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Theorem psrbas 16217
Description: The base set of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
psrbas.s  |-  S  =  ( I mPwSer  R )
psrbas.k  |-  K  =  ( Base `  R
)
psrbas.d  |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
psrbas.b  |-  B  =  ( Base `  S
)
psrbas.i  |-  ( ph  ->  I  e.  V )
Assertion
Ref Expression
psrbas  |-  ( ph  ->  B  =  ( K  ^m  D ) )
Distinct variable group:    f, I
Allowed substitution hints:    ph( f)    B( f)    D( f)    R( f)    S( f)    K( f)    V( f)

Proof of Theorem psrbas
Dummy variables  g  h  k  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 psrbas.s . . . . 5  |-  S  =  ( I mPwSer  R )
2 psrbas.k . . . . 5  |-  K  =  ( Base `  R
)
3 eqid 2358 . . . . 5  |-  ( +g  `  R )  =  ( +g  `  R )
4 eqid 2358 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
5 eqid 2358 . . . . 5  |-  ( TopOpen `  R )  =  (
TopOpen `  R )
6 psrbas.d . . . . 5  |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
7 eqidd 2359 . . . . 5  |-  ( (
ph  /\  R  e.  _V )  ->  ( K  ^m  D )  =  ( K  ^m  D
) )
8 eqid 2358 . . . . 5  |-  (  o F ( +g  `  R
)  |`  ( ( K  ^m  D )  X.  ( K  ^m  D
) ) )  =  (  o F ( +g  `  R )  |`  ( ( K  ^m  D )  X.  ( K  ^m  D ) ) )
9 eqid 2358 . . . . 5  |-  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D
)  |->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  { y  e.  D  |  y  o R  <_  k }  |->  ( ( g `  x ) ( .r
`  R ) ( h `  ( k  o F  -  x
) ) ) ) ) ) )  =  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D ) 
|->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  o R  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  o F  -  x ) ) ) ) ) ) )
10 eqid 2358 . . . . 5  |-  ( x  e.  K ,  g  e.  ( K  ^m  D )  |->  ( ( D  X.  { x } )  o F ( .r `  R
) g ) )  =  ( x  e.  K ,  g  e.  ( K  ^m  D
)  |->  ( ( D  X.  { x }
)  o F ( .r `  R ) g ) )
11 eqidd 2359 . . . . 5  |-  ( (
ph  /\  R  e.  _V )  ->  ( Xt_ `  ( D  X.  {
( TopOpen `  R ) } ) )  =  ( Xt_ `  ( D  X.  { ( TopOpen `  R ) } ) ) )
12 psrbas.i . . . . . 6  |-  ( ph  ->  I  e.  V )
1312adantr 451 . . . . 5  |-  ( (
ph  /\  R  e.  _V )  ->  I  e.  V )
14 simpr 447 . . . . 5  |-  ( (
ph  /\  R  e.  _V )  ->  R  e. 
_V )
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14psrval 16203 . . . 4  |-  ( (
ph  /\  R  e.  _V )  ->  S  =  ( { <. ( Base `  ndx ) ,  ( K  ^m  D
) >. ,  <. ( +g  `  ndx ) ,  (  o F ( +g  `  R )  |`  ( ( K  ^m  D )  X.  ( K  ^m  D ) ) ) >. ,  <. ( .r `  ndx ) ,  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D ) 
|->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  o R  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  o F  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  g  e.  ( K  ^m  D )  |->  ( ( D  X.  {
x } )  o F ( .r `  R ) g ) ) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } ) )
1615fveq2d 5609 . . 3  |-  ( (
ph  /\  R  e.  _V )  ->  ( Base `  S )  =  (
Base `  ( { <. ( Base `  ndx ) ,  ( K  ^m  D ) >. ,  <. ( +g  `  ndx ) ,  (  o F
( +g  `  R )  |`  ( ( K  ^m  D )  X.  ( K  ^m  D ) ) ) >. ,  <. ( .r `  ndx ) ,  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D ) 
|->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  o R  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  o F  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  g  e.  ( K  ^m  D )  |->  ( ( D  X.  {
x } )  o F ( .r `  R ) g ) ) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } ) ) )
17 psrbas.b . . 3  |-  B  =  ( Base `  S
)
18 ovex 5967 . . . 4  |-  ( K  ^m  D )  e. 
