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Theorem frlmsplit2 27222
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 958 . . . . . 6  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  R  e.  Ring )
2 simp2 959 . . . . . 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 2438 . . . . . . 7  |-  ( LSubSp `  ( (ringLMod `  R
)  ^s  U ) )  =  ( LSubSp `  ( (ringLMod `  R )  ^s  U ) )
63, 4, 5frlmlss 27198 . . . . . 6  |-  ( ( R  e.  Ring  /\  U  e.  X )  ->  B  e.  ( LSubSp `  ( (ringLMod `  R )  ^s  U ) ) )
71, 2, 6syl2anc 644 . . . . 5  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  B  e.  ( LSubSp `  ( (ringLMod `  R )  ^s  U ) ) )
8 eqid 2438 . . . . . 6  |-  ( Base `  ( (ringLMod `  R
)  ^s  U ) )  =  ( Base `  (
(ringLMod `  R )  ^s  U
) )
98, 5lssss 16015 . . . . 5  |-  ( B  e.  ( LSubSp `  (
(ringLMod `  R )  ^s  U
) )  ->  B  C_  ( Base `  (
(ringLMod `  R )  ^s  U
) ) )
10 resmpt 5193 . . . . 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 19 . . . 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 2488 . . 3  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  (
( x  e.  (
Base `  ( (ringLMod `  R )  ^s  U ) )  |->  ( x  |`  V ) )  |`  B )  =  F )
14 rlmlmod 16278 . . . . . 6  |-  ( R  e.  Ring  ->  (ringLMod `  R
)  e.  LMod )
15 eqid 2438 . . . . . . 7  |-  ( (ringLMod `  R )  ^s  U )  =  ( (ringLMod `  R
)  ^s  U )
16 eqid 2438 . . . . . . 7  |-  ( (ringLMod `  R )  ^s  V )  =  ( (ringLMod `  R
)  ^s  V )
17 eqid 2438 . . . . . . 7  |-  ( Base `  ( (ringLMod `  R
)  ^s  V ) )  =  ( Base `  (
(ringLMod `  R )  ^s  V
) )
18 eqid 2438 . . . . . . 7  |-  ( x  e.  ( Base `  (
(ringLMod `  R )  ^s  U
) )  |->  ( x  |`  V ) )  =  ( x  e.  (
Base `  ( (ringLMod `  R )  ^s  U ) )  |->  ( x  |`  V ) )
1915, 16, 8, 17, 18pwssplit3 27169 . . . . . 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 1218 . . . . 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 2438 . . . . . 6  |-  ( ( (ringLMod `  R )  ^s  U )s  B )  =  ( ( (ringLMod `  R
)  ^s  U )s  B )
225, 21reslmhm 16130 . . . . 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 644 . . . 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 979 . . . . . 6  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  (ringLMod `  R )  e.  LMod )
25 simp3 960 . . . . . . 7  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  V  C_  U )
262, 25ssexd 4352 . . . . . 6  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  V  e.  _V )
2716pwslmod 16048 . . . . . 6  |-  ( ( (ringLMod `  R )  e.  LMod  /\  V  e.  _V )  ->  ( (ringLMod `  R )  ^s  V )  e.  LMod )
2824, 26, 27syl2anc 644 . . . . 5  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  (
(ringLMod `  R )  ^s  V
