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Theorem pj1lmhm 16099
Description: The left projection function is a linear operator. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
pj1lmhm.l  |-  L  =  ( LSubSp `  W )
pj1lmhm.s  |-  .(+)  =  (
LSSum `  W )
pj1lmhm.z  |-  .0.  =  ( 0g `  W )
pj1lmhm.p  |-  P  =  ( proj 1 `  W )
pj1lmhm.1  |-  ( ph  ->  W  e.  LMod )
pj1lmhm.2  |-  ( ph  ->  T  e.  L )
pj1lmhm.3  |-  ( ph  ->  U  e.  L )
pj1lmhm.4  |-  ( ph  ->  ( T  i^i  U
)  =  {  .0.  } )
Assertion
Ref Expression
pj1lmhm  |-  ( ph  ->  ( T P U )  e.  ( ( Ws  ( T  .(+)  U ) ) LMHom  W ) )

Proof of Theorem pj1lmhm
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2387 . . 3  |-  ( +g  `  W )  =  ( +g  `  W )
2 pj1lmhm.s . . 3  |-  .(+)  =  (
LSSum `  W )
3 pj1lmhm.z . . 3  |-  .0.  =  ( 0g `  W )
4 eqid 2387 . . 3  |-  (Cntz `  W )  =  (Cntz `  W )
5 pj1lmhm.1 . . . . 5  |-  ( ph  ->  W  e.  LMod )
6 pj1lmhm.l . . . . . 6  |-  L  =  ( LSubSp `  W )
76lsssssubg 15961 . . . . 5  |-  ( W  e.  LMod  ->  L  C_  (SubGrp `  W ) )
85, 7syl 16 . . . 4  |-  ( ph  ->  L  C_  (SubGrp `  W
) )
9 pj1lmhm.2 . . . 4  |-  ( ph  ->  T  e.  L )
108, 9sseldd 3292 . . 3  |-  ( ph  ->  T  e.  (SubGrp `  W ) )
11 pj1lmhm.3 . . . 4  |-  ( ph  ->  U  e.  L )
128, 11sseldd 3292 . . 3  |-  ( ph  ->  U  e.  (SubGrp `  W ) )
13 pj1lmhm.4 . . 3  |-  ( ph  ->  ( T  i^i  U
)  =  {  .0.  } )
14 lmodabl 15918 . . . . 5  |-  ( W  e.  LMod  ->  W  e. 
Abel )
155, 14syl 16 . . . 4  |-  ( ph  ->  W  e.  Abel )
164, 15, 10, 12ablcntzd 15399 . . 3  |-  ( ph  ->  T  C_  ( (Cntz `  W ) `  U
) )
17 pj1lmhm.p . . 3  |-  P  =  ( proj 1 `  W )
181, 2, 3, 4, 10, 12, 13, 16, 17pj1ghm 15262 . 2  |-  ( ph  ->  ( T P U )  e.  ( ( Ws  ( T  .(+)  U ) )  GrpHom  W ) )
19 eqid 2387 . . 3  |-  (Scalar `  W )  =  (Scalar `  W )
2019a1i 11 . 2  |-  ( ph  ->  (Scalar `  W )  =  (Scalar `  W )
)
211, 2, 3, 4, 10, 12, 13, 16, 17pj1id 15258 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( T  .(+)  U ) )  ->  y  =  ( ( ( T P U ) `  y ) ( +g  `  W ) ( ( U P T ) `
 y ) ) )
2221adantrl 697 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  y  =  ( ( ( T P U ) `  y
) ( +g  `  W
) ( ( U P T ) `  y ) ) )
2322oveq2d 6036 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  ( x ( .s `  W ) y )  =  ( x ( .s `  W ) ( ( ( T P U ) `  y ) ( +g  `  W
) ( ( U P T ) `  y ) ) ) )
245adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  W  e.  LMod )
25 simprl 733 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  x  e.  (
Base `  (Scalar `  W
) ) )
269adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  T  e.  L
)
27 eqid 2387 . . . . . . . . . . 11  |-  ( Base `  W )  =  (
Base `  W )
2827, 6lssss 15940 . . . . . . . . . 10  |-  ( T  e.  L  ->  T  C_  ( Base `  W
) )
2926, 28syl 16 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  T  C_  ( Base `  W ) )
