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Theorem mpllsslemOLD 18075
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 18073 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 18021 . 2  |-  ( ph  ->  R  =  (Scalar `  S ) )
5 eqidd 2444 . 2  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  R ) )
6 mplsubglemOLD.b . . 3  |-  B  =  ( Base `  S
)
76a1i 11 . 2  |-  ( ph  ->  B  =  ( Base `  S ) )
8 eqidd 2444 . 2  |-  ( ph  ->  ( +g  `  S
)  =  ( +g  `  S ) )
9 eqidd 2444 . 2  |-  ( ph  ->  ( .s `  S
)  =  ( .s
`  S ) )
10 eqidd 2444 . 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 17182 . . . . 5  |-  ( R  e.  Ring  ->  R  e. 
Grp )
183, 17syl 16 . . . 4  |-  ( ph  ->  R  e.  Grp )
191, 6, 11, 12, 2, 13, 14, 15, 16, 18mplsubglemOLD 18074 . . 3  |-  ( ph  ->  U  e.  (SubGrp `  S ) )
206subgss 16181 . . 3  |-  ( U  e.  (SubGrp `  S
)  ->  U  C_  B
)
2119, 20syl 16 . 2  |-  ( ph  ->  U  C_  B )
22 eqid 2443 . . . 4  |-  ( 0g
`  S )  =  ( 0g `  S
)
2322subg0cl 16188 . . 3  |-  ( U  e.  (SubGrp `  S
)  ->  ( 0g `  S )  e.  U
)
24 ne0i 3776 . . 3  |-  ( ( 0g `  S )  e.  U  ->  U  =/=  (/) )
2519, 23, 243syl 20 . 2  |-  ( ph  ->  U  =/=  (/) )
2619adantr 465 . . 3  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U  /\  w  e.  U
) )  ->  U  e.  (SubGrp `  S )
)
27 eqid 2443 . . . . . 6  |-  ( .s
`  S )  =  ( .s `  S
)
28 eqid 2443 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
293adantr 465 . . . . . 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 757 . . . . . . . 8  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
v  e.  U )
3216adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  ->  U  =  { g  e.  B  |  ( `' g " ( _V  \  {  .0.  }
) )  e.  A } )
3332eleq2d 2513 . . . . . . . . 9  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( v  e.  U  <->  v  e.  { g  e.  B  |  ( `' g " ( _V 
\  {  .0.  }
) )  e.  A } ) )
34 cnveq 5166 . . . . . . . . . . . 12  |-  ( g  =  v  ->  `' g  =  `' v
)
3534imaeq1d 5326 . . . . . . . . . . 11  |-  ( g  =  v  ->  ( `' g " ( _V  \  {  .0.  }
) )  =  ( `' v " ( _V  \  {  .0.  }
) ) )
3635eleq1d 2512 . . . . . . . . . 10  |-  ( g  =  v  ->  (
( `' g "
( _V  \  {  .0.  } ) )  e.  A  <->  ( `' v
" ( _V  \  {  .0.  } ) )  e.  A ) )
3736elrab 3243 . . . . . . . . 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 459 . . . . . 6  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
v  e.  B )
411, 27, 28, 6, 29, 30, 40psrvscacl 18025 . . . . 5  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( u ( .s
`  S ) v )  e.  B )
4239simprd 463 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( `' v "
( _V  \  {  .0.  } ) )  e.  A )
431, 28, 12, 6, 41psrelbas 18011 . . . . . . . 8  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( u ( .s
`  S ) v ) : D --> ( Base `  R ) )
44 eqid 2443 . . . . . . . . . 10  |-  ( .r
`  R )  =  ( .r `  R
)
4530adantr 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  ( Base `  R )  /\  v  e.  U ) )  /\  k  e.  ( D  \  ( `' v "
( _V  \  {  .0.  } ) ) ) )  ->  u  e.  ( Base `  R )
)
4640adantr 465 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  ( Base `  R )  /\  v  e.  U ) )  /\  k  e.  ( D  \  ( `' v "
( _V  \  {  .0.  } ) ) ) )  ->  v  e.  B )
47 eldifi 3611 . . . . . . . . . . 11  |-  ( k  e.  ( D  \ 
( `' v "
( _V  \  {  .0.  } ) ) )  ->  k  e.  D
)
4847adantl 466 . . . . . . . . . 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 18024 . . . . . . . . 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 18011 . . . . . . . . . . 11  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
v : D --> ( Base `  R ) )
51 ssid 3508 . . . . . . . . . . . 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 6006 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  ( Base `  R )  /\  v  e.  U ) )  /\  k  e.  ( D  \  ( `' v "
( _V  \  {  .0.  } ) ) ) )  ->  ( v `  k )  =  .0.  )
5453oveq2d 6297 . . . . . . . . 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 17215 . . . . . . . . . . 11  |-  ( ( R  e.  Ring  /\  u  e.  ( Base `  R
) )  ->  (
u ( .r `  R )  .0.  )  =  .0.  )
5629, 30, 55syl2anc 661 . . . . . . . . . 10  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( u ( .r
`  R )  .0.  )  =  .0.  )
5756adantr 465 . . . . . . . . 9  |-  ( ( ( ph  /\  (
u  e.  ( Base `  R )  /\  v  e.  U ) )  /\  k  e.  ( D  \  ( `' v "
( _V  \  {  .0.  } ) ) ) )  ->  ( u
( .r `  R
)  .0.  )  =  .0.  )
5849, 54, 573eqtrd 2488 . . . . . . . 8  |-  ( ( ( ph  /\  (
u  e.  ( Base `  R )  /\  v  e.  U ) )  /\  k  e.  ( D  \  ( `' v "
( _V  \  {  .0.  } ) ) ) )  ->  ( (
u ( .s `  S ) v ) `
 k )  =  .0.  )
5943, 58suppssOLD 6005 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( `' ( u ( .s `  S
) v ) "
( _V  \  {  .0.  } ) )  C_  ( `' v " ( _V  \  {  .0.  }
) ) )
6042, 59ssexd 4584 . . . . . 6  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( `' ( u ( .s `  S
) v ) "
( _V  \  {  .0.  } ) )  e. 