)  e.  LMod )
29 frlmsplit2.z . . . . . . 7  |-  Z  =  ( R freeLMod  V )
30 frlmsplit2.c . . . . . . 7  |-  C  =  ( Base `  Z
)
31 eqid 2438 . . . . . . 7  |-  ( LSubSp `  ( (ringLMod `  R
)  ^s  V ) )  =  ( LSubSp `  ( (ringLMod `  R )  ^s  V ) )
3229, 30, 31frlmlss 27198 . . . . . 6  |-  ( ( R  e.  Ring  /\  V  e.  _V )  ->  C  e.  ( LSubSp `  ( (ringLMod `  R )  ^s  V ) ) )
331, 26, 32syl2anc 644 . . . . 5  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  C  e.  ( LSubSp `  ( (ringLMod `  R )  ^s  V ) ) )
3411rneqd 5099 . . . . . 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 ) ) )
35 eqid 2438 . . . . . . . . . . . . 13  |-  ( Base `  R )  =  (
Base `  R )
363, 35, 4frlmbasf 27207 . . . . . . . . . . . 12  |-  ( ( U  e.  X  /\  x  e.  B )  ->  x : U --> ( Base `  R ) )
372, 36sylan 459 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  x : U --> ( Base `  R ) )
38 simpl3 963 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  V  C_  U )
39 fssres 5612 . . . . . . . . . . 11  |-  ( ( x : U --> ( Base `  R )  /\  V  C_  U )  ->  (
x  |`  V ) : V --> ( Base `  R
) )
4037, 38, 39syl2anc 644 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  ( x  |`  V ) : V --> ( Base `  R ) )
41 fvex 5744 . . . . . . . . . . . 12  |-  ( Base `  R )  e.  _V
42 elmapg 7033 . . . . . . . . . . . 12  |-  ( ( ( Base `  R
)  e.  _V  /\  V  e.  _V )  ->  ( ( x  |`  V )  e.  ( ( Base `  R
)  ^m  V )  <->  ( x  |`  V ) : V --> ( Base `  R
) ) )
4341, 26, 42sylancr 646 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  (
( x  |`  V )  e.  ( ( Base `  R )  ^m  V
)  <->  ( x  |`  V ) : V --> ( Base `  R )
) )
4443adantr 453 . . . . . . . . . 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
) ) )
4540, 44mpbird 225 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  ( x  |`  V )  e.  ( ( Base `  R )  ^m  V
) )
46 cnvresima 5361 . . . . . . . . . 10  |-  ( `' ( x  |`  V )
" ( _V  \  { ( 0g `  R ) } ) )  =  ( ( `' x " ( _V 
\  { ( 0g
`  R ) } ) )  i^i  V
)
47 eqid 2438 . . . . . . . . . . . . 13  |-  ( 0g
`  R )  =  ( 0g `  R
)
483, 47, 4frlmbassup 27205 . . . . . . . . . . . 12  |-  ( ( U  e.  X  /\  x  e.  B )  ->  ( `' x "
( _V  \  {
( 0g `  R
) } ) )  e.  Fin )
492, 48sylan 459 . . . . . . . . . . 11  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  ( `' x "
( _V  \  {
( 0g `  R
) } ) )  e.  Fin )
50 inss1 3563 . . . . . . . . . . 11  |-  ( ( `' x " ( _V 
\  { ( 0g
`  R ) } ) )  i^i  V
)  C_  ( `' x " ( _V  \  { ( 0g `  R ) } ) )
51 ssfi 7331 . . . . . . . . . . 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 )
5249, 50, 51sylancl 645 . . . . . . . . . 10  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  ( ( `' x " ( _V  \  {
( 0g `  R
) } ) )  i^i  V )  e. 
