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Theorem mpllsslem 16491
Description: If  A is an ideal of subsets (a nonempty collection closed under subset and binary union) of the set  D of finite bags (the primary applications being  A  =  Fin and  A  =  ~P B for some  B), then the set of all power series whose coefficient functions are supported on an element of  A is a linear subspace of the set of all power series. (Contributed by Mario Carneiro, 12-Jan-2015.)
Hypotheses
Ref Expression
mplsubglem.s  |-  S  =  ( I mPwSer  R )
mplsubglem.b  |-  B  =  ( Base `  S
)
mplsubglem.z  |-  .0.  =  ( 0g `  R )
mplsubglem.d  |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
mplsubglem.i  |-  ( ph  ->  I  e.  W )
mplsubglem.0  |-  ( ph  -> 
(/)  e.  A )
mplsubglem.a  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( x  u.  y
)  e.  A )
mplsubglem.y  |-  ( (
ph  /\  ( x  e.  A  /\  y  C_  x ) )  -> 
y  e.  A )
mplsubglem.u  |-  ( ph  ->  U  =  { g  e.  B  |  ( `' g " ( _V  \  {  .0.  }
) )  e.  A } )
mpllsslem.r  |-  ( ph  ->  R  e.  Ring )
Assertion
Ref Expression
mpllsslem  |-  ( ph  ->  U  e.  ( LSubSp `  S ) )
Distinct variable groups:    f, g, x, y,  .0.    A, f, g, x, y    B, f, g    D, g    f, I    ph, x, y    S, f, g, y
Allowed substitution hints:    ph( f, g)    B( x, y)    D( x, y, f)    R( x, y, f, g)    S( x)    U( x, y, f, g)    I( x, y, g)    W( x, y, f, g)

Proof of Theorem mpllsslem
Dummy variables  k  u  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mplsubglem.s . . 3  |-  S  =  ( I mPwSer  R )
2 mplsubglem.i . . 3  |-  ( ph  ->  I  e.  W )
3 mpllsslem.r . . 3  |-  ( ph  ->  R  e.  Ring )
41, 2, 3psrsca 16445 . 2  |-  ( ph  ->  R  =  (Scalar `  S ) )
5 eqidd 2436 . 2  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  R ) )
6 mplsubglem.b . . 3  |-  B  =  ( Base `  S
)
76a1i 11 . 2  |-  ( ph  ->  B  =  ( Base `  S ) )
8 eqidd 2436 . 2  |-  ( ph  ->  ( +g  `  S
)  =  ( +g  `  S ) )
9 eqidd 2436 . 2  |-  ( ph  ->  ( .s `  S
)  =  ( .s
`  S ) )
10 eqidd 2436 . 2  |-  ( ph  ->  ( LSubSp `  S )  =  ( LSubSp `  S
) )
11 mplsubglem.z . . . 4  |-  .0.  =  ( 0g `  R )
12 mplsubglem.d . . . 4  |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
13 mplsubglem.0 . . . 4  |-  ( ph  -> 
(/)  e.  A )
14 mplsubglem.a . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( x  u.  y
)  e.  A )
15 mplsubglem.y . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  y  C_  x ) )  -> 
y  e.  A )
16 mplsubglem.u . . . 4  |-  ( ph  ->  U  =  { g  e.  B  |  ( `' g " ( _V  \  {  .0.  }
) )  e.  A } )
17 rnggrp 15661 . . . . 5  |-  ( R  e.  Ring  ->  R  e. 