3010adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  T  e.  (SubGrp `  W ) )
3112adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  U  e.  (SubGrp `  W ) )
3213adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  ( T  i^i  U )  =  {  .0.  } )
3316adantr 452 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  T  C_  (
(Cntz `  W ) `  U ) )
341, 2, 3, 4, 30, 31, 32, 33, 17pj1f 15256 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  ( T P U ) : ( T  .(+)  U ) --> T )
35 simprr 734 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  y  e.  ( T  .(+)  U )
)
3634, 35ffvelrnd 5810 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  ( ( T P U ) `  y )  e.  T
)
3729, 36sseldd 3292 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  ( ( T P U ) `  y )  e.  (
Base `  W )
)
3811adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  U  e.  L
)
3927, 6lssss 15940 . . . . . . . . . 10  |-  ( U  e.  L  ->  U  C_  ( Base `  W
) )
4038, 39syl 16 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  U  C_  ( Base `  W ) )
411, 2, 3, 4, 30, 31, 32, 33, 17pj2f 15257 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  ( U P T ) : ( T  .(+)  U ) --> U )
4241, 35ffvelrnd 5810 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  ( ( U P T ) `  y )  e.  U
)
4340, 42sseldd 3292 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  ( ( U P T ) `  y )  e.  (
Base `  W )
)
44 eqid 2387 . . . . . . . . 9  |-  ( .s
`  W )  =  ( .s `  W
)
45 eqid 2387 . . . . . . . . 9  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
4627, 1, 19, 44, 45lmodvsdi 15900 . . . . . . . 8  |-  ( ( W  e.  LMod  /\  (
x  e.  ( Base `  (Scalar `  W )
)  /\  ( ( T P U ) `  y )  e.  (
Base `  W )  /\  ( ( U P T ) `  y
)  e.  ( Base `  W ) ) )  ->  ( x ( .s `  W ) ( ( ( T P U ) `  y ) ( +g  `  W ) ( ( U P T ) `
 y ) ) )  =  ( ( x ( .s `  W ) ( ( T P U ) `
 y ) ) ( +g  `  W
) ( x ( .s `  W ) ( ( U P T ) `  y
) ) ) )
4724, 25, 37, 43, 46syl13anc 1186 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  ( x ( .s `  W ) ( ( ( T P U ) `  y ) ( +g  `  W ) ( ( U P T ) `
 y ) ) )  =  ( ( x ( .s `  W ) ( ( T P U ) `
 y ) ) ( +g  `  W
) ( x ( .s `  W ) ( ( U P T ) `  y
) ) ) )
4823, 47eqtrd 2419 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  ( x ( .s `  W ) y )  =  ( ( x ( .s
`  W ) ( ( T P U ) `  y ) ) ( +g  `  W
) ( x ( .s `  W ) ( ( U P T ) `  y
) ) ) )
496, 2lsmcl 16082 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  T  e.  L  /\  U  e.  L )  ->  ( T  .(+)  U )  e.  L )
505, 9, 11, 49syl3anc 1184 . . . . . . . . 9  |-  ( ph  ->  ( T  .(+)  U )  e.  L )
5150adantr 452 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  ( T  .(+)  U )  e.  L )
5219, 44, 45, 6lssvscl 15958 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  ( T  .(+)  U )  e.  L )  /\  ( x  e.  ( Base `  (Scalar `  W
) )  /\  y  e.  ( T  .(+)  U ) ) )  ->  (
x ( .s `  W ) y )  e.  ( T  .(+)  U ) )