Fin )
5346, 52syl5eqel 2522 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  ( `' ( x  |`  V ) " ( _V  \  { ( 0g
`  R ) } ) )  e.  Fin )
5429, 35, 47, 30frlmelbas 27203 . . . . . . . . . . 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 ) ) )
551, 26, 54syl2anc 644 . . . . . . . . . 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 ) ) )
5655adantr 453 . . . . . . . . 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 )
) )
5745, 53, 56mpbir2and 890 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  /\  x  e.  B )  ->  ( x  |`  V )  e.  C )
58 eqid 2438 . . . . . . . 8  |-  ( x  e.  B  |->  ( x  |`  V ) )  =  ( x  e.  B  |->  ( x  |`  V ) )
5957, 58fmptd 5895 . . . . . . 7  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  (
x  e.  B  |->  ( x  |`  V )
) : B --> C )
60 frn 5599 . . . . . . 7  |-  ( ( x  e.  B  |->  ( x  |`  V )
) : B --> C  ->  ran  ( x  e.  B  |->  ( x  |`  V ) )  C_  C )
6159, 60syl 16 . . . . . 6  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  ran  ( x  e.  B  |->  ( x  |`  V ) )  C_  C )
6234, 61eqsstrd 3384 . . . . 5  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  ran  ( ( x  e.  ( Base `  (
(ringLMod `  R )  ^s  U
) )  |->  ( x  |`  V ) )  |`  B )  C_  C
)
63 eqid 2438 . . . . . 6  |-  ( ( (ringLMod `  R )  ^s  V )s  C )  =  ( ( (ringLMod `  R
)  ^s  V )s  C )
6463, 31reslmhm2b 16132 . . . . 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 ) ) ) )
6528, 33, 62, 64syl3anc 1185 . . . 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 ) ) ) )
6623, 65mpbid 203 . . 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 ) ) )
6713, 66eqeltrrd 2513 . 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 ) ) )
683, 4frlmpws 27197 . . . 4  |-  ( ( R  e.  Ring  /\  U  e.  X )  ->  Y  =  ( ( (ringLMod `  R )  ^s  U )s  B ) )
691, 2, 68syl2anc 644 . . 3  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  Y  =  ( ( (ringLMod `  R )  ^s  U )s  B ) )
7029, 30frlmpws 27197 . . . 4  |-  ( ( R  e.  Ring  /\  V  e.  _V )  ->  Z  =  ( ( (ringLMod `  R )  ^s  V )s  C ) )
711, 26, 70syl2anc 644 . . 3  |-  ( ( R  e.  Ring  /\  U  e.  X  /\  V  C_  U )  ->  Z  =  ( ( (ringLMod `  R )  ^s  V )s  C ) )
7269, 71oveq12d 6101 . 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 ) ) )
7367, 72eleqtrrd 2515 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   _Vcvv 2958    \ cdif 3319    i^i cin 3321    C_ wss 3322   {csn 3816    e. cmpt 4268   `'ccnv 4879   ran crn 4881    |` cres 4882   "cima 4883   -->wf 5452   ` cfv 5456  (class class class)co 6083    ^m cmap 7020   Fincfn 7111   Basecbs 13471   ↾s cress 13472    ^s cpws 13672   0gc0g 13725   Ringcrg 15662   LModclmod 15952   LSubSpclss 16010   LMHom clmhm 16097  ringLModcrglmod 16243   freeLMod cfrlm 27191
This theorem is referenced by:  frlmsslss  27223
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-of 6307  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-oadd 6730  df-er 6907  df-map 7022  df-ixp 7066  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-2 10060  df-3 10061  df-4 10062  df-5 10063  df-6 10064  df-7 10065  df-8 10066  df-9 10067  df-10 10068  df-n0 10224  df-z 10285  df-dec 10385  df-uz 10491  df-fz 11046  df-struct 13473  df-ndx 13474  df-slot 13475  df-base 13476  df-sets 13477  df-ress 13478  df-plusg 13544  df-mulr 13545  df-sca 13547  df-vsca 13548  df-tset 13550  df-ple 13551  df-ds 13553  df-hom 13555  df-cco 13556  df-prds 13673  df-pws 13675  df-0g 13729  df-mnd 14692  df-mhm 14740  df-submnd 14741  df-grp 14814  df-minusg 14815  df-sbg 14816  df-subg 14943  df-ghm 15006  df-mgp 15651  df-rng 15665  df-ur 15667  df-subrg 15868  df-lmod 15954  df-lss 16011  df-lmhm 16100  df-sra 16246  df-rgmod 16247  df-dsmm 27177  df-frlm 27193
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