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Theorem pj1lmhm 15869
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 2296 . . 3  |-  ( +g  `  W )  =  ( +g  `  W )
2 pj1lmhm.s . . 3  |-  .(+)  =  (
LSSum `  W )
3 pj1lmhm.z . . 3  |-  .0.  =  ( 0g `  W )
4 eqid 2296 . . 3  |-  (Cntz `  W )  =  (Cntz `  W )
5 pj1lmhm.1 . . . . 5  |-  ( ph  ->  W  e.  LMod )
6 pj1lmhm.l . . . . . 6  |-  L  =  ( LSubSp `  W )
76lsssssubg 15731 . . . . 5  |-  ( W  e.  LMod  ->  L  C_  (SubGrp `  W ) )
85, 7syl 15 . . . 4  |-  ( ph  ->  L  C_  (SubGrp `  W
) )
9 pj1lmhm.2 . . . 4  |-  ( ph  ->  T  e.  L )
108, 9sseldd 3194 . . 3  |-  ( ph  ->  T  e.  (SubGrp `  W ) )
11 pj1lmhm.3 . . . 4  |-  ( ph  ->  U  e.  L )
128, 11sseldd 3194 . . 3  |-  ( ph  ->  U  e.  (SubGrp `  W ) )
13 pj1lmhm.4 . . 3  |-  ( ph  ->  ( T  i^i  U
)  =  {  .0.  } )
14 lmodabl 15688 . . . . 5  |-  ( W  e.  LMod  ->  W  e. 
Abel )
155, 14syl 15 . . . 4  |-  ( ph  ->  W  e.  Abel )
164, 15, 10, 12ablcntzd 15165 . . 3  |-  ( ph  ->  T  C_  ( (Cntz `  W ) `  U
) )
17 pj1lmhm.p . . 3  |-  P  =  ( proj 1 `  W )
181, 2, 3, 4, 10, 12, 13, 16, 17pj1ghm 15028 . 2  |-  ( ph  ->  ( T P U )  e.  ( ( Ws  ( T  .(+)  U ) )  GrpHom  W ) )
19 eqid 2296 . . 3  |-  (Scalar `  W )  =  (Scalar `  W )
2019a1i 10 . 2  |-  ( ph  ->  (Scalar `  W )  =  (Scalar `  W )
)
211, 2, 3, 4, 10, 12, 13, 16, 17pj1id 15024 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( T  .(+)  U ) )  ->  y  =  ( ( ( T P U ) `  y ) ( +g  `  W ) ( ( U P T ) `
 y ) ) )
2221adantrl 696 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  y  =  ( ( ( T P U ) `  y
) ( +g  `  W
) ( ( U P T ) `  y ) ) )
2322oveq2d 5890 . . . . . . 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 451 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  W  e.  LMod )
25 simprl 732 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  x  e.  (
Base `  (Scalar `  W
) ) )
269adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  T  e.  L
)
27 eqid 2296 . . . . . . . . . . 11  |-  ( Base `  W )  =  (
Base `  W )
2827, 6lssss 15710 . . . . . . . . . 10  |-  ( T  e.  L  ->  T  C_  ( Base `  W
) )
2926, 28syl 15 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  T  C_  ( Base `  W ) )
3010adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  T  e.  (SubGrp `  W ) )
3112adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  U  e.  (SubGrp `  W ) )
3213adantr 451 . . . . . . . . . . 11  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  ( T  i^i  U )  =  {  .0.  } )
3316adantr 451 . . . . . . . . . . 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 15022 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  ( T P U ) : ( T  .(+)  U ) --> T )
35 simprr 733 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  y  e.  ( T  .(+)  U )
)
36 ffvelrn 5679 . . . . . . . . . 10  |-  ( ( ( T P U ) : ( T 
.(+)  U ) --> T  /\  y  e.  ( T  .(+) 
U ) )  -> 
( ( T P U ) `  y
)  e.  T )
3734, 35, 36syl2anc 642 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  ( ( T P U ) `  y )  e.  T
)
3829, 37sseldd 3194 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  ( ( T P U ) `  y )  e.  (
Base `  W )
)
3911adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  U  e.  L
)
4027, 6lssss 15710 . . . . . . . . . 10  |-  ( U  e.  L  ->  U  C_  ( Base `  W
) )
4139, 40syl 15 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  U  C_  ( Base `  W ) )
421, 2, 3, 4, 30, 31, 32, 33, 17pj2f 15023 . . . . . . . . . 10  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  ( U P T ) : ( T  .(+)  U ) --> U )
43 ffvelrn 5679 . . . . . . . . . 10  |-  ( ( ( U P T ) : ( T 
.(+)  U ) --> U  /\  y  e.  ( T  .(+) 
U ) )  -> 
( ( U P T ) `  y
)  e.  U )
4442, 35, 43syl2anc 642 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  ( ( U P T ) `  y )  e.  U
)
4541, 44sseldd 3194 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  ( ( U P T ) `  y )  e.  (
Base `  W )
)
46 eqid 2296 . . . . . . . . 9  |-  ( .s
`  W )  =  ( .s `  W
)
47 eqid 2296 . . . . . . . . 9  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
4827, 1, 19, 46, 47lmodvsdi 15666 . . . . . . . 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
) ) ) )
4924, 25, 38, 45, 48syl13anc 1184 . . . . . . 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
) ) ) )
5023, 49eqtrd 2328 . . . . . 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
