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Theorem pj1lmhm 16164
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 2435 . . 3  |-  ( +g  `  W )  =  ( +g  `  W )
2 pj1lmhm.s . . 3  |-  .(+)  =  (
LSSum `  W )
3 pj1lmhm.z . . 3  |-  .0.  =  ( 0g `  W )
4 eqid 2435 . . 3  |-  (Cntz `  W )  =  (Cntz `  W )
5 pj1lmhm.1 . . . . 5  |-  ( ph  ->  W  e.  LMod )
6 pj1lmhm.l . . . . . 6  |-  L  =  ( LSubSp `  W )
76lsssssubg 16026 . . . . 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 3341 . . 3  |-  ( ph  ->  T  e.  (SubGrp `  W ) )
11 pj1lmhm.3 . . . 4  |-  ( ph  ->  U  e.  L )
128, 11sseldd 3341 . . 3  |-  ( ph  ->  U  e.  (SubGrp `  W ) )
13 pj1lmhm.4 . . 3  |-  ( ph  ->  ( T  i^i  U
)  =  {  .0.  } )
14 lmodabl 15983 . . . . 5  |-  ( W  e.  LMod  ->  W  e. 
Abel )
155, 14syl 16 . . . 4  |-  ( ph  ->  W  e.  Abel )
164, 15, 10, 12ablcntzd 15464 . . 3  |-  ( ph  ->  T  C_  ( (Cntz `  W ) `  U
) )
17 pj1lmhm.p . . 3  |-  P  =  ( proj 1 `  W )
181, 2, 3, 4, 10, 12, 13, 16, 17pj1ghm 15327 . 2  |-  ( ph  ->  ( T P U )  e.  ( ( Ws  ( T  .(+)  U ) )  GrpHom  W ) )
19 eqid 2435 . . 3  |-  (Scalar `  W )  =  (Scalar `  W )
2019a1i 11 . 2  |-  ( ph  ->  (Scalar `  W )  =  (Scalar `  W )
)
211, 2, 3, 4, 10, 12, 13, 16, 17pj1id 15323 . . . . . . . . 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 6089 . . . . . . 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 2435 . . . . . . . . . . 11  |-  ( Base `  W )  =  (
Base `  W )
2827, 6lssss 16005 . . . . . . . . . 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 15321 . . . . . . . . . 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 5863 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  ( ( T P U ) `  y )  e.  T
)
3729, 36sseldd 3341 . . . . . . . 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 16005 . . . . . . . . . 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 15322 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  ( U P T ) : ( T  .(+)  U ) --> U )
4241, 35ffvelrnd 5863 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  ( ( U P T ) `  y )  e.  U
)
4340, 42sseldd 3341 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  ( ( U P T ) `  y )  e.  (
Base `  W )
)
44 eqid 2435 . . . . . . . . 9  |-  ( .s
`  W )  =  ( .s `  W
)
45 eqid 2435 . . . . . . . . 9  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
4627, 1, 19, 44, 45lmodvsdi 15965 . . . . . . . 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 2467 . . . . . 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 16147 . . . . . . . . . 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 16023 . . . . . . . 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 16023 . . . . . . . 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 16023 . . . . . . . 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 15324 . . . . . 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 2790 . . 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 3341 . . . . . 6  |-  ( ph  ->  ( T  .(+)  U )  e.  (SubGrp `  W
) )
63 eqid 2435 . . . . . . 7  |-  ( Ws  ( T  .(+)  U )
)  =  ( Ws  ( T  .(+)  U )
)
6463subgbas 14940 . . . . . 6  |-  ( ( T  .(+)  U )  e.  (SubGrp `  W )  ->  ( T  .(+)  U )  =  ( Base `  ( Ws  ( T  .(+)  U ) ) ) )
6562, 64syl 16 . . . . 5  |-  ( ph  ->  ( T  .(+)  U )  =  ( Base `  ( Ws  ( T  .(+)  U ) ) ) )
6665raleqdv 2902 . . . 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 2717 . . 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 16028 . . . 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 6098 . . . . 5  |-  ( T 
.(+)  U )  e.  _V
7263, 19resssca 13596 . . . . 5  |-  ( ( T  .(+)  U )  e.  _V  ->  (Scalar `  W
)  =  (Scalar `  ( Ws  ( T  .(+)  U ) ) ) )
7371, 72ax-mp 8 . . . 4  |-  (Scalar `  W )  =  (Scalar `  ( Ws  ( T  .(+)  U ) ) )
74 eqid 2435 . . . 4  |-  ( Base `  ( Ws  ( T  .(+)  U ) ) )  =  ( Base `  ( Ws  ( T  .(+)  U ) ) )
7563, 44ressvsca 13597 . . . . 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 16096 . . 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 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948    i^i cin 3311    C_ wss 3312   {csn 3806   ` cfv 5446  (class class class)co 6073   Basecbs 13461   ↾s cress 13462   +g cplusg 13521  Scalarcsca 13524   .scvsca 13525   0gc0g 13715  SubGrpcsubg 14930    GrpHom cghm 14995  Cntzccntz 15106   LSSumclsm 15260   proj
1cpj1 15261   Abelcabel 15405   LModclmod 15942   LSubSpclss 16000   LMHom clmhm 16087
This theorem is referenced by:  pj1lmhm2  16165  pjff  16931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-sca 13537  df-vsca 13538  df-0g 13719  df-mnd 14682  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-subg 14933  df-ghm 14996  df-cntz 15108  df-lsm 15262  df-pj1 15263  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-ur 15657  df-lmod 15944  df-lss 16001  df-lmhm 16090
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