Grp )
183, 17syl 16 . . . 4  |-  ( ph  ->  R  e.  Grp )
191, 6, 11, 12, 2, 13, 14, 15, 16, 18mplsubglem 16490 . . 3  |-  ( ph  ->  U  e.  (SubGrp `  S ) )
206subgss 14937 . . 3  |-  ( U  e.  (SubGrp `  S
)  ->  U  C_  B
)
2119, 20syl 16 . 2  |-  ( ph  ->  U  C_  B )
22 eqid 2435 . . . 4  |-  ( 0g
`  S )  =  ( 0g `  S
)
2322subg0cl 14944 . . 3  |-  ( U  e.  (SubGrp `  S
)  ->  ( 0g `  S )  e.  U
)
24 ne0i 3626 . . 3  |-  ( ( 0g `  S )  e.  U  ->  U  =/=  (/) )
2519, 23, 243syl 19 . 2  |-  ( ph  ->  U  =/=  (/) )
2619adantr 452 . . 3  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U  /\  w  e.  U
) )  ->  U  e.  (SubGrp `  S )
)
27 eqid 2435 . . . . . 6  |-  ( .s
`  S )  =  ( .s `  S
)
28 eqid 2435 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
293adantr 452 . . . . . 6  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  ->  R  e.  Ring )
30 simprl 733 . . . . . 6  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  ->  u  e.  ( Base `  R ) )
31 simprr 734 . . . . . . . 8  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
v  e.  U )
3216adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  ->  U  =  { g  e.  B  |  ( `' g " ( _V  \  {  .0.  }
) )  e.  A } )
3332eleq2d 2502 . . . . . . . . 9  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( v  e.  U  <->  v  e.  { g  e.  B  |  ( `' g " ( _V 
\  {  .0.  }
) )  e.  A } ) )
34 cnveq 5038 . . . . . . . . . . . 12  |-  ( g  =  v  ->  `' g  =  `' v
)
3534imaeq1d 5194 . . . . . . . . . . 11  |-  ( g  =  v  ->  ( `' g " ( _V  \  {  .0.  }
) )  =  ( `' v " ( _V  \  {  .0.  }
) ) )
3635eleq1d 2501 . . . . . . . . . 10  |-  ( g  =  v  ->  (
( `' g "
( _V  \  {  .0.  } ) )  e.  A  <->  ( `' v
" ( _V  \  {  .0.  } ) )  e.  A ) )
3736elrab 3084 . . . . . . . . 9  |-  ( v  e.  { g  e.  B  |  ( `' g " ( _V 
\  {  .0.  }
) )  e.  A } 
<->  ( v  e.  B  /\  ( `' v "
( _V  \  {  .0.  } ) )  e.  A ) )
3833, 37syl6bb 253 . . . . . . . 8  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( v  e.  U  <->  ( v  e.  B  /\  ( `' v " ( _V  \  {  .0.  }
) )  e.  A
) ) )
3931, 38mpbid 202 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( v  e.  B  /\  ( `' v "
( _V  \  {  .0.  } ) )  e.  A ) )
4039simpld 446 . . . . . 6  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
v  e.  B )
411, 27, 28, 6, 29, 30, 40psrvscacl 16449 . . . . 5  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( u ( .s
`  S ) v )  e.  B )
4239simprd 450 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( `' v "
( _V  \  {  .0.  } ) )  e.  A )
431, 28, 12, 6, 41psrelbas 16436 . . . . . . . 8  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( u ( .s
`  S ) v ) : D --> ( Base `  R ) )
44 eqid 2435 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
4530adantr 452 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  ( Base `  R )  /\  v  e.  U ) )  /\  k  e.  ( D  \  ( `' v "
( _V  \  {  .0.  } ) ) ) )  ->  u  e.  ( Base `  R )
)
4640adantr 452 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  ( Base `  R )  /\  v  e.  U ) )  /\  k  e.  ( D  \  ( `' v "
( _V  \  {  .0.  } ) ) ) )  ->  v  e.  B )
47 eldifi 3461 . . . . . . . . . . 11  |-  ( k  e.  ( D  \ 
( `' v "
( _V  \  {  .0.  } ) ) )  ->  k  e.  D
)
4847adantl 453 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  ( Base `  R )  /\  v  e.  U ) )  /\  k  e.  ( D  \  ( `' v "
( _V  \  {  .0.  } ) ) ) )  ->  k  e.  D )
491, 27, 28, 6, 44, 12, 45, 46, 48psrvscaval 16448 . . . . . . . . 9  |-  ( ( ( ph  /\  (
u  e.  ( Base `  R )  /\  v  e.  U ) )  /\  k  e.  ( D  \  ( `' v "
( _V  \  {  .0.  } ) ) ) )  ->  ( (
u ( .s `  S ) v ) `
 k )  =  ( u ( .r
`  R ) ( v `  k ) ) )
