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Theorem mpllsslem 16428
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 16382 . 2  |-  ( ph  ->  R  =  (Scalar `  S ) )
5 eqidd 2390 . 2  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  R ) )
6 mplsubglem.b . . 3  |-  B  =  ( Base `  S
)
76a1i 11 . 2  |-  ( ph  ->  B  =  ( Base `  S ) )
8 eqidd 2390 . 2  |-  ( ph  ->  ( +g  `  S
)  =  ( +g  `  S ) )
9 eqidd 2390 . 2  |-  ( ph  ->  ( .s `  S
)  =  ( .s
`  S ) )
10 eqidd 2390 . 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 15598 . . . . 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 16427 . . 3  |-  ( ph  ->  U  e.  (SubGrp `  S ) )
206subgss 14874 . . 3  |-  ( U  e.  (SubGrp `  S
)  ->  U  C_  B
)
2119, 20syl 16 . 2  |-  ( ph  ->  U  C_  B )
22 eqid 2389 . . . 4  |-  ( 0g
`  S )  =  ( 0g `  S
)
2322subg0cl 14881 . . 3  |-  ( U  e.  (SubGrp `  S
)  ->  ( 0g `  S )  e.  U
)
24 ne0i 3579 . . 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 2389 . . . . . 6  |-  ( .s
`  S )  =  ( .s `  S
)
28 eqid 2389 . . . . . 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 2456 . . . . . . . . 9  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( v  e.  U  <->  v  e.  { g  e.  B  |  ( `' g " ( _V 
\  {  .0.  }
) )  e.  A } ) )
34 cnveq 4988 . . . . . . . . . . . 12  |-  ( g  =  v  ->  `' g  =  `' v
)
3534imaeq1d 5144 . . . . . . . . . . 11  |-  ( g  =  v  ->  ( `' g " ( _V  \  {  .0.  }
) )  =  ( `' v " ( _V  \  {  .0.  }
) ) )
3635eleq1d 2455 . . . . . . . . . 10  |-  ( g  =  v  ->  (
( `' g "
( _V  \  {  .0.  } ) )  e.  A  <->  ( `' v
" ( _V  \  {  .0.  } ) )  e.  A ) )
3736elrab 3037 . . . . . . . . 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 16386 . . . . 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 16373 . . . . . . . 8  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( u ( .s
`  S ) v ) : D --> ( Base `  R ) )
44 eqid 2389 . . . . . . . . . 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 3414 . . . . . . . . . . 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 16385 . . . . . . . . 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 16373 . . . . . . . . . . 11  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
v : D --> ( Base `  R ) )
51 ssid 3312 . . . . . . . . . . . 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 5805 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  ( Base `  R )  /\  v  e.  U ) )  /\  k  e.  ( D  \  ( `' v "
( _V  \  {  .0.  } ) ) ) )  ->  ( v `  k )  =  .0.  )
5453oveq2d 6038 . . . . . . . . 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 15630 . . . . . . . . . . 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 2425 . . . . . . . 8  |-  ( ( ( ph  /\  (
u  e.  ( Base `  R )  /\  v  e.  U ) )  /\  k  e.  ( D  \  ( `' v "
( _V  \  {  .0.  } ) ) ) )  ->  ( (
u ( .s `  S ) v ) `
 k )  =  .0.  )
5943, 58suppss 5804 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( `' ( u ( .s `  S
) v ) "
( _V  \  {  .0.  } ) )  C_  ( `' v " ( _V  \  {  .0.  }
) ) )
6042, 59ssexd 4293 . . . . . 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 1638 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  A. y
( y  C_  x  ->  y  e.  A ) )
6362ralrimiva 2734 . . . . . . . 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 3315 . . . . . . . . . 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 1632 . . . . . . . 8  |-  ( x  =  ( `' v
" ( _V  \  {  .0.  } ) )  ->  ( A. y
( y  C_  x  ->  y  e.  A )  <->  A. y ( y  C_  ( `' v " ( _V  \  {  .0.  }
) )  ->  y  e.  A ) ) )
6867rspcv 2993 . . . . . . 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 3314 . . . . . . . 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 2449 . . . . . . . 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 2981 . . . . . 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 2456 . . . . . 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 4988 . . . . . . . . 9  |-  ( g  =  ( u ( .s `  S ) v )  ->  `' g  =  `' (
u ( .s `  S ) v ) )
7776imaeq1d 5144 . . . . . . . 8  |-  ( g  =  ( u ( .s `  S ) v )  ->  ( `' g " ( _V  \  {  .0.  }
) )  =  ( `' ( u ( .s `  S ) v ) " ( _V  \  {  .0.  }
) ) )
7877eleq1d 2455 . . . . . . 7  |-  ( g  =  ( u ( .s `  S ) v )  ->  (
( `' g "
( _V  \  {  .0.  } ) )  e.  A  <->  ( `' ( u ( .s `  S ) v )
" ( _V  \  {  .0.  } ) )  e.  A ) )
7978elrab 3037 . . . . . 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 2389 . . . 4  |-  ( +g  `  S )  =  ( +g  `  S )
8584subgcl 14883 . . 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 15941 1  |-  ( ph  ->  U  e.  ( LSubSp `  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   A.wal 1546    = wceq 1649    e. wcel 1717    =/= wne 2552   A.wral 2651   {crab 2655   _Vcvv 2901    \ cdif 3262    u. cun 3263    C_ wss 3265   (/)c0 3573   {csn 3759   `'ccnv 4819   "cima 4823   ` cfv 5396  (class class class)co 6022    ^m cmap 6956   Fincfn 7047   NNcn 9934   NN0cn0 10155   Basecbs 13398   +g cplusg 13458   .rcmulr 13459   .scvsca 13462   0gc0g 13652   Grpcgrp 14614  SubGrpcsubg 14867   Ringcrg 15589   LSubSpclss 15937   mPwSer cmps 16335
This theorem is referenced by:  mpllss  16430
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 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-of 6246  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-map 6958  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-3 9993  df-4 9994  df-5 9995  df-6 9996  df-7 9997  df-8 9998  df-9 9999  df-n0 10156  df-z 10217  df-uz 10423  df-fz 10978  df-struct 13400  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-mulr 13472  df-sca 13474  df-vsca 13475  df-tset 13477  df-0g 13656  df-mnd 14619  df-grp 14741  df-minusg 14742  df-subg 14870  df-mgp 15578  df-rng 15592  df-lss 15938  df-psr 16346
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