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Theorem frlmsplit2 27243
Description: Restriction is homomoprhic on free modules. (Contributed by Stefan O'Rear, 3-Feb-2015.)
Hypotheses
Ref Expression
frlmsplit2.y  |-  Y  =  ( R freeLMod  U )
frlmsplit2.z  |-  Z  =  ( R freeLMod  V )
frlmsplit2.b  |-  B  =  ( Base `  Y
)
frlmsplit2.c  |-  C  =  ( Base `  Z
)
frlmsplit2.f  |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )
Assertion
Ref Expression
frlmsplit2  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  F  e.  ( Y LMHom  Z ) )
Distinct variable groups:    x, Y    x, R    x, U    x, Z    x, V    x, B    x, C    x, X
Allowed substitution hint:    F( x)

Proof of Theorem frlmsplit2
StepHypRef Expression
1 simp1 955 . . . . . 6  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  R  e.  Ring )
2 simp2 956 . . . . . 6  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  U  e.  X )
3 frlmsplit2.y . . . . . . 7  |-  Y  =  ( R freeLMod  U )
4 frlmsplit2.b . . . . . . 7  |-  B  =  ( Base `  Y
)
5 eqid 2283 . . . . . . 7  |-  ( LSubSp `  ( (ringLMod `  R
)  ^s  U ) )  =  ( LSubSp `  ( (ringLMod `  R )  ^s  U ) )
63, 4, 5frlmlss 27219 . . . . . 6  |-  ( ( R  e.  Ring  /\  U  e.  X )  ->  B  e.  ( LSubSp `  ( (ringLMod `  R )  ^s  U ) ) )
71, 2, 6syl2anc 642 . . . . 5  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  B  e.  ( LSubSp `  ( (ringLMod `  R )  ^s  U ) ) )
8 eqid 2283 . . . . . 6  |-  ( Base `  ( (ringLMod `  R
)  ^s  U ) )  =  ( Base `  (
(ringLMod `  R )  ^s  U
) )
98, 5lssss 15694 . . . . 5  |-  ( B  e.  ( LSubSp `  (
(ringLMod `  R )  ^s  U
) )  ->  B  C_  ( Base `  (
(ringLMod `  R )  ^s  U
) ) )
10 resmpt 5000 . . . . 5  |-  ( B 
C_  ( Base `  (
(ringLMod `  R )  ^s  U
) )  ->  (
( x  e.  (
Base `  ( (ringLMod `  R )  ^s  U ) )  |->  ( x  |`  V ) )  |`  B )  =  ( x  e.  B  |->  ( x  |`  V )
) )
117, 9, 103syl 18 . . . 4  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  (
( x  e.  (
Base `  ( (ringLMod `  R )  ^s  U ) )  |->  ( x  |`  V ) )  |`  B )  =  ( x  e.  B  |->  ( x  |`  V )
) )
12 frlmsplit2.f . . . 4  |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )
1311, 12syl6eqr 2333 . . 3  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  (
( x  e.  (
Base `  ( (ringLMod `  R )  ^s  U ) )  |->  ( x  |`  V ) )  |`  B )  =  F )
14 rlmlmod 15957 . . . . . 6  |-  ( R  e.  Ring  ->  (ringLMod `  R
)  e.  LMod )
15 eqid 2283 . . . . . . 7  |-  ( (ringLMod `  R )  ^s  U )  =  ( (ringLMod `  R
)  ^s  U )
16 eqid 2283 . . . . . . 7  |-  ( (ringLMod `  R )  ^s  V )  =  ( (ringLMod `  R
)  ^s  V )
17 eqid 2283 . . . . . . 7  |-  ( Base `  ( (ringLMod `  R
)  ^s  V ) )  =  ( Base `  (
(ringLMod `  R )  ^s  V
) )
18 eqid 2283 . . . . . . 7  |-  ( x  e.  ( Base `  (
(ringLMod `  R )  ^s  U
) )  |->  ( x  |`  V ) )  =  ( x  e.  (
Base `  ( (ringLMod `  R )  ^s  U ) )  |->  ( x  |`  V ) )
1915, 16, 8, 17, 18pwssplit3 27190 . . . . . 6  |-  ( ( (ringLMod `  R )  e.  LMod  /\  U  e.  X  /\  V  C_  U
)  ->  ( x  e.  ( Base `  (
(ringLMod `  R )  ^s  U
) )  |->  ( x  |`  V ) )  e.  ( ( (ringLMod `  R
)  ^s  U ) LMHom  ( (ringLMod `  R )  ^s  V ) ) )
