ~P  ran  F ( a  e. 
Fin  /\  ( ( LSpan `  T ) `  a )  =  ran  F ) )
12 simpl1 991 . . . . 5  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e. 
~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  ->  F  e.  ( S LMHom  T ) )
13 eqid 2443 . . . . . 6  |-  ( Base `  S )  =  (
Base `  S )
14 eqid 2443 . . . . . 6  |-  ( Base `  T )  =  (
Base `  T )
1513, 14lmhmf 17137 . . . . 5  |-  ( F  e.  ( S LMHom  T
)  ->  F :
( Base `  S ) --> ( Base `  T )
)
16 ffn 5580 . . . . 5  |-  ( F : ( Base `  S
) --> ( Base `  T
)  ->  F  Fn  ( Base `  S )
)
1712, 15, 163syl 20 . . . 4  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e. 
~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  ->  F  Fn  ( Base `  S )
)
18 elpwi 3890 . . . . 5  |-  ( a  e.  ~P ran  F  ->  a  C_  ran  F )
1918ad2antrl 727 . . . 4  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e. 
~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  ->  a  C_  ran  F )
20 simprrl 763 . . . 4  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e. 
~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  ->  a  e.  Fin )
21 fipreima 7638 . . . 4  |-  ( ( F  Fn  ( Base `  S )  /\  a  C_ 
ran  F  /\  a  e.  Fin )  ->  E. b  e.  ( ~P ( Base `  S )  i^i  Fin ) ( F "
b )  =  a )
2217, 19, 20, 21syl3anc 1218 . . 3  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e. 
~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  ->  E. b  e.  ( ~P ( Base `  S )  i^i  Fin ) ( F "
b )  =  a )
23 eqid 2443 . . . . . . 7  |-  ( LSubSp `  S )  =  (
LSubSp `  S )
24 eqid 2443 . . . . . . 7  |-  ( LSSum `  S )  =  (
LSSum `  S )
25 lmhmfgsplit.z . . . . . . 7  |-  .0.  =  ( 0g `  T )
26 lmhmfgsplit.k . . . . . . 7  |-  K  =  ( `' F " {  .0.  } )
27 simpll1 1027 . . . . . . 7  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  ->  F  e.  ( S LMHom  T ) )
28 lmhmlmod1 17136 . . . . . . . . . 10  |-  ( F  e.  ( S LMHom  T
)  ->  S  e.  LMod )
29283ad2ant1 1009 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  S  e.  LMod )
3029ad2antrr 725 . . . . . . . 8  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  ->  S  e.  LMod )
31 inss1 3591 . . . . . . . . . . 11  |-  ( ~P ( Base `  S
)  i^i  Fin )  C_ 
~P ( Base `  S
)
3231sseli 3373 . . . . . . . . . 10  |-  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  ->  b  e.  ~P ( Base `  S
) )
33 elpwi 3890 . . . . . . . . . 10  |-  ( b  e.  ~P ( Base `  S )  ->  b  C_  ( Base `  S
) )
3432, 33syl 16 . . . . . . . . 9  |-  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  ->  b  C_  ( Base `  S
) )
3534ad2antrl 727 . . . . . . . 8  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
b  C_  ( Base `  S ) )
36 eqid 2443 . . . . . . . . 9  |-  ( LSpan `  S )  =  (
LSpan `  S )
3713, 23, 36lspcl 17079 . . . . . . . 8  |-  ( ( S  e.  LMod  /\  b  C_  ( Base `  S
) )  ->  (
( LSpan `  S ) `  b )  e.  (
LSubSp `  S ) )
3830, 35, 37syl2anc 661 . . . . . . 7  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
( ( LSpan `  S
) `  b )  e.  ( LSubSp `  S )
)
3913, 36, 8lmhmlsp 17152 . . . . . . . . 9  |-  ( ( F  e.  ( S LMHom 
T )  /\  b  C_  ( Base `  S
) )  ->  ( F " ( ( LSpan `  S ) `  b
) )  =  ( ( LSpan `  T ) `  ( F " b
) ) )
4027, 35, 39syl2anc 661 . . . . . . . 8  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
( F " (
( LSpan `  S ) `  b ) )  =  ( ( LSpan `  T
) `  ( F " b ) ) )
41 fveq2 5712 . . . . . . . . 9  |-  ( ( F " b )  =  a  ->  (
( LSpan `  T ) `  ( F " b
) )  =  ( ( LSpan `  T ) `  a ) )
4241ad2antll 728 . . . . . . . 8  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
( ( LSpan `  T
) `  ( F " b ) )  =  ( ( LSpan `  T
) `  a )
)
43 simp2rr 1058 . . . . . . . . 9  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e. 
