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Theorem mpllsslemOLD 18416
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.) Obsolete version of mpllsslem 18414 as of 16-Jul-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
mplsubglemOLD.s  |-  S  =  ( I mPwSer  R )
mplsubglemOLD.b  |-  B  =  ( Base `  S
)
mplsubglemOLD.z  |-  .0.  =  ( 0g `  R )
mplsubglemOLD.d  |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
mplsubglemOLD.i  |-  ( ph  ->  I  e.  W )
mplsubglemOLD.0  |-  ( ph  -> 
(/)  e.  A )
mplsubgOLD.a  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( x  u.  y
)  e.  A )
mplsubglemOLD.y  |-  ( (
ph  /\  ( x  e.  A  /\  y  C_  x ) )  -> 
y  e.  A )
mplsubglemOLD.u  |-  ( ph  ->  U  =  { g  e.  B  |  ( `' g " ( _V  \  {  .0.  }
) )  e.  A } )
mpllsslemOLD.r  |-  ( ph  ->  R  e.  Ring )
Assertion
Ref Expression
mpllsslemOLD  |-  ( 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 mpllsslemOLD
Dummy variables  k  u  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mplsubglemOLD.s . . 3  |-  S  =  ( I mPwSer  R )
2 mplsubglemOLD.i . . 3  |-  ( ph  ->  I  e.  W )
3 mpllsslemOLD.r . . 3  |-  ( ph  ->  R  e.  Ring )
41, 2, 3psrsca 18362 . 2  |-  ( ph  ->  R  =  (Scalar `  S ) )
5 eqidd 2403 . 2  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  R ) )
6 mplsubglemOLD.b . . 3  |-  B  =  ( Base `  S
)
76a1i 11 . 2  |-  ( ph  ->  B  =  ( Base `  S ) )
8 eqidd 2403 . 2  |-  ( ph  ->  ( +g  `  S
)  =  ( +g  `  S ) )
9 eqidd 2403 . 2  |-  ( ph  ->  ( .s `  S
)  =  ( .s
`  S ) )
10 eqidd 2403 . 2  |-  ( ph  ->  ( LSubSp `  S )  =  ( LSubSp `  S
) )
11 mplsubglemOLD.z . . . 4  |-  .0.  =  ( 0g `  R )
12 mplsubglemOLD.d . . . 4  |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
13 mplsubglemOLD.0 . . . 4  |-  ( ph  -> 
(/)  e.  A )
14 mplsubgOLD.a . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  y  e.  A ) )  -> 
( x  u.  y
)  e.  A )
15 mplsubglemOLD.y . . . 4  |-  ( (
ph  /\  ( x  e.  A  /\  y  C_  x ) )  -> 
y  e.  A )
16 mplsubglemOLD.u . . . 4  |-  ( ph  ->  U  =  { g  e.  B  |  ( `' g " ( _V  \  {  .0.  }
) )  e.  A } )
17 ringgrp 17523 . . . . 5  |-  ( R  e.  Ring  ->  R  e. 
Grp )
183, 17syl 17 . . . 4  |-  ( ph  ->  R  e.  Grp )
191, 6, 11, 12, 2, 13, 14, 15, 16, 18mplsubglemOLD 18415 . . 3  |-  ( ph  ->  U  e.  (SubGrp `  S ) )
206subgss 16526 . . 3  |-  ( U  e.  (SubGrp `  S
)  ->  U  C_  B
)
2119, 20syl 17 . 2  |-  ( ph  ->  U  C_  B )
22 eqid 2402 . . . 4  |-  ( 0g
`  S )  =  ( 0g `  S
)
2322subg0cl 16533 . . 3  |-  ( U  e.  (SubGrp `  S
)  ->  ( 0g `  S )  e.  U
)
24 ne0i 3744 . . 3  |-  ( ( 0g `  S )  e.  U  ->  U  =/=  (/) )
2519, 23, 243syl 18 . 2  |-  ( ph  ->  U  =/=  (/) )
2619adantr 463 . . 3  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U  /\  w  e.  U
) )  ->  U  e.  (SubGrp `  S )
)
27 eqid 2402 . . . . . 6  |-  ( .s
`  S )  =  ( .s `  S
)
28 eqid 2402 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
293adantr 463 . . . . . 6  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  ->  R  e.  Ring )
30 simprl 756 . . . . . 6  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  ->  u  e.  ( Base `  R ) )
31 simprr 758 . . . . . . . 8  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
v  e.  U )
3216adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  ->  U  =  { g  e.  B  |  ( `' g " ( _V  \  {  .0.  }
) )  e.  A } )
3332eleq2d 2472 . . . . . . . . 9  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( v  e.  U  <->  v  e.  { g  e.  B  |  ( `' g " ( _V 