2014, 19syl3an1 1215 . . . . 5  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  (
x  e.  ( Base `  ( (ringLMod `  R
)  ^s  U ) )  |->  ( x  |`  V )
)  e.  ( ( (ringLMod `  R )  ^s  U ) LMHom  ( (ringLMod `  R )  ^s  V ) ) )
21 eqid 2283 . . . . . 6  |-  ( ( (ringLMod `  R )  ^s  U )s  B )  =  ( ( (ringLMod `  R
)  ^s  U )s  B )
225, 21reslmhm 15809 . . . . 5  |-  ( ( ( x  e.  (
Base `  ( (ringLMod `  R )  ^s  U ) )  |->  ( x  |`  V ) )  e.  ( ( (ringLMod `  R
)  ^s  U ) LMHom  ( (ringLMod `  R )  ^s  V ) )  /\  B  e.  ( LSubSp `  ( (ringLMod `  R )  ^s  U ) ) )  ->  (
( x  e.  (
Base `  ( (ringLMod `  R )  ^s  U ) )  |->  ( x  |`  V ) )  |`  B )  e.  ( ( ( (ringLMod `  R
)  ^s  U )s  B ) LMHom  ( (ringLMod `  R )  ^s  V ) ) )
2320, 7, 22syl2anc 642 . . . 4  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  (
( x  e.  (
Base `  ( (ringLMod `  R )  ^s  U ) )  |->  ( x  |`  V ) )  |`  B )  e.  ( ( ( (ringLMod `  R
)  ^s  U )s  B ) LMHom  ( (ringLMod `  R )  ^s  V ) ) )
24143ad2ant1 976 . . . . . 6  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  (ringLMod `  R )  e.  LMod )
25 simp3 957 . . . . . . 7  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  V  C_  U )
26 ssexg 4160 . . . . . . 7  |-  ( ( V  C_  U  /\  U  e.  X )  ->  V  e.  _V )
2725, 2, 26syl2anc 642 . . . . . 6  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  V  e.  _V )
2816pwslmod 15727 . . . . . 6  |-  ( ( (ringLMod `  R )  e.  LMod  /\  V  e.  _V )  ->  ( (ringLMod `  R )  ^s  V )  e.  LMod )
2924, 27, 28syl2anc 642 . . . . 5  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  (
(ringLMod `  R )  ^s  V
)  e.  LMod )
30 frlmsplit2.z . . . . . . 7  |-  Z  =  ( R freeLMod  V )
31 frlmsplit2.c . . . . . . 7  |-  C  =  ( Base `  Z
)
32 eqid 2283 . . . . . . 7  |-  ( LSubSp `  ( (ringLMod `  R
)  ^s  V ) )  =  ( LSubSp `  ( (ringLMod `  R )  ^s  V ) )
3330, 31, 32frlmlss 27219 . . . . . 6  |-  ( ( R  e.  Ring  /\  V  e.  _V )  ->  C  e.  ( LSubSp `  ( (ringLMod `  R )  ^s  V ) ) )
341, 27, 33syl2anc 642 . . . . 5  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  C  e.  ( LSubSp `  ( (ringLMod `  R )  ^s  V ) ) )
3511rneqd 4906 . . . . . 6  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  ran  ( ( x  e.  ( Base `  (
(ringLMod `  R )  ^s  U
) )  |->  ( x  |`  V ) )  |`  B )  =  ran  ( x  e.  B  |->  ( x  |`  V ) ) )
36 eqid 2283 . . . . . . . . . . . . 13  |-  ( Base `  R )  =  (
Base `  R )
373, 36, 4frlmbasf 27228 . . . . . . . . . . . 12  |-  ( ( U  e.  X  /\  x  e.  B )  ->  x : U --> ( Base `  R ) )
382, 37sylan 457 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  x : U --> ( Base `  R ) )
39 simpl3 960 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  V  C_  U )
40 fssres 5408 . . . . . . . . . . 11  |-  ( ( x : U --> ( Base `  R )  /\  V  C_  U )  ->  (
x  |`  V ) : V --> ( Base `  R
) )
4138, 39, 40syl2anc 642 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  ( x  |`  V ) : V --> ( Base `  R ) )
42 fvex 5539 . . . . . . . . . . . 12  |-  ( Base `  R )  e.  _V
43 elmapg 6785 . . . . . . . . . . . 12  |-  ( ( ( Base `  R
)  e.  _V  /\  V  e.  _V )  ->  ( ( x  |`  V )  e.  ( ( Base `  R