_V
19 psrvalstr 16204 . . . . 5  |-  ( {
<. ( Base `  ndx ) ,  ( K  ^m  D ) >. ,  <. ( +g  `  ndx ) ,  (  o F
( +g  `  R )  |`  ( ( K  ^m  D )  X.  ( K  ^m  D ) ) ) >. ,  <. ( .r `  ndx ) ,  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D ) 
|->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  o R  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  o F  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  g  e.  ( K  ^m  D )  |->  ( ( D  X.  {
x } )  o F ( .r `  R ) g ) ) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } ) Struct  <. 1 ,  9 >.
20 baseid 13281 . . . . 5  |-  Base  = Slot  ( Base `  ndx )
21 snsstp1 3845 . . . . . 6  |-  { <. (
Base `  ndx ) ,  ( K  ^m  D
) >. }  C_  { <. (
Base `  ndx ) ,  ( K  ^m  D
) >. ,  <. ( +g  `  ndx ) ,  (  o F ( +g  `  R )  |`  ( ( K  ^m  D )  X.  ( K  ^m  D ) ) ) >. ,  <. ( .r `  ndx ) ,  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D ) 
|->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  o R  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  o F  -  x ) ) ) ) ) ) ) >. }
22 ssun1 3414 . . . . . 6  |-  { <. (
Base `  ndx ) ,  ( K  ^m  D
) >. ,  <. ( +g  `  ndx ) ,  (  o F ( +g  `  R )  |`  ( ( K  ^m  D )  X.  ( K  ^m  D ) ) ) >. ,  <. ( .r `  ndx ) ,  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D ) 
|->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  o R  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  o F  -  x ) ) ) ) ) ) ) >. }  C_  ( { <. ( Base `  ndx ) ,  ( K  ^m  D ) >. ,  <. ( +g  `  ndx ) ,  (  o F
( +g  `  R )  |`  ( ( K  ^m  D )  X.  ( K  ^m  D ) ) ) >. ,  <. ( .r `  ndx ) ,  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D ) 
|->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  o R  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  o F  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  g  e.  ( K  ^m  D )  |->  ( ( D  X.  {
x } )  o F ( .r `  R ) g ) ) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } )
2321, 22sstri 3264 . . . . 5  |-  { <. (
Base `  ndx ) ,  ( K  ^m  D
) >. }  C_  ( { <. ( Base `  ndx ) ,  ( K  ^m  D ) >. ,  <. ( +g  `  ndx ) ,  (  o F
( +g  `  R )  |`  ( ( K  ^m  D )  X.  ( K  ^m  D ) ) ) >. ,  <. ( .r `  ndx ) ,  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D ) 
|->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  o R  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  o F  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  g  e.  ( K  ^m  D )  |->  ( ( D  X.  {
x } )  o F ( .r `  R ) g ) ) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } )
2419, 20, 23strfv 13271 . . . 4  |-  ( ( K  ^m  D )  e.  _V  ->  ( K  ^m  D )  =  ( Base `  ( { <. ( Base `  ndx ) ,  ( K  ^m  D ) >. ,  <. ( +g  `  ndx ) ,  (  o F
( +g  `  R )  |`  ( ( K  ^m  D )  X.  ( K  ^m  D ) ) ) >. ,  <. ( .r `  ndx ) ,  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D ) 