\  {  .0.  }
) )  e.  A } ) )
34 cnveq 4997 . . . . . . . . . . . 12  |-  ( g  =  v  ->  `' g  =  `' v
)
3534imaeq1d 5156 . . . . . . . . . . 11  |-  ( g  =  v  ->  ( `' g " ( _V  \  {  .0.  }
) )  =  ( `' v " ( _V  \  {  .0.  }
) ) )
3635eleq1d 2471 . . . . . . . . . 10  |-  ( g  =  v  ->  (
( `' g "
( _V  \  {  .0.  } ) )  e.  A  <->  ( `' v
" ( _V  \  {  .0.  } ) )  e.  A ) )
3736elrab 3207 . . . . . . . . 9  |-  ( v  e.  { g  e.  B  |  ( `' g " ( _V 
\  {  .0.  }
) )  e.  A } 
<->  ( v  e.  B  /\  ( `' v "
( _V  \  {  .0.  } ) )  e.  A ) )
3833, 37syl6bb 261 . . . . . . . 8  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( v  e.  U  <->  ( v  e.  B  /\  ( `' v " ( _V  \  {  .0.  }
) )  e.  A
) ) )
3931, 38mpbid 210 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( v  e.  B  /\  ( `' v "
( _V  \  {  .0.  } ) )  e.  A ) )
4039simpld 457 . . . . . 6  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
v  e.  B )
411, 27, 28, 6, 29, 30, 40psrvscacl 18366 . . . . 5  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( u ( .s
`  S ) v )  e.  B )
4239simprd 461 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( `' v "
( _V  \  {  .0.  } ) )  e.  A )
431, 28, 12, 6, 41psrelbas 18352 . . . . . . . 8  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( u ( .s
`  S ) v ) : D --> ( Base `  R ) )
44 eqid 2402 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
4530adantr 463 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  ( Base `  R )  /\  v  e.  U ) )  /\  k  e.  ( D  \  ( `' v "
( _V  \  {  .0.  } ) ) ) )  ->  u  e.  ( Base `  R )
)
4640adantr 463 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  ( Base `  R )  /\  v  e.  U ) )  /\  k  e.  ( D  \  ( `' v "
( _V  \  {  .0.  } ) ) ) )  ->  v  e.  B )
47 eldifi 3565 . . . . . . . . . . 11  |-  ( k  e.  ( D  \ 
( `' v "
( _V  \  {  .0.  } ) ) )  ->  k  e.  D
)
4847adantl 464 . . . . . . . . . 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 18365 . . . . . . . . 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 18352 . . . . . . . . . . 11  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
v : D --> ( Base `  R ) )
51 ssid 3461 . . . . . . . . . . . 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, 52suppssrOLD 5999 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  ( Base `  R )  /\  v  e.  U ) )  /\  k  e.  ( D  \  ( `' v "
( _V  \  {  .0.  } ) ) ) )  ->  ( v `  k )  =  .0.  )
5453oveq2d 6294 . . . . . . . . 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, 11ringrz 17556 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  u  e.  ( Base `  R
) )  ->  (
u ( .r `  R )  .0.  )  =  .0.  )
5629, 30, 55syl2anc 659 . . . . . . . . . 10  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( u ( .r
`  R )  .0.  )  =  .0.  )
5756adantr 463 . . . . . . . . 9  |-  ( ( ( ph  /\  (
u  e.  ( Base `  R )  /\  v  e.  U ) )  /\  k  e.  ( D  \  ( `' v "
( _V  \  {  .0.  } ) ) ) )  ->  ( u
( .r `  R
)  .0.  )  =  .0.  )
5849, 54, 573eqtrd 2447 . . . . . . . 8  |-  ( ( ( ph  /\  (
u  e.  ( Base `  R )  /\  v  e.  U ) )  /\  k  e.  ( D  \  ( `' v "
( _V  \  {  .0.  } ) ) ) )  ->  ( (
u ( .s `  S ) v ) `
 k )  =  .0.  )
5943, 58suppssOLD 5998 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( `' ( u ( .s `  S
) v ) "
( _V  \  {  .0.  } ) )  C_  ( `' v " ( _V  \  {  .0.  }
) ) )
6042, 59ssexd 4541 . . . . . 6  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( `' ( u ( .s `  S
) v ) "
( _V  \  {  .0.  } ) )  e. 