)  ^m  V )  <->  ( x  |`  V ) : V --> ( Base `  R
) ) )
4442, 27, 43sylancr 644 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  (
( x  |`  V )  e.  ( ( Base `  R )  ^m  V
)  <->  ( x  |`  V ) : V --> ( Base `  R )
) )
4544adantr 451 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  ( ( x  |`  V )  e.  ( ( Base `  R
)  ^m  V )  <->  ( x  |`  V ) : V --> ( Base `  R
) ) )
4641, 45mpbird 223 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  ( x  |`  V )  e.  ( ( Base `  R )  ^m  V
) )
47 cnvresima 5162 . . . . . . . . . 10  |-  ( `' ( x  |`  V )
" ( _V  \  { ( 0g `  R ) } ) )  =  ( ( `' x " ( _V 
\  { ( 0g
`  R ) } ) )  i^i  V
)
48 eqid 2283 . . . . . . . . . . . . 13  |-  ( 0g
`  R )  =  ( 0g `  R
)
493, 48, 4frlmbassup 27226 . . . . . . . . . . . 12  |-  ( ( U  e.  X  /\  x  e.  B )  ->  ( `' x "
( _V  \  {
( 0g `  R
) } ) )  e.  Fin )
502, 49sylan 457 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  ( `' x "
( _V  \  {
( 0g `  R
) } ) )  e.  Fin )
51 inss1 3389 . . . . . . . . . . 11  |-  ( ( `' x " ( _V 
\  { ( 0g
`  R ) } ) )  i^i  V
)  C_  ( `' x " ( _V  \  { ( 0g `  R ) } ) )
52 ssfi 7083 . . . . . . . . . . 11  |-  ( ( ( `' x "
( _V  \  {
( 0g `  R
) } ) )  e.  Fin  /\  (
( `' x "
( _V  \  {
( 0g `  R
) } ) )  i^i  V )  C_  ( `' x " ( _V 
\  { ( 0g
`  R ) } ) ) )  -> 
( ( `' x " ( _V  \  {
( 0g `  R
) } ) )  i^i  V )  e. 
Fin )
5350, 51, 52sylancl 643 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  ( ( `' x " ( _V  \  {
( 0g `  R
) } ) )  i^i  V )  e. 
Fin )
5447, 53syl5eqel 2367 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  ( `' ( x  |`  V ) " ( _V  \  { ( 0g
`  R ) } ) )  e.  Fin )
5530, 36, 48, 31frlmelbas 27224 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  V  e.  _V )  ->  (
( x  |`  V )  e.  C  <->  ( (
x  |`  V )  e.  ( ( Base `  R
)  ^m  V )  /\  ( `' ( x  |`  V ) " ( _V  \  { ( 0g
`  R ) } ) )  e.  Fin ) ) )
561, 27, 55syl2anc 642 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  (
( x  |`  V )  e.  C  <->  ( (
x  |`  V )  e.  ( ( Base `  R
)  ^m  V )  /\  ( `' ( x  |`  V ) " ( _V  \  { ( 0g
`  R ) } ) )  e.  Fin ) ) )
5756adantr 451 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  ( ( x  |`  V )  e.  C  <->  ( ( x  |`  V )  e.  ( ( Base `  R )  ^m  V
)  /\  ( `' ( x  |`  V )
" ( _V  \  { ( 0g `  R ) } ) )  e.  Fin )
) )
5846, 54, 57mpbir2and 888 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  ( x  |`  V )  e.  C )
59 eqid 2283 . . . . . . . 8  |-  ( x  e.  B  |->  ( x  |`  V ) )  =  ( x  e.  B  |->  ( x  |`  V ) )
6058, 59fmptd 5684 . . . . . . 7  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  (
x  e.  B  |->  ( x  |`  V )
) : B --> C )
61 frn 5395 . . . . . . 7  |-  ( ( x  e.  B  |->  ( x  |`  V )
) : B --> C  ->  ran  ( x  e.  B  |->  ( x  |`  V ) )  C_  C )
6260, 61syl 15 . . . . . 6  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  ran  ( x  e.  B  |->  ( x  |`  V ) )  C_  C )
6335, 62eqsstrd 3212 . . . . 5  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  ran  ( ( x  e.  ( Base `  (
(ringLMod `  R )  ^s  U
) )  |->  ( x  |`  V ) )  |`  B )  C_  C
)