5324, 51, 25, 35, 52syl22anc 1185 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  ( x ( .s `  W ) y )  e.  ( T  .(+)  U )
)
5419, 44, 45, 6lssvscl 15958 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  T  e.  L )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  ( ( T P U ) `  y
)  e.  T ) )  ->  ( x
( .s `  W
) ( ( T P U ) `  y ) )  e.  T )
5524, 26, 25, 36, 54syl22anc 1185 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  ( x ( .s `  W ) ( ( T P U ) `  y
) )  e.  T
)
5619, 44, 45, 6lssvscl 15958 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  U  e.  L )  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  ( ( U P T ) `  y
)  e.  U ) )  ->  ( x
( .s `  W
) ( ( U P T ) `  y ) )  e.  U )
5724, 38, 25, 42, 56syl22anc 1185 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  ( x ( .s `  W ) ( ( U P T ) `  y
) )  e.  U
)
581, 2, 3, 4, 30, 31, 32, 33, 17, 53, 55, 57pj1eq 15259 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  ( ( x ( .s `  W
) y )  =  ( ( x ( .s `  W ) ( ( T P U ) `  y
) ) ( +g  `  W ) ( x ( .s `  W
) ( ( U P T ) `  y ) ) )  <-> 
( ( ( T P U ) `  ( x ( .s
`  W ) y ) )  =  ( x ( .s `  W ) ( ( T P U ) `
 y ) )  /\  ( ( U P T ) `  ( x ( .s
`  W ) y ) )  =  ( x ( .s `  W ) ( ( U P T ) `
 y ) ) ) ) )
5948, 58mpbid 202 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  ( ( ( T P U ) `
 ( x ( .s `  W ) y ) )  =  ( x ( .s
`  W ) ( ( T P U ) `  y ) )  /\  ( ( U P T ) `
 ( x ( .s `  W ) y ) )  =  ( x ( .s
`  W ) ( ( U P T ) `  y ) ) ) )
6059simpld 446 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  ( ( T P U ) `  ( x ( .s
`  W ) y ) )  =  ( x ( .s `  W ) ( ( T P U ) `
 y ) ) )
6160ralrimivva 2741 . . 3  |-  ( ph  ->  A. x  e.  (
Base `  (Scalar `  W
) ) A. y  e.  ( T  .(+)  U ) ( ( T P U ) `  (
x ( .s `  W ) y ) )  =  ( x ( .s `  W
) ( ( T P U ) `  y ) ) )
628, 50sseldd 3292 . . . . . 6  |-  ( ph  ->  ( T  .(+)  U )  e.  (SubGrp `  W
) )
63 eqid 2387 . . . . . . 7  |-  ( Ws  ( T  .(+)  U )
)  =  ( Ws  ( T  .(+)  U )
)
6463subgbas 14875 . . . . . 6  |-  ( ( T  .(+)  U )  e.  (SubGrp `  W )  ->  ( T  .(+)  U )  =  ( Base `  ( Ws  ( T  .(+)  U ) ) ) )
6562, 64syl 16 . . . . 5  |-  ( ph  ->  ( T  .(+)  U )  =  ( Base `  ( Ws  ( T  .(+)  U ) ) ) )
6665raleqdv 2853 . . . 4  |-  ( ph  ->  ( A. y  e.  ( T  .(+)  U ) ( ( T P U ) `  (
x ( .s `  W ) y ) )  =  ( x ( .s `  W
) ( ( T P U ) `  y ) )  <->  A. y  e.  ( Base `  ( Ws  ( T  .(+)  U ) ) ) ( ( T P U ) `
 ( x ( .s `  W ) y ) )  =  ( x ( .s
`  W ) ( ( T P U ) `  y ) ) ) )
6766ralbidv 2669 . . 3  |-  ( ph  ->  ( A. x  e.  ( Base `  (Scalar `  W ) ) A. y  e.  ( T  .(+) 
U ) ( ( T P U ) `
 ( x ( .s `  W ) y ) )  =  ( x ( .s
`  W ) ( ( T P U ) `  y ) )  <->  A. x  e.  (
Base `  (Scalar `  W
) ) A. y  e.  ( Base `  ( Ws  ( T  .(+)  U ) ) ) ( ( T P U ) `
 ( x ( .s `  W ) y ) )  =  ( x ( .s
`  W ) ( ( T P U ) `  y ) ) ) )