|->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  o R  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  o F  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  g  e.  ( K  ^m  D )  |->  ( ( D  X.  {
x } )  o F ( .r `  R ) g ) ) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } ) ) )
2518, 24ax-mp 8 . . 3  |-  ( K  ^m  D )  =  ( Base `  ( { <. ( Base `  ndx ) ,  ( K  ^m  D ) >. ,  <. ( +g  `  ndx ) ,  (  o F
( +g  `  R )  |`  ( ( K  ^m  D )  X.  ( K  ^m  D ) ) ) >. ,  <. ( .r `  ndx ) ,  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D ) 
|->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  o R  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  o F  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  g  e.  ( K  ^m  D )  |->  ( ( D  X.  {
x } )  o F ( .r `  R ) g ) ) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } ) )
2616, 17, 253eqtr4g 2415 . 2  |-  ( (
ph  /\  R  e.  _V )  ->  B  =  ( K  ^m  D
) )
27 reldmpsr 16202 . . . . . . . 8  |-  Rel  dom mPwSer
2827ovprc2 5971 . . . . . . 7  |-  ( -.  R  e.  _V  ->  ( I mPwSer  R )  =  (/) )
2928adantl 452 . . . . . 6  |-  ( (
ph  /\  -.  R  e.  _V )  ->  (
I mPwSer  R )  =  (/) )
301, 29syl5eq 2402 . . . . 5  |-  ( (
ph  /\  -.  R  e.  _V )  ->  S  =  (/) )
3130fveq2d 5609 . . . 4  |-  ( (
ph  /\  -.  R  e.  _V )  ->  ( Base `  S )  =  ( Base `  (/) ) )
32 base0 13276 . . . 4  |-  (/)  =  (
Base `  (/) )
3331, 17, 323eqtr4g 2415 . . 3  |-  ( (
ph  /\  -.  R  e.  _V )  ->  B  =  (/) )
34 fvprc 5599 . . . . . 6  |-  ( -.  R  e.  _V  ->  (
Base `  R )  =  (/) )
3534adantl 452 . . . . 5  |-  ( (
ph  /\  -.  R  e.  _V )  ->  ( Base `  R )  =  (/) )
362, 35syl5eq 2402 . . . 4  |-  ( (
ph  /\  -.  R  e.  _V )  ->  K  =  (/) )
37 0nn0 10069 . . . . . . . 8  |-  0  e.  NN0
3837a1i 10 . . . . . . 7  |-  ( ( ( ph  /\  -.  R  e.  _V )  /\  x  e.  I
)  ->  0  e.  NN0 )
39 eqid 2358 . . . . . . 7  |-  ( x  e.  I  |->  0 )  =  ( x  e.  I  |->  0 )
4038, 39fmptd 5764 . . . . . 6  |-  ( (
ph  /\  -.  R  e.  _V )  ->  (
x  e.  I  |->  0 ) : I --> NN0 )
41 0fin 7174 . . . . . . 7  |-  (/)  e.  Fin
42 nn0supp 10106 . . . . . . . . 9  |-  ( ( x  e.  I  |->  0 ) : I --> NN0  ->  ( `' ( x  e.  I  |->  0 ) "
( _V  \  {
0 } ) )  =  ( `' ( x  e.  I  |->  0 ) " NN ) )
4340, 42syl 15 . . . . . . . 8  |-  ( (
ph  /\  -.  R  e.  _V )  ->  ( `' ( x  e.  I  |->  0 ) "
( _V  \  {
0 } ) )  =  ( `' ( x  e.  I  |->  0 ) " NN ) )
44 eqidd 2359 . . . . . . . . 9  |-  ( ( ( ph  /\  -.  R  e.  _V )  /\  x  e.  (
I  \  (/) ) )  ->  0  =  0 )
4544suppss2 6157 . . . . . . . 8  |-  ( (
ph  /\  -.  R  e.  _V )  ->  ( `' ( x  e.  I  |->  0 ) "
( _V  \  {
0 } ) ) 
C_  (/) )
4643, 45eqsstr3d 3289 . . . . . . 7  |-  ( (
ph  /\  -.  R  e.  _V )  ->  ( `' ( x  e.  I  |->  0 ) " NN )  C_  (/) )
47 ssfi 7168 . . . . . . 7  |-  ( (
(/)  e.  Fin  /\  ( `' ( x  e.  I  |->  0 ) " NN )  C_  (/) )  -> 
( `' ( x  e.  I  |->  0 )
" NN )  e. 