_V )
6115expr 615 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  (
y  C_  x  ->  y  e.  A ) )
6261alrimiv 1706 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  A. y
( y  C_  x  ->  y  e.  A ) )
6362ralrimiva 2857 . . . . . . . 8  |-  ( ph  ->  A. x  e.  A  A. y ( y  C_  x  ->  y  e.  A
) )
6463adantr 465 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  ->  A. x  e.  A  A. y ( y  C_  x  ->  y  e.  A
) )
65 sseq2 3511 . . . . . . . . . 10  |-  ( x  =  ( `' v
" ( _V  \  {  .0.  } ) )  ->  ( y  C_  x 
<->  y  C_  ( `' v " ( _V  \  {  .0.  } ) ) ) )
6665imbi1d 317 . . . . . . . . 9  |-  ( x  =  ( `' v
" ( _V  \  {  .0.  } ) )  ->  ( ( y 
C_  x  ->  y  e.  A )  <->  ( y  C_  ( `' v "
( _V  \  {  .0.  } ) )  -> 
y  e.  A ) ) )
6766albidv 1700 . . . . . . . 8  |-  ( x  =  ( `' v
" ( _V  \  {  .0.  } ) )  ->  ( A. y
( y  C_  x  ->  y  e.  A )  <->  A. y ( y  C_  ( `' v " ( _V  \  {  .0.  }
) )  ->  y  e.  A ) ) )
6867rspcv 3192 . . . . . . 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 60 . . . . . 6  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  ->  A. y ( y  C_  ( `' v " ( _V  \  {  .0.  }
) )  ->  y  e.  A ) )
70 sseq1 3510 . . . . . . . 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 2515 . . . . . . . 8  |-  ( y  =  ( `' ( u ( .s `  S ) v )
" ( _V  \  {  .0.  } ) )  ->  ( y  e.  A  <->  ( `' ( u ( .s `  S ) v )
" ( _V  \  {  .0.  } ) )  e.  A ) )
7270, 71imbi12d 320 . . . . . . 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 3180 . . . . . 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 61 . . . . 5  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( `' ( u ( .s `  S
) v ) "
( _V  \  {  .0.  } ) )  e.  A )
7532eleq2d 2513 . . . . . 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 5166 . . . . . . . . 9  |-  ( g  =  ( u ( .s `  S ) v )  ->  `' g  =  `' (
u ( .s `  S ) v ) )
7776imaeq1d 5326 . . . . . . . 8  |-  ( g  =  ( u ( .s `  S ) v )  ->  ( `' g " ( _V  \  {  .0.  }
) )  =  ( `' ( u ( .s `  S ) v ) " ( _V  \  {  .0.  }
) ) )
7877eleq1d 2512 . . . . . . 7  |-  ( g  =  ( u ( .s `  S ) v )  ->  (
( `' g "
( _V  \  {  .0.  } ) )  e.  A  <->  ( `' ( u ( .s `  S ) v )
" ( _V  \  {  .0.  } ) )  e.  A ) )
7978elrab 3243 . . . . . 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 922 . . . 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 2443 . . . 4  |-  ( +g  `  S )  =  ( +g  `  S )
8584subgcl 16190 . . 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 1229 . 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 17561 1  |-  ( ph  ->  U  e.  ( LSubSp `  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 974   A.wal 1381    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   {crab 2797   _Vcvv 3095    \ cdif 3458    u. cun 3459    C_ wss 3461   (/)c0 3770   {csn 4014   `'ccnv 4988   "cima 4992   ` cfv 5578  (class class class)co 6281    ^m cmap 7422   Fincfn 7518   NNcn 10543   NN0cn0 10802   Basecbs 14614   +g cplusg 14679   .rcmulr 14680   .scvsca 14683   0gc0g 14819   Grpcgrp 16032  SubGrpcsubg 16174   Ringcrg 17177   LSubSpclss 17557   mPwSer cmps 17979
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-n0 10803  df-z 10872  df-uz 11093  df-fz 11684  df-struct 14616  df-ndx 14617  df-slot 14618  df-base 14619  df-sets 14620  df-ress 14621  df-plusg 14692  df-mulr 14693  df-sca 14695  df-vsca 14696  df-tset 14698  df-0g 14821  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-grp 16036  df-minusg 16037  df-subg 16177  df-mgp 17121  df-ring 17179  df-lss 17558  df-psr 17984
This theorem is referenced by: (None)
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