501, 28, 12, 6, 40psrelbas 16436 . . . . . . . . . . 11  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
v : D --> ( Base `  R ) )
51 ssid 3359 . . . . . . . . . . . 12  |-  ( `' v " ( _V 
\  {  .0.  }
) )  C_  ( `' v " ( _V  \  {  .0.  }
) )
5251a1i 11 . . . . . . . . . . 11  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( `' v "
( _V  \  {  .0.  } ) )  C_  ( `' v " ( _V  \  {  .0.  }
) ) )
5350, 52suppssr 5856 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  ( Base `  R )  /\  v  e.  U ) )  /\  k  e.  ( D  \  ( `' v "
( _V  \  {  .0.  } ) ) ) )  ->  ( v `  k )  =  .0.  )
5453oveq2d 6089 . . . . . . . . 9  |-  ( ( ( ph  /\  (
u  e.  ( Base `  R )  /\  v  e.  U ) )  /\  k  e.  ( D  \  ( `' v "
( _V  \  {  .0.  } ) ) ) )  ->  ( u
( .r `  R
) ( v `  k ) )  =  ( u ( .r
`  R )  .0.  ) )
5528, 44, 11rngrz 15693 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  u  e.  ( Base `  R
) )  ->  (
u ( .r `  R )  .0.  )  =  .0.  )
5629, 30, 55syl2anc 643 . . . . . . . . . 10  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( u ( .r
`  R )  .0.  )  =  .0.  )
5756adantr 452 . . . . . . . . 9  |-  ( ( ( ph  /\  (
u  e.  ( Base `  R )  /\  v  e.  U ) )  /\  k  e.  ( D  \  ( `' v "
( _V  \  {  .0.  } ) ) ) )  ->  ( u
( .r `  R
)  .0.  )  =  .0.  )
5849, 54, 573eqtrd 2471 . . . . . . . 8  |-  ( ( ( ph  /\  (
u  e.  ( Base `  R )  /\  v  e.  U ) )  /\  k  e.  ( D  \  ( `' v "
( _V  \  {  .0.  } ) ) ) )  ->  ( (
u ( .s `  S ) v ) `
 k )  =  .0.  )
5943, 58suppss 5855 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( `' ( u ( .s `  S
) v ) "
( _V  \  {  .0.  } ) )  C_  ( `' v " ( _V  \  {  .0.  }
) ) )
6042, 59ssexd 4342 . . . . . 6  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( `' ( u ( .s `  S
) v ) "
( _V  \  {  .0.  } ) )  e. 
_V )
6115expr 599 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  (
y  C_  x  ->  y  e.  A ) )
6261alrimiv 1641 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  A. y
( y  C_  x  ->  y  e.  A ) )
6362ralrimiva 2781 . . . . . . . 8  |-  ( ph  ->  A. x  e.  A  A. y ( y  C_  x  ->  y  e.  A
) )
6463adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  ->  A. x  e.  A  A. y ( y  C_  x  ->  y  e.  A
) )
65 sseq2 3362 . . . . . . . . . 10  |-  ( x  =  ( `' v
" ( _V  \  {  .0.  } ) )  ->  ( y  C_  x 
<->  y  C_  ( `' v " ( _V  \  {  .0.  } ) ) ) )
6665imbi1d 309 . . . . . . . . 9  |-  ( x  =  ( `' v
" ( _V  \  {  .0.  } ) )  ->  ( ( y 
C_  x  ->  y  e.  A )  <->  ( y  C_  ( `' v "
( _V  \  {  .0.  } ) )  -> 
y  e.  A ) ) )
6766albidv 1635 . . . . . . . 8  |-  ( x  =  ( `' v
" ( _V  \  {  .0.  } ) )  ->  ( A. y
( y  C_  x  ->  y  e.  A )  <->  A. y ( y  C_  ( `' v " ( _V  \  {  .0.  }
) )  ->  y  e.  A ) ) )
6867rspcv 3040 . . . . . . 7  |-  ( ( `' v " ( _V  \  {  .0.  }
) )  e.  A  ->  ( A. x  e.  A  A. y ( y  C_  x  ->  y  e.  A )  ->  A. y ( y  C_  ( `' v " ( _V  \  {  .0.  }
) )  ->  y  e.  A ) ) )
6942, 64, 68sylc 58 . . . . . 6  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  ->  A. y ( y  C_  ( `' v " ( _V  \  {  .0.  }
) )  ->  y  e.  A ) )
70 sseq1 3361 . . . . . . . 8  |-  ( y  =  ( `' ( u ( .s `  S ) v )
" ( _V  \  {  .0.  } ) )  ->  ( y  C_  ( `' v " ( _V  \  {  .0.  }
) )  <->  ( `' ( u ( .s
`  S ) v ) " ( _V 
\  {  .0.  }
) )  C_  ( `' v " ( _V  \  {  .0.  }
) ) ) )
71 eleq1 2495 . . . . . . . 8  |-  ( y  =  ( `' ( u ( .s `  S ) v )
" ( _V  \  {  .0.  } ) )  ->  ( y  e.  A  <->  ( `' ( u ( .s `  S ) v )
" ( _V  \  {  .0.  } ) )  e.  A ) )
7270, 71imbi12d 312 . . . . . . 7  |-  ( y  =  ( `' ( u ( .s `  S ) v )
" ( _V  \  {  .0.  } ) )  ->  ( ( y 
C_  ( `' v
" ( _V  \  {  .0.  } ) )  ->  y  e.  A