~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i  Fin )  /\  ( F "
b )  =  a ) )  ->  (
( LSpan `  T ) `  a )  =  ran  F )
44433expa 1187 . . . . . . . 8  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
( ( LSpan `  T
) `  a )  =  ran  F )
4540, 42, 443eqtrd 2479 . . . . . . 7  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
( F " (
( LSpan `  S ) `  b ) )  =  ran  F )
4623, 24, 25, 26, 13, 27, 38, 45kercvrlsm 29462 . . . . . 6  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
( K ( LSSum `  S ) ( (
LSpan `  S ) `  b ) )  =  ( Base `  S
) )
4746oveq2d 6128 . . . . 5  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
( Ss  ( K (
LSSum `  S ) ( ( LSpan `  S ) `  b ) ) )  =  ( Ss  ( Base `  S ) ) )
4813ressid 14254 . . . . . . 7  |-  ( S  e.  LMod  ->  ( Ss  (
Base `  S )
)  =  S )
4929, 48syl 16 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  ( Ss  ( Base `  S
) )  =  S )
5049ad2antrr 725 . . . . 5  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
( Ss  ( Base `  S
) )  =  S )
5147, 50eqtr2d 2476 . . . 4  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  ->  S  =  ( Ss  ( K ( LSSum `  S
) ( ( LSpan `  S ) `  b
) ) ) )
52 lmhmfgsplit.u . . . . 5  |-  U  =  ( Ss  K )
53 eqid 2443 . . . . 5  |-  ( Ss  ( ( LSpan `  S ) `  b ) )  =  ( Ss  ( ( LSpan `  S ) `  b
) )
54 eqid 2443 . . . . 5  |-  ( Ss  ( K ( LSSum `  S
) ( ( LSpan `  S ) `  b
) ) )  =  ( Ss  ( K (
LSSum `  S ) ( ( LSpan `  S ) `  b ) ) )
5526, 25, 23lmhmkerlss 17154 . . . . . . 7  |-  ( F  e.  ( S LMHom  T
)  ->  K  e.  ( LSubSp `  S )
)
56553ad2ant1 1009 . . . . . 6  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  K  e.  ( LSubSp `  S ) )
5756ad2antrr 725 . . . . 5  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  ->  K  e.  ( LSubSp `  S ) )
58 simpll2 1028 . . . . 5  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  ->  U  e. LFinGen )
59 inss2 3592 . . . . . . . 8  |-  ( ~P ( Base `  S
)  i^i  Fin )  C_ 
Fin
6059sseli 3373 . . . . . . 7  |-  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  ->  b  e.  Fin )
6160ad2antrl 727 . . . . . 6  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
b  e.  Fin )
6236, 13, 53islssfgi 29451 . . . . . 6  |-  ( ( S  e.  LMod  /\  b  C_  ( Base `  S
)  /\  b  e.  Fin )  ->  ( Ss  ( ( LSpan `  S ) `  b ) )  e. LFinGen )
6330, 35, 61, 62syl3anc 1218 . . . . 5  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
( Ss  ( ( LSpan `  S ) `  b
) )  e. LFinGen )
6423, 24, 52, 53, 54, 30, 57, 38, 58, 63lsmfgcl 29453 . . . 4  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  -> 
( Ss  ( K (
LSSum `  S ) ( ( LSpan `  S ) `  b ) ) )  e. LFinGen )
6551, 64eqeltrd 2517 . . 3  |-  ( ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e.  ~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  /\  ( b  e.  ( ~P ( Base `  S )  i^i 
Fin )  /\  ( F " b )  =  a ) )  ->  S  e. LFinGen )
6622, 65rexlimddv 2866 . 2  |-  ( ( ( F  e.  ( S LMHom  T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  /\  ( a  e. 
~P ran  F  /\  ( a  e.  Fin  /\  ( ( LSpan `  T
) `  a )  =  ran  F ) ) )  ->  S  e. LFinGen )
6711, 66rexlimddv 2866 1  |-  ( ( F  e.  ( S LMHom 
T )  /\  U  e. LFinGen  /\  V  e. LFinGen )  ->  S  e. LFinGen )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2737    i^i cin 3348    C_ wss 3349   ~Pcpw 3881   {csn 3898   `'ccnv 4860   ran crn 4862   "cima 4864    Fn wfn 5434   -->wf 5435   ` cfv 5439  (class class class)co 6112   Fincfn 7331   Basecbs 14195   ↾s cress 14196   0gc0g 14399   LSSumclsm 16154   LModclmod 16970   LSubSpclss 17035   LSpanclspn 17074   LMHom clmhm 17122  LFinGenclfig 29446
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-sca 14275  df-vsca 14276  df-0g 14401  df-mnd 15436  df-submnd 15486  df-grp 15566  df-minusg 15567  df-sbg 15568  df-subg 15699  df-ghm 15766  df-cntz 15856  df-lsm 16156  df-cmn 16300  df-abl 16301  df-mgp 16614  df-ur 16626  df-rng 16669  df-lmod 16972  df-lss 17036  df-lsp 17075  df-lmhm 17125  df-lfig 29447
This theorem is referenced by:  lmhmlnmsplit  29466
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