Fin )
4841, 46, 47sylancr 644 . . . . . 6  |-  ( (
ph  /\  -.  R  e.  _V )  ->  ( `' ( x  e.  I  |->  0 ) " NN )  e.  Fin )
496psrbag 16205 . . . . . . . 8  |-  ( I  e.  V  ->  (
( x  e.  I  |->  0 )  e.  D  <->  ( ( x  e.  I  |->  0 ) : I --> NN0  /\  ( `' ( x  e.  I  |->  0 ) " NN )  e.  Fin )
) )
5012, 49syl 15 . . . . . . 7  |-  ( ph  ->  ( ( x  e.  I  |->  0 )  e.  D  <->  ( ( x  e.  I  |->  0 ) : I --> NN0  /\  ( `' ( x  e.  I  |->  0 ) " NN )  e.  Fin ) ) )
5150adantr 451 . . . . . 6  |-  ( (
ph  /\  -.  R  e.  _V )  ->  (
( x  e.  I  |->  0 )  e.  D  <->  ( ( x  e.  I  |->  0 ) : I --> NN0  /\  ( `' ( x  e.  I  |->  0 ) " NN )  e.  Fin )
) )
5240, 48, 51mpbir2and 888 . . . . 5  |-  ( (
ph  /\  -.  R  e.  _V )  ->  (
x  e.  I  |->  0 )  e.  D )
53 ne0i 3537 . . . . 5  |-  ( ( x  e.  I  |->  0 )  e.  D  ->  D  =/=  (/) )
5452, 53syl 15 . . . 4  |-  ( (
ph  /\  -.  R  e.  _V )  ->  D  =/=  (/) )
55 fvex 5619 . . . . . 6  |-  ( Base `  R )  e.  _V
562, 55eqeltri 2428 . . . . 5  |-  K  e. 
_V
57 ovex 5967 . . . . . . 7  |-  ( NN0 
^m  I )  e. 
_V
5857rabex 4244 . . . . . 6  |-  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  e.  _V
596, 58eqeltri 2428 . . . . 5  |-  D  e. 
_V
6056, 59map0 6893 . . . 4  |-  ( ( K  ^m  D )  =  (/)  <->  ( K  =  (/)  /\  D  =/=  (/) ) )
6136, 54, 60sylanbrc 645 . . 3  |-  ( (
ph  /\  -.  R  e.  _V )  ->  ( K  ^m  D )  =  (/) )
6233, 61eqtr4d 2393 . 2  |-  ( (
ph  /\  -.  R  e.  _V )  ->  B  =  ( K  ^m  D ) )
6326, 62pm2.61dan 766 1  |-  ( ph  ->  B  =  ( K  ^m  D ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521   {crab 2623   _Vcvv 2864    \ cdif 3225    u. cun 3226    C_ wss 3228   (/)c0 3531   {csn 3716   {ctp 3718   <.cop 3719   class class class wbr 4102    e. cmpt 4156    X. cxp 4766   `'ccnv 4767    |` cres 4770   "cima 4771   -->wf 5330   ` cfv 5334  (class class class)co 5942    e. cmpt2 5944    o Fcof 6160    o Rcofr 6161    ^m cmap 6857   Fincfn 6948   0cc0 8824   1c1 8825    <_ cle 8955    - cmin 9124   NNcn 9833   9c9 9889   NN0cn0 10054   ndxcnx 13236   Basecbs 13239   +g cplusg 13299   .rcmulr 13300  Scalarcsca 13302   .scvsca 13303  TopSetcts 13305   TopOpenctopn 13419   Xt_cpt 13436    gsumg cgsu 13494   mPwSer cmps 16180
This theorem is referenced by:  psrelbas  16218  psrplusg  16219  psraddcl  16221  psrmulr  16222  psrmulcllem  16225  psrsca  16227  psrvscafval  16228  psrvscacl  16231  psr0cl  16232  psrnegcl  16234  psr1cl  16240  resspsrbas  16252  resspsradd  16253  resspsrmul  16254  subrgpsr  16256  mvrf  16262  mplmon  16300  mplcoe1  16302  opsrtoslem2  16319  psr1bas  16363  psrbaspropd  16405  ply1plusgfvi  16413
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-of 6162  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-oadd 6567  df-er 6744  df-map 6859  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-nn 9834  df-2 9891  df-3 9892  df-4 9893  df-5 9894  df-6 9895  df-7 9896  df-8 9897  df-9 9898  df-n0 10055  df-z 10114  df-uz 10320  df-fz 10872  df-struct 13241  df-ndx 13242  df-slot 13243  df-base 13244  df-plusg 13312  df-mulr 13313  df-sca 13315  df-vsca 13316  df-tset 13318  df-psr 16191
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