)  <->  ( ( `' ( u ( .s
`  S ) v ) " ( _V 
\  {  .0.  }
) )  C_  ( `' v " ( _V  \  {  .0.  }
) )  ->  ( `' ( u ( .s `  S ) v ) " ( _V  \  {  .0.  }
) )  e.  A
) ) )
7372spcgv 3028 . . . . . 6  |-  ( ( `' ( u ( .s `  S ) v ) " ( _V  \  {  .0.  }
) )  e.  _V  ->  ( A. y ( y  C_  ( `' v " ( _V  \  {  .0.  } ) )  ->  y  e.  A
)  ->  ( ( `' ( u ( .s `  S ) v ) " ( _V  \  {  .0.  }
) )  C_  ( `' v " ( _V  \  {  .0.  }
) )  ->  ( `' ( u ( .s `  S ) v ) " ( _V  \  {  .0.  }
) )  e.  A
) ) )
7460, 69, 59, 73syl3c 59 . . . . 5  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( `' ( u ( .s `  S
) v ) "
( _V  \  {  .0.  } ) )  e.  A )
7532eleq2d 2502 . . . . . 6  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( ( u ( .s `  S ) v )  e.  U  <->  ( u ( .s `  S ) v )  e.  { g  e.  B  |  ( `' g " ( _V 
\  {  .0.  }
) )  e.  A } ) )
76 cnveq 5038 . . . . . . . . 9  |-  ( g  =  ( u ( .s `  S ) v )  ->  `' g  =  `' (
u ( .s `  S ) v ) )
7776imaeq1d 5194 . . . . . . . 8  |-  ( g  =  ( u ( .s `  S ) v )  ->  ( `' g " ( _V  \  {  .0.  }
) )  =  ( `' ( u ( .s `  S ) v ) " ( _V  \  {  .0.  }
) ) )
7877eleq1d 2501 . . . . . . 7  |-  ( g  =  ( u ( .s `  S ) v )  ->  (
( `' g "
( _V  \  {  .0.  } ) )  e.  A  <->  ( `' ( u ( .s `  S ) v )
" ( _V  \  {  .0.  } ) )  e.  A ) )
7978elrab 3084 . . . . . 6  |-  ( ( u ( .s `  S ) v )  e.  { g  e.  B  |  ( `' g " ( _V 
\  {  .0.  }
) )  e.  A } 
<->  ( ( u ( .s `  S ) v )  e.  B  /\  ( `' ( u ( .s `  S
) v ) "
( _V  \  {  .0.  } ) )  e.  A ) )
8075, 79syl6bb 253 . . . . 5  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( ( u ( .s `  S ) v )  e.  U  <->  ( ( u ( .s
`  S ) v )  e.  B  /\  ( `' ( u ( .s `  S ) v ) " ( _V  \  {  .0.  }
) )  e.  A
) ) )
8141, 74, 80mpbir2and 889 . . . 4  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( u ( .s
`  S ) v )  e.  U )
82813adantr3 1118 . . 3  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U  /\  w  e.  U
) )  ->  (
u ( .s `  S ) v )  e.  U )
83 simpr3 965 . . 3  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U  /\  w  e.  U
) )  ->  w  e.  U )
84 eqid 2435 . . . 4  |-  ( +g  `  S )  =  ( +g  `  S )
8584subgcl 14946 . . 3  |-  ( ( U  e.  (SubGrp `  S )  /\  (
u ( .s `  S ) v )  e.  U  /\  w  e.  U )  ->  (
( u ( .s
`  S ) v ) ( +g  `  S
) w )  e.  U )
8626, 82, 83, 85syl3anc 1184 . 2  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U  /\  w  e.  U
) )  ->  (
( u ( .s
`  S ) v ) ( +g  `  S
) w )  e.  U )
874, 5, 7, 8, 9, 10, 21, 25, 86islssd 16004 1  |-  ( ph  ->  U  e.  ( LSubSp `  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   A.wal 1549    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   {crab 2701   _Vcvv 2948    \ cdif 3309    u. cun 3310    C_ wss 3312   (/)c0 3620   {csn 3806   `'ccnv 4869   "cima 4873   ` cfv 5446  (class class class)co 6073    ^m cmap 7010   Fincfn 7101   NNcn 9992   NN0cn0 10213   Basecbs 13461   +g cplusg 13521   .rcmulr 13522   .scvsca 13525   0gc0g 13715   Grpcgrp 14677  SubGrpcsubg 14930   Ringcrg 15652   LSubSpclss 16000   mPwSer cmps 16398
This theorem is referenced by:  mpllss  16493
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-int 4043  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-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  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-7 10055  df-8 10056  df-9 10057  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-sca 13537  df-vsca 13538  df-tset 13540  df-0g 13719  df-mnd 14682  df-grp 14804  df-minusg 14805  df-subg 14933  df-mgp 15641  df-rng 15655  df-lss 16001  df-psr 16409
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