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Theorem mpllsslemOLD 17864
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 17862 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 17810 . 2  |-  ( ph  ->  R  =  (Scalar `  S ) )
5 eqidd 2468 . 2  |-  ( ph  ->  ( Base `  R
)  =  ( Base `  R ) )
6 mplsubglemOLD.b . . 3  |-  B  =  ( Base `  S
)
76a1i 11 . 2  |-  ( ph  ->  B  =  ( Base `  S ) )
8 eqidd 2468 . 2  |-  ( ph  ->  ( +g  `  S
)  =  ( +g  `  S ) )
9 eqidd 2468 . 2  |-  ( ph  ->  ( .s `  S
)  =  ( .s
`  S ) )
10 eqidd 2468 . 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 rnggrp 16988 . . . . 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 17863 . . 3  |-  ( ph  ->  U  e.  (SubGrp `  S ) )
206subgss 15994 . . 3  |-  ( U  e.  (SubGrp `  S
)  ->  U  C_  B
)
2119, 20syl 16 . 2  |-  ( ph  ->  U  C_  B )
22 eqid 2467 . . . 4  |-  ( 0g
`  S )  =  ( 0g `  S
)
2322subg0cl 16001 . . 3  |-  ( U  e.  (SubGrp `  S
)  ->  ( 0g `  S )  e.  U
)
24 ne0i 3791 . . 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 2467 . . . . . 6  |-  ( .s
`  S )  =  ( .s `  S
)
28 eqid 2467 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
293adantr 465 . . . . . 6  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  ->  R  e.  Ring )
30 simprl 755 . . . . . 6  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  ->  u  e.  ( Base `  R ) )
31 simprr 756 . . . . . . . 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 2537 . . . . . . . . 9  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( v  e.  U  <->  v  e.  { g  e.  B  |  ( `' g " ( _V 
\  {  .0.  }
) )  e.  A } ) )
34 cnveq 5174 . . . . . . . . . . . 12  |-  ( g  =  v  ->  `' g  =  `' v
)
3534imaeq1d 5334 . . . . . . . . . . 11  |-  ( g  =  v  ->  ( `' g " ( _V  \  {  .0.  }
) )  =  ( `' v " ( _V  \  {  .0.  }
) ) )
3635eleq1d 2536 . . . . . . . . . 10  |-  ( g  =  v  ->  (
( `' g "
( _V  \  {  .0.  } ) )  e.  A  <->  ( `' v
" ( _V  \  {  .0.  } ) )  e.  A ) )
3736elrab 3261 . . . . . . . . 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 17814 . . . . 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 17800 . . . . . . . 8  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( u ( .s
`  S ) v ) : D --> ( Base `  R ) )
44 eqid 2467 . . . . . . . . . 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 3626 . . . . . . . . . . 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 17813 . . . . . . . . 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 17800 . . . . . . . . . . 11  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
v : D --> ( Base `  R ) )
51 ssid 3523 . . . . . . . . . . . 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 6013 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
u  e.  ( Base `  R )  /\  v  e.  U ) )  /\  k  e.  ( D  \  ( `' v "
( _V  \  {  .0.  } ) ) ) )  ->  ( v `  k )  =  .0.  )
5453oveq2d 6298 . . . . . . . . 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 17020 . . . . . . . . . . 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 2512 . . . . . . . 8  |-  ( ( ( ph  /\  (
u  e.  ( Base `  R )  /\  v  e.  U ) )  /\  k  e.  ( D  \  ( `' v "
( _V  \  {  .0.  } ) ) ) )  ->  ( (
u ( .s `  S ) v ) `
 k )  =  .0.  )
5943, 58suppssOLD 6012 . . . . . . 7  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( `' ( u ( .s `  S
) v ) "
( _V  \  {  .0.  } ) )  C_  ( `' v " ( _V  \  {  .0.  }
) ) )
6042, 59ssexd 4594 . . . . . 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 1695 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  A )  ->  A. y
( y  C_  x  ->  y  e.  A ) )
6362ralrimiva 2878 . . . . . . . 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 3526 . . . . . . . . . 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 1689 . . . . . . . 8  |-  ( x  =  ( `' v
" ( _V  \  {  .0.  } ) )  ->  ( A. y
( y  C_  x  ->  y  e.  A )  <->  A. y ( y  C_  ( `' v " ( _V  \  {  .0.  }
) )  ->  y  e.  A ) ) )
6867rspcv 3210 . . . . . . 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 3525 . . . . . . . 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 2539 . . . . . . . 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 3198 . . . . . 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 2537 . . . . . 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 5174 . . . . . . . . 9  |-  ( g  =  ( u ( .s `  S ) v )  ->  `' g  =  `' (
u ( .s `  S ) v ) )
7776imaeq1d 5334 . . . . . . . 8  |-  ( g  =  ( u ( .s `  S ) v )  ->  ( `' g " ( _V  \  {  .0.  }
) )  =  ( `' ( u ( .s `  S ) v ) " ( _V  \  {  .0.  }
) ) )
7877eleq1d 2536 . . . . . . 7  |-  ( g  =  ( u ( .s `  S ) v )  ->  (
( `' g "
( _V  \  {  .0.  } ) )  e.  A  <->  ( `' ( u ( .s `  S ) v )
" ( _V  \  {  .0.  } ) )  e.  A ) )
7978elrab 3261 . . . . . 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 920 . . . 4  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U ) )  -> 
( u ( .s
`  S ) v )  e.  U )
82813adantr3 1157 . . 3  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U  /\  w  e.  U
) )  ->  (
u ( .s `  S ) v )  e.  U )
83 simpr3 1004 . . 3  |-  ( (
ph  /\  ( u  e.  ( Base `  R
)  /\  v  e.  U  /\  w  e.  U
) )  ->  w  e.  U )
84 eqid 2467 . . . 4  |-  ( +g  `  S )  =  ( +g  `  S )
8584subgcl 16003 . . 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 1228 . 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 17362 1  |-  ( ph  ->  U  e.  ( LSubSp `  S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973   A.wal 1377    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   {crab 2818   _Vcvv 3113    \ cdif 3473    u. cun 3474    C_ wss 3476   (/)c0 3785   {csn 4027   `'ccnv 4998   "cima 5002   ` cfv 5586  (class class class)co 6282    ^m cmap 7417   Fincfn 7513   NNcn 10532   NN0cn0 10791   Basecbs 14483   +g cplusg 14548   .rcmulr 14549   .scvsca 14552   0gc0g 14688   Grpcgrp 15720  SubGrpcsubg 15987   Ringcrg 16983   LSubSpclss 17358   mPwSer cmps 17768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-struct 14485  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-mulr 14562  df-sca 14564  df-vsca 14565  df-tset 14567  df-0g 14690  df-mnd 15725  df-grp 15855  df-minusg 15856  df-subg 15990  df-mgp 16929  df-rng 16985  df-lss 17359  df-psr 17773
This theorem is referenced by: (None)
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