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Theorem frlmsplit2 27346
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 2296 . . . . . . 7  |-  ( LSubSp `  ( (ringLMod `  R
)  ^s  U ) )  =  ( LSubSp `  ( (ringLMod `  R )  ^s  U ) )
63, 4, 5frlmlss 27322 . . . . . 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 2296 . . . . . 6  |-  ( Base `  ( (ringLMod `  R
)  ^s  U ) )  =  ( Base `  (
(ringLMod `  R )  ^s  U
) )
98, 5lssss 15710 . . . . 5  |-  ( B  e.  ( LSubSp `  (
(ringLMod `  R )  ^s  U
) )  ->  B  C_  ( Base `  (
(ringLMod `  R )  ^s  U
) ) )
10 resmpt 5016 . . . . 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 2346 . . 3  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  (
( x  e.  (
Base `  ( (ringLMod `  R )  ^s  U ) )  |->  ( x  |`  V ) )  |`  B )  =  F )
14 rlmlmod 15973 . . . . . 6  |-  ( R  e.  Ring  ->  (ringLMod `  R
)  e.  LMod )
15 eqid 2296 . . . . . . 7  |-  ( (ringLMod `  R )  ^s  U )  =  ( (ringLMod `  R
)  ^s  U )
16 eqid 2296 . . . . . . 7  |-  ( (ringLMod `  R )  ^s  V )  =  ( (ringLMod `  R
)  ^s  V )
17 eqid 2296 . . . . . . 7  |-  ( Base `  ( (ringLMod `  R
)  ^s  V ) )  =  ( Base `  (
(ringLMod `  R )  ^s  V
) )
18 eqid 2296 . . . . . . 7  |-  ( x  e.  ( Base `  (
(ringLMod `  R )  ^s  U
) )  |->  ( x  |`  V ) )  =  ( x  e.  (
Base `  ( (ringLMod `  R )  ^s  U ) )  |->  ( x  |`  V ) )
1915, 16, 8, 17, 18pwssplit3 27293 . . . . . 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 2296 . . . . . 6  |-  ( ( (ringLMod `  R )  ^s  U )s  B )  =  ( ( (ringLMod `  R
)  ^s  U )s  B )
225, 21reslmhm 15825 . . . . 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 4176 . . . . . . 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 15743 . . . . . 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 2296 . . . . . . 7  |-  ( LSubSp `  ( (ringLMod `  R
)  ^s  V ) )  =  ( LSubSp `  ( (ringLMod `  R )  ^s  V ) )
3330, 31, 32frlmlss 27322 . . . . . 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 4922 . . . . . 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 2296 . . . . . . . . . . . . 13  |-  ( Base `  R )  =  (
Base `  R )
373, 36, 4frlmbasf 27331 . . . . . . . . . . . 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 5424 . . . . . . . . . . 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 5555 . . . . . . . . . . . 12  |-  ( Base `  R )  e.  _V
43 elmapg 6801 . . . . . . . . . . . 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 5178 . . . . . . . . . 10  |-  ( `' ( x  |`  V )
" ( _V  \  { ( 0g `  R ) } ) )  =  ( ( `' x " ( _V 
\  { ( 0g
`  R ) } ) )  i^i  V
)
48 eqid 2296 . . . . . . . . . . . . 13  |-  ( 0g
`  R )  =  ( 0g `  R
)
493, 48, 4frlmbassup 27329 . . . . . . . . . . . 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 3402 . . . . . . . . . . 11  |-  ( ( `' x " ( _V 
\  { ( 0g
`  R ) } ) )  i^i  V
)  C_  ( `' x " ( _V  \  { ( 0g `  R ) } ) )
52 ssfi 7099 . . . . . . . . . . 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 2380 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  ( `' ( x  |`  V ) " ( _V  \  { ( 0g
`  R ) } ) )  e.  Fin )
5530, 36, 48, 31frlmelbas 27327 . . . . . . . . . . 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 2296 . . . . . . . 8  |-  ( x  e.  B  |->  ( x  |`  V ) )  =  ( x  e.  B  |->  ( x  |`  V ) )
6058, 59fmptd 5700 . . . . . . 7  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  (
x  e.  B  |->  ( x  |`  V )
) : B --> C )
61 frn 5411 . . . . . . 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 3225 . . . . 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 2296 . . . . . 6  |-  ( ( (ringLMod `  R )  ^s  V )s  C )  =  ( ( (ringLMod `  R
)  ^s  V )s  C )
6564, 32reslmhm2b 15827 . . . . 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 2371 . 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 27321 . . . 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 27321 . . . 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 5892 . 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 2373 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 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162    i^i cin 3164    C_ wss 3165   {csn 3653    e. cmpt 4093   `'ccnv 4704   ran crn 4706    |` cres 4707   "cima 4708   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   Fincfn 6879   Basecbs 13164   ↾s cress 13165    ^s cpws 13363   0gc0g 13416   Ringcrg 15353   LModclmod 15643   LSubSpclss 15705   LMHom clmhm 15792  ringLModcrglmod 15938   freeLMod cfrlm 27315
This theorem is referenced by:  frlmsslss  27347
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-prds 13364  df-pws 13366  df-0g 13420  df-mnd 14383  df-mhm 14431  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-ghm 14697  df-mgp 15342  df-rng 15356  df-ur 15358  df-subrg 15559  df-lmod 15645  df-lss 15706  df-lmhm 15795  df-sra 15941  df-rgmod 15942  df-dsmm 27301  df-frlm 27317
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