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Theorem psrbas 18228
Description: The base set of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.) (Proof shortened by AV, 8-Jul-2019.)
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 2454 . . . . 5  |-  ( +g  `  R )  =  ( +g  `  R )
4 eqid 2454 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
5 eqid 2454 . . . . 5  |-  ( TopOpen `  R )  =  (
TopOpen `  R )
6 psrbas.d . . . . 5  |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
7 eqidd 2455 . . . . 5  |-  ( (
ph  /\  R  e.  _V )  ->  ( K  ^m  D )  =  ( K  ^m  D
) )
8 eqid 2454 . . . . 5  |-  (  oF ( +g  `  R
)  |`  ( ( K  ^m  D )  X.  ( K  ^m  D
) ) )  =  (  oF ( +g  `  R )  |`  ( ( K  ^m  D )  X.  ( K  ^m  D ) ) )
9 eqid 2454 . . . . 5  |-  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D
)  |->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  { y  e.  D  |  y  oR  <_  k }  |->  ( ( g `  x ) ( .r
`  R ) ( h `  ( k  oF  -  x
) ) ) ) ) ) )  =  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D ) 
|->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  oR  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  oF  -  x ) ) ) ) ) ) )
10 eqid 2454 . . . . 5  |-  ( x  e.  K ,  g  e.  ( K  ^m  D )  |->  ( ( D  X.  { x } )  oF ( .r `  R
) g ) )  =  ( x  e.  K ,  g  e.  ( K  ^m  D
)  |->  ( ( D  X.  { x }
)  oF ( .r `  R ) g ) )
11 eqidd 2455 . . . . 5  |-  ( (
ph  /\  R  e.  _V )  ->  ( Xt_ `  ( D  X.  {
( TopOpen `  R ) } ) )  =  ( Xt_ `  ( D  X.  { ( TopOpen `  R ) } ) ) )
12 psrbas.i . . . . . 6  |-  ( ph  ->  I  e.  V )
1312adantr 463 . . . . 5  |-  ( (
ph  /\  R  e.  _V )  ->  I  e.  V )
14 simpr 459 . . . . 5  |-  ( (
ph  /\  R  e.  _V )  ->  R  e. 
_V )
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14psrval 18209 . . . 4  |-  ( (
ph  /\  R  e.  _V )  ->  S  =  ( { <. ( Base `  ndx ) ,  ( K  ^m  D
) >. ,  <. ( +g  `  ndx ) ,  (  oF ( +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  oR  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  oF  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  g  e.  ( K  ^m  D )  |->  ( ( D  X.  {
x } )  oF ( .r `  R ) g ) ) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } ) )
1615fveq2d 5852 . . 3  |-  ( (
ph  /\  R  e.  _V )  ->  ( Base `  S )  =  (
Base `  ( { <. ( Base `  ndx ) ,  ( K  ^m  D ) >. ,  <. ( +g  `  ndx ) ,  (  oF
( +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  oR  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  oF  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  g  e.  ( K  ^m  D )  |->  ( ( D  X.  {
x } )  oF ( .r `  R ) g ) ) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } ) ) )
17 psrbas.b . . 3  |-  B  =  ( Base `  S
)
18 ovex 6298 . . . 4  |-  ( K  ^m  D )  e. 
_V
19 psrvalstr 18210 . . . . 5  |-  ( {
<. ( Base `  ndx ) ,  ( K  ^m  D ) >. ,  <. ( +g  `  ndx ) ,  (  oF
( +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  oR  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  oF  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  g  e.  ( K  ^m  D )  |->  ( ( D  X.  {
x } )  oF ( .r `  R ) g ) ) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } ) Struct  <. 1 ,  9 >.
20 baseid 14767 . . . . 5  |-  Base  = Slot  ( Base `  ndx )
21 snsstp1 4167 . . . . . 6  |-  { <. (
Base `  ndx ) ,  ( K  ^m  D
) >. }  C_  { <. (
Base `  ndx ) ,  ( K  ^m  D
) >. ,  <. ( +g  `  ndx ) ,  (  oF ( +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  oR  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  oF  -  x ) ) ) ) ) ) ) >. }
22 ssun1 3653 . . . . . 6  |-  { <. (
Base `  ndx ) ,  ( K  ^m  D
) >. ,  <. ( +g  `  ndx ) ,  (  oF ( +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  oR  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  oF  -  x ) ) ) ) ) ) ) >. }  C_  ( { <. ( Base `  ndx ) ,  ( K  ^m  D ) >. ,  <. ( +g  `  ndx ) ,  (  oF
( +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  oR  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  oF  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  g  e.  ( K  ^m  D )  |->  ( ( D  X.  {
x } )  oF ( .r `  R ) g ) ) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } )
2321, 22sstri 3498 . . . . 5  |-  { <. (
Base `  ndx ) ,  ( K  ^m  D
) >. }  C_  ( { <. ( Base `  ndx ) ,  ( K  ^m  D ) >. ,  <. ( +g  `  ndx ) ,  (  oF