6861, 67mpbid 202 . 2  |-  ( ph  ->  A. x  e.  (
Base `  (Scalar `  W
) ) A. y  e.  ( Base `  ( Ws  ( T  .(+)  U ) ) ) ( ( T P U ) `
 ( x ( .s `  W ) y ) )  =  ( x ( .s
`  W ) ( ( T P U ) `  y ) ) )
6963, 6lsslmod 15963 . . . 4  |-  ( ( W  e.  LMod  /\  ( T  .(+)  U )  e.  L )  ->  ( Ws  ( T  .(+)  U ) )  e.  LMod )
705, 50, 69syl2anc 643 . . 3  |-  ( ph  ->  ( Ws  ( T  .(+)  U ) )  e.  LMod )
71 ovex 6045 . . . . 5  |-  ( T 
.(+)  U )  e.  _V
7263, 19resssca 13531 . . . . 5  |-  ( ( T  .(+)  U )  e.  _V  ->  (Scalar `  W
)  =  (Scalar `  ( Ws  ( T  .(+)  U ) ) ) )
7371, 72ax-mp 8 . . . 4  |-  (Scalar `  W )  =  (Scalar `  ( Ws  ( T  .(+)  U ) ) )
74 eqid 2387 . . . 4  |-  ( Base `  ( Ws  ( T  .(+)  U ) ) )  =  ( Base `  ( Ws  ( T  .(+)  U ) ) )
7563, 44ressvsca 13532 . . . . 5  |-  ( ( T  .(+)  U )  e.  _V  ->  ( .s `  W )  =  ( .s `  ( Ws  ( T  .(+)  U )
) ) )
7671, 75ax-mp 8 . . . 4  |-  ( .s
`  W )  =  ( .s `  ( Ws  ( T  .(+)  U ) ) )
7773, 19, 45, 74, 76, 44islmhm3 16031 . . 3  |-  ( ( ( Ws  ( T  .(+)  U ) )  e.  LMod  /\  W  e.  LMod )  ->  ( ( T P U )  e.  ( ( Ws  ( T  .(+)  U ) ) LMHom  W )  <-> 
( ( T P U )  e.  ( ( Ws  ( T  .(+)  U ) )  GrpHom  W )  /\  (Scalar `  W
)  =  (Scalar `  W )  /\  A. x  e.  ( Base `  (Scalar `  W )
) A. y  e.  ( Base `  ( Ws  ( T  .(+)  U ) ) ) ( ( T P U ) `
 ( x ( .s `  W ) y ) )  =  ( x ( .s
`  W ) ( ( T P U ) `  y ) ) ) ) )
7870, 5, 77syl2anc 643 . 2  |-  ( ph  ->  ( ( T P U )  e.  ( ( Ws  ( T  .(+)  U ) ) LMHom  W )  <-> 
( ( T P U )  e.  ( ( Ws  ( T  .(+)  U ) )  GrpHom  W )  /\  (Scalar `  W
)  =  (Scalar `  W )  /\  A. x  e.  ( Base `  (Scalar `  W )
) A. y  e.  ( Base `  ( Ws  ( T  .(+)  U ) ) ) ( ( T P U ) `
 ( x ( .s `  W ) y ) )  =  ( x ( .s
`  W ) ( ( T P U ) `  y ) ) ) ) )
7918, 20, 68, 78mpbir3and 1137 1  |-  ( ph  ->  ( T P U )  e.  ( ( Ws  ( T  .(+)  U ) ) LMHom  W ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2649   _Vcvv 2899    i^i cin 3262    C_ wss 3263   {csn 3757   ` cfv 5394  (class class class)co 6020   Basecbs 13396   ↾s cress 13397   +g cplusg 13456  Scalarcsca 13459   .scvsca 13460   0gc0g 13650  SubGrpcsubg 14865    GrpHom cghm 14930  Cntzccntz 15041   LSSumclsm 15195   proj
1cpj1 15196   Abelcabel 15340   LModclmod 15877   LSubSpclss 15935   LMHom clmhm 16022
This theorem is referenced by:  pj1lmhm2  16100  pjff  16862
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-nn 9933  df-2 9990  df-3 9991  df-4 9992  df-5 9993  df-6 9994  df-ndx 13399  df-slot 13400  df-base 13401  df-sets 13402  df-ress 13403  df-plusg 13469  df-sca 13472  df-vsca 13473  df-0g 13654  df-mnd 14617  df-submnd 14666  df-grp 14739  df-minusg 14740  df-sbg 14741  df-subg 14868  df-ghm 14931  df-cntz 15043  df-lsm 15197  df-pj1 15198  df-cmn 15341  df-abl 15342  df-mgp 15576  df-rng 15590  df-ur 15592  df-lmod 15879  df-lss 15936  df-lmhm 16025
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