) ) ) )
516, 2lsmcl 15852 . . . . . . . . . 10  |-  ( ( W  e.  LMod  /\  T  e.  L  /\  U  e.  L )  ->  ( T  .(+)  U )  e.  L )
525, 9, 11, 51syl3anc 1182 . . . . . . . . 9  |-  ( ph  ->  ( T  .(+)  U )  e.  L )
5352adantr 451 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  ( T  .(+)  U )  e.  L )
5419, 46, 47, 6lssvscl 15728 . . . . . . . 8  |-  ( ( ( W  e.  LMod  /\  ( T  .(+)  U )  e.  L )  /\  ( x  e.  ( Base `  (Scalar `  W
) )  /\  y  e.  ( T  .(+)  U ) ) )  ->  (
x ( .s `  W ) y )  e.  ( T  .(+)  U ) )
5524, 53, 25, 35, 54syl22anc 1183 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  ( x ( .s `  W ) y )  e.  ( T  .(+)  U )
)
5619, 46, 47, 6lssvscl 15728 . . . . . . . 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 )
5724, 26, 25, 37, 56syl22anc 1183 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  ( x ( .s `  W ) ( ( T P U ) `  y
) )  e.  T
)
5819, 46, 47, 6lssvscl 15728 . . . . . . . 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 )
5924, 39, 25, 44, 58syl22anc 1183 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  ( Base `  (Scalar `  W ) )  /\  y  e.  ( T  .(+) 
U ) ) )  ->  ( x ( .s `  W ) ( ( U P T ) `  y
) )  e.  U
)
601, 2, 3, 4, 30, 31, 32, 33, 17, 55, 57, 59pj1eq 15025 . . . . . 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 ) ) ) ) )
6150, 60mpbid 201 . . . . 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 ) ) ) )
6261simpld 445 . . . 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 ) ) )
6362ralrimivva 2648 . . 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 ) ) )
648, 52sseldd 3194 . . . . . 6  |-  ( ph  ->  ( T  .(+)  U )  e.  (SubGrp `  W
) )
65 eqid 2296 . . . . . . 7  |-  ( Ws  ( T  .(+)  U )
)  =  ( Ws  ( T  .(+)  U )
)
6665subgbas 14641 . . . . . 6  |-  ( ( T  .(+)  U )  e.  (SubGrp `  W )  ->  ( T  .(+)  U )  =  ( Base `  ( Ws  ( T  .(+)  U ) ) ) )
6764, 66syl 15 . . . . 5  |-  ( ph  ->  ( T  .(+)  U )  =  ( Base `  ( Ws  ( T  .(+)  U ) ) ) )
6867raleqdv 2755 . . . 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 ) ) ) )
6968ralbidv 2576 . . 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 ) ) ) )
7063, 69mpbid 201 . 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 ) ) )
7165, 6lsslmod 15733 . . . 4  |-  ( ( W  e.  LMod  /\  ( T  .(+)  U )  e.  L )  ->  ( Ws  ( T  .(+)  U ) )  e.  LMod )
725, 52, 71syl2anc 642 . . 3  |-  ( ph  ->  ( Ws  ( T  .(+)  U ) )  e.  LMod )
73 ovex 5899 . . . . 5  |-  ( T 
.(+)  U )  e.  _V
7465, 19resssca 13299 . . . . 5  |-  ( ( T  .(+)  U )  e.  _V  ->  (Scalar `  W
)  =  (Scalar `  ( Ws  ( T  .(+)  U ) ) ) )
7573, 74ax-mp 8 . . . 4  |-  (Scalar `  W )  =  (Scalar `  ( Ws  ( T  .(+)  U ) ) )
76 eqid 2296 . . . 4  |-  ( Base `  ( Ws  ( T  .(+)  U ) ) )  =  ( Base `  ( Ws  ( T  .(+)  U ) ) )
7765, 46ressvsca 13300 . . . . 5  |-  ( ( T  .(+)  U )  e.  _V  ->  ( .s `  W )  =  ( .s `  ( Ws  ( T  .(+)  U )
) ) )
7873, 77ax-mp 8 . . . 4  |-  ( .s
`  W )  =  ( .s `  ( Ws  ( T  .(+)  U ) ) )
7975, 19, 47, 76, 78, 46islmhm3 15801 . . 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 ) ) ) ) )
8072, 5, 79syl2anc 642 . 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 ) ) ) ) )
8118, 20, 70, 80mpbir3and 1135 1  |-  ( ph  ->  ( T P U )  e.  ( ( Ws  ( T  .(+)  U ) ) LMHom  W ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    i^i cin 3164    C_ wss 3165   {csn 3653   -->wf 5267   ` cfv 5271  (class class class)co 5874   Basecbs 13164   ↾s cress 13165   +g cplusg 13224  Scalarcsca 13227   .scvsca 13228   0gc0g 13416  SubGrpcsubg 14631    GrpHom cghm 14696  Cntzccntz 14807   LSSumclsm 14961   proj
1cpj1 14962   Abelcabel 15106   LModclmod 15643   LSubSpclss 15705   LMHom clmhm 15792
This theorem is referenced by:  pj1lmhm2  15870  pjff  16628
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-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-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  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-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-sca 13240  df-vsca 13241  df-0g 13420  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-ghm 14697  df-cntz 14809  df-lsm 14963  df-pj1 14964  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-lmod 15645  df-lss 15706  df-lmhm 15795
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