_V )
6115expr 613 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  (
y  C_  x  ->  y  e.  A ) )
6261alrimiv 1740 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  A. y
( y  C_  x  ->  y  e.  A ) )
6362ralrimiva 2818 . . . . . . . 8  |-  ( ph  ->  A. x  e.  A  A. y ( y  C_  x  ->  y  e.  A
) )
6463adantr 463 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  ->  A. x  e.  A  A. y ( y  C_  x  ->  y  e.  A
) )
65 sseq2 3464 . . . . . . . . . 10  |-  ( x  =  ( `' v
" ( _V  \  {  .0.  } ) )  ->  ( y  C_  x 
<->  y  C_  ( `' v " ( _V  \  {  .0.  } ) ) ) )
6665imbi1d 315 . . . . . . . . 9  |-  ( x  =  ( `' v
" ( _V  \  {  .0.  } ) )  ->  ( ( y 
C_  x  ->  y  e.  A )  <->  ( y  C_  ( `' v "
( _V  \  {  .0.  } ) )  -> 
y  e.  A ) ) )
6766albidv 1734 . . . . . . . 8  |-  ( x  =  ( `' v
" ( _V  \  {  .0.  } ) )  ->  ( A. y
( y  C_  x  ->  y  e.  A )  <->  A. y ( y  C_  ( `' v " ( _V  \  {  .0.  }
) )  ->  y  e.  A ) ) )
6867rspcv 3156 . . . . . . 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 59 . . . . . 6  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  ->  A. y ( y  C_  ( `' v " ( _V  \  {  .0.  }
) )  ->  y  e.  A ) )
70 sseq1 3463 . . . . . . . 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 2474 . . . . . . . 8  |-  ( y  =  ( `' ( u ( .s `  S ) v )
" ( _V  \  {  .0.  } ) )  ->  ( y  e.  A  <->  ( `' ( u ( .s `  S ) v )
" ( _V  \  {  .0.  } ) )  e.  A ) )
7270, 71imbi12d 318 . . . . . . 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 3144 . . . . . 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 60 . . . . 5  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( `' ( u ( .s `  S
) v ) "
( _V  \  {  .0.  } ) )  e.  A )
7532eleq2d 2472 . . . . . 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 4997 . . . . . . . . 9  |-  ( g  =  ( u ( .s `  S ) v )  ->  `' g  =  `' (
u ( .s `  S ) v ) )
7776imaeq1d 5156 . . . . . . . 8  |-  ( g  =  ( u ( .s `  S ) v )  ->  ( `' g " ( _V  \  {  .0.  }
) )  =  ( `' ( u ( .s `  S ) v ) " ( _V  \  {  .0.  }
) ) )
7877eleq1d 2471 . . . . . . 7  |-  ( g  =  ( u ( .s `  S ) v )  ->  (
( `' g "
( _V  \  {  .0.  } ) )  e.  A  <->  ( `' ( u ( .s `  S ) v )
" ( _V  \  {  .0.  } ) )  e.  A ) )
7978elrab 3207 . . . . . 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 261 . . . . 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 923 . . . 4  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( u ( .s
`  S ) v )  e.  U )
82813adantr3 1158 . . 3  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U  /\  w  e.  U
) )  ->  (
u ( .s `  S ) v )  e.  U )
83 simpr3 1005 . . 3  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U  /\  w  e.  U
) )  ->  w  e.  U )
84 eqid 2402 . . . 4  |-  ( +g  `  S )  =  ( +g  `  S )
8584subgcl 16535 . . 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 1230 . 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 17902 1  |-  ( ph  ->  U  e.  ( LSubSp `  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 974   A.wal 1403    = wceq 1405    e. wcel 1842    =/= wne 2598   A.wral 2754   {crab 2758   _Vcvv 3059    \ cdif 3411    u. cun 3412    C_ wss 3414   (/)c0 3738   {csn 3972   `'ccnv 4822   "cima 4826   ` cfv 5569  (class class class)co 6278    ^m cmap 7457   Fincfn 7554   NNcn 10576   NN0cn0 10836   Basecbs 14841   +g cplusg 14909   .rcmulr 14910   .scvsca 14913   0gc0g 15054   Grpcgrp 16377  SubGrpcsubg 16519   Ringcrg 17518   LSubSpclss 17898   mPwSer cmps 18320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-n0 10837  df-z 10906  df-uz 11128  df-fz 11727  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-sca 14925  df-vsca 14926  df-tset 14928  df-0g 15056  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-grp 16381  df-minusg 16382  df-subg 16522  df-mgp 17462  df-ring 17520  df-lss 17899  df-psr 18325
This theorem is referenced by: (None)
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