( +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  oR  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  oF  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  g  e.  ( K  ^m  D )  |->  ( ( D  X.  {
x } )  oF ( .r `  R ) g ) ) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } )
2419, 20, 23strfv 14755 . . . 4  |-  ( ( K  ^m  D )  e.  _V  ->  ( K  ^m  D )  =  ( Base `  ( { <. ( Base `  ndx ) ,  ( K  ^m  D ) >. ,  <. ( +g  `  ndx ) ,  (  oF
( +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  oR  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  oF  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  g  e.  ( K  ^m  D )  |->  ( ( D  X.  {
x } )  oF ( .r `  R ) g ) ) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } ) ) )
2518, 24ax-mp 5 . . 3  |-  ( K  ^m  D )  =  ( Base `  ( { <. ( Base `  ndx ) ,  ( K  ^m  D ) >. ,  <. ( +g  `  ndx ) ,  (  oF
( +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  oR  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  oF  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  g  e.  ( K  ^m  D )  |->  ( ( D  X.  {
x } )  oF ( .r `  R ) g ) ) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } ) )
2616, 17, 253eqtr4g 2520 . 2  |-  ( (
ph  /\  R  e.  _V )  ->  B  =  ( K  ^m  D
) )
27 reldmpsr 18208 . . . . . . . 8  |-  Rel  dom mPwSer
2827ovprc2 6302 . . . . . . 7  |-  ( -.  R  e.  _V  ->  ( I mPwSer  R )  =  (/) )
2928adantl 464 . . . . . 6  |-  ( (
ph  /\  -.  R  e.  _V )  ->  (
I mPwSer  R )  =  (/) )
301, 29syl5eq 2507 . . . . 5  |-  ( (
ph  /\  -.  R  e.  _V )  ->  S  =  (/) )
3130fveq2d 5852 . . . 4  |-  ( (
ph  /\  -.  R  e.  _V )  ->  ( Base `  S )  =  ( Base `  (/) ) )
32 base0 14760 . . . 4  |-  (/)  =  (
Base `  (/) )
3331, 17, 323eqtr4g 2520 . . 3  |-  ( (
ph  /\  -.  R  e.  _V )  ->  B  =  (/) )
34 fvprc 5842 . . . . . 6  |-  ( -.  R  e.  _V  ->  (
Base `  R )  =  (/) )
3534adantl 464 . . . . 5  |-  ( (
ph  /\  -.  R  e.  _V )  ->  ( Base `  R )  =  (/) )
362, 35syl5eq 2507 . . . 4  |-  ( (
ph  /\  -.  R  e.  _V )  ->  K  =  (/) )
376fczpsrbag 18214 . . . . . . 7  |-  ( I  e.  V  ->  (
x  e.  I  |->  0 )  e.  D )
3812, 37syl 16 . . . . . 6  |-  ( ph  ->  ( x  e.  I  |->  0 )  e.  D
)
3938adantr 463 . . . . 5  |-  ( (
ph  /\  -.  R  e.  _V )  ->  (
x  e.  I  |->  0 )  e.  D )
40 ne0i 3789 . . . . 5  |-  ( ( x  e.  I  |->  0 )  e.  D  ->  D  =/=  (/) )
4139, 40syl 16 . . . 4  |-  ( (
ph  /\  -.  R  e.  _V )  ->  D  =/=  (/) )
42 fvex 5858 . . . . . 6  |-  ( Base `  R )  e.  _V
432, 42eqeltri 2538 . . . . 5  |-  K  e. 
_V
44 ovex 6298 . . . . . 6  |-  ( NN0 
^m  I )  e. 
_V
456, 44rabex2 4590 . . . . 5  |-  D  e. 
_V
4643, 45map0 7452 . . . 4  |-  ( ( K  ^m  D )  =  (/)  <->  ( K  =  (/)  /\  D  =/=  (/) ) )
4736, 41, 46sylanbrc 662 . . 3  |-  ( (
ph  /\  -.  R  e.  _V )  ->  ( K  ^m  D )  =  (/) )
4833, 47eqtr4d 2498 . 2  |-  ( (
ph  /\  -.  R  e.  _V )  ->  B  =  ( K  ^m  D ) )
4926, 48pm2.61dan 789 1  |-  ( ph  ->  B  =  ( K  ^m  D ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   {crab 2808   _Vcvv 3106    u. cun 3459   (/)c0 3783   {csn 4016   {ctp 4020   <.cop 4022   class class class wbr 4439    |-> cmpt 4497    X. cxp 4986   `'ccnv 4987    |` cres 4990   "cima 4991   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272    oFcof 6511    oRcofr 6512    ^m cmap 7412   Fincfn 7509   0cc0 9481   1c1 9482    <_ cle 9618    - cmin 9796   NNcn 10531   9c9 10588   NN0cn0 10791   ndxcnx 14716   Basecbs 14719   +g cplusg 14787   .rcmulr 14788  Scalarcsca 14790   .scvsca 14791  TopSetcts 14793   TopOpenctopn 14914   Xt_cpt 14931    gsumg cgsu 14933   mPwSer cmps 18198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  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 11083  df-fz 11676  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-plusg 14800  df-mulr 14801  df-sca 14803  df-vsca 14804  df-tset 14806  df-psr 18203
This theorem is referenced by:  psrelbas  18230  psrplusg  18232  psraddcl  18234  psrmulr  18235  psrmulcllem  18238  psrsca  18240  psrvscafval  18241  psrvscacl  18244  psr0cl  18245  psrnegcl  18247  psr1cl  18253  resspsrbas  18268  resspsradd  18269  resspsrmul  18270  subrgpsr  18272  mvrf  18278  mplmon  18323  mplcoe1  18325  opsrtoslem2  18347  psr1bas  18428  psrbaspropd  18474  ply1plusgfvi  18481
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