64 eqid 2283 . . . . . 6  |-  ( ( (ringLMod `  R )  ^s  V )s  C )  =  ( ( (ringLMod `  R
)  ^s  V )s  C )
6564, 32reslmhm2b 15811 . . . . 5  |-  ( ( ( (ringLMod `  R
)  ^s  V )  e.  LMod  /\  C  e.  ( LSubSp `  ( (ringLMod `  R
)  ^s  V ) )  /\  ran  ( ( x  e.  ( Base `  (
(ringLMod `  R )  ^s  U
) )  |->  ( x  |`  V ) )  |`  B )  C_  C
)  ->  ( (
( x  e.  (
Base `  ( (ringLMod `  R )  ^s  U ) )  |->  ( x  |`  V ) )  |`  B )  e.  ( ( ( (ringLMod `  R
)  ^s  U )s  B ) LMHom  ( (ringLMod `  R )  ^s  V ) )  <->  ( ( x  e.  ( Base `  (
(ringLMod `  R )  ^s  U
) )  |->  ( x  |`  V ) )  |`  B )  e.  ( ( ( (ringLMod `  R
)  ^s  U )s  B ) LMHom  ( ( (ringLMod `  R )  ^s  V )s  C ) ) ) )
6629, 34, 63, 65syl3anc 1182 . . . 4  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  (
( ( x  e.  ( Base `  (
(ringLMod `  R )  ^s  U
) )  |->  ( x  |`  V ) )  |`  B )  e.  ( ( ( (ringLMod `  R
)  ^s  U )s  B ) LMHom  ( (ringLMod `  R )  ^s  V ) )  <->  ( ( x  e.  ( Base `  (
(ringLMod `  R )  ^s  U
) )  |->  ( x  |`  V ) )  |`  B )  e.  ( ( ( (ringLMod `  R
)  ^s  U )s  B ) LMHom  ( ( (ringLMod `  R )  ^s  V )s  C ) ) ) )
6723, 66mpbid 201 . . 3  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  (
( x  e.  (
Base `  ( (ringLMod `  R )  ^s  U ) )  |->  ( x  |`  V ) )  |`  B )  e.  ( ( ( (ringLMod `  R
)  ^s  U )s  B ) LMHom  ( ( (ringLMod `  R )  ^s  V )s  C ) ) )
6813, 67eqeltrrd 2358 . 2  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  F  e.  ( ( ( (ringLMod `  R )  ^s  U )s  B ) LMHom  ( ( (ringLMod `  R )  ^s  V )s  C ) ) )
693, 4frlmpws 27218 . . . 4  |-  ( ( R  e.  Ring  /\  U  e.  X )  ->  Y  =  ( ( (ringLMod `  R )  ^s  U )s  B ) )
701, 2, 69syl2anc 642 . . 3  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  Y  =  ( ( (ringLMod `  R )  ^s  U )s  B ) )
7130, 31frlmpws 27218 . . . 4  |-  ( ( R  e.  Ring  /\  V  e.  _V )  ->  Z  =  ( ( (ringLMod `  R )  ^s  V )s  C ) )
721, 27, 71syl2anc 642 . . 3  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  Z  =  ( ( (ringLMod `  R )  ^s  V )s  C ) )
7370, 72oveq12d 5876 . 2  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  ( Y LMHom  Z )  =  ( ( ( (ringLMod `  R
)  ^s  U )s  B ) LMHom  ( ( (ringLMod `  R )  ^s  V )s  C ) ) )
7468, 73eleqtrrd 2360 1  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  F  e.  ( Y LMHom  Z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    i^i cin 3151    C_ wss 3152   {csn 3640    e. cmpt 4077   `'ccnv 4688   ran crn 4690    |` cres 4691   "cima 4692   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   Fincfn 6863   Basecbs 13148   ↾s cress 13149    ^s cpws 13347   0gc0g 13400   Ringcrg 15337   LModclmod 15627   LSubSpclss 15689   LMHom clmhm 15776  ringLModcrglmod 15922   freeLMod cfrlm 27212
This theorem is referenced by:  frlmsslss  27244
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-prds 13348  df-pws 13350  df-0g 13404  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-ghm 14681  df-mgp 15326  df-rng 15340  df-ur 15342  df-subrg 15543  df-lmod 15629  df-lss 15690  df-lmhm 15779  df-sra 15925  df-rgmod 15926  df-dsmm 27198  df-frlm 27214
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