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Theorem psrbas 17794
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 2460 . . . . 5  |-  ( +g  `  R )  =  ( +g  `  R )
4 eqid 2460 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
5 eqid 2460 . . . . 5  |-  ( TopOpen `  R )  =  (
TopOpen `  R )
6 psrbas.d . . . . 5  |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
7 eqidd 2461 . . . . 5  |-  ( (
ph  /\  R  e.  _V )  ->  ( K  ^m  D )  =  ( K  ^m  D
) )
8 eqid 2460 . . . . 5  |-  (  oF ( +g  `  R
)  |`  ( ( K  ^m  D )  X.  ( K  ^m  D
) ) )  =  (  oF ( +g  `  R )  |`  ( ( K  ^m  D )  X.  ( K  ^m  D ) ) )
9 eqid 2460 . . . . 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 2460 . . . . 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 2461 . . . . 5  |-  ( (
ph  /\  R  e.  _V )  ->  ( Xt_ `  ( D  X.  {
( TopOpen `  R ) } ) )  =  ( Xt_ `  ( D  X.  { ( TopOpen `  R ) } ) ) )
12 psrbas.i . . . . . 6  |-  ( ph  ->  I  e.  V )
1312adantr 465 . . . . 5  |-  ( (
ph  /\  R  e.  _V )  ->  I  e.  V )
14 simpr 461 . . . . 5  |-  ( (
ph  /\  R  e.  _V )  ->  R  e. 
_V )
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14psrval 17775 . . . 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 5861 . . 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 6300 . . . 4  |-  ( K  ^m  D )  e. 
_V
19 psrvalstr 17776 . . . . 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 14525 . . . . 5  |-  Base  = Slot  ( Base `  ndx )
21 snsstp1 4171 . . . . . 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 3660 . . . . . 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 3506 . . . . 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 14513 . . . 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 2526 . 2  |-  ( (
ph  /\  R  e.  _V )  ->  B  =  ( K  ^m  D
) )
27 reldmpsr 17774 . . . . . . . 8  |-  Rel  dom mPwSer
2827ovprc2 6304 . . . . . . 7  |-  ( -.  R  e.  _V  ->  ( I mPwSer  R )  =  (/) )
2928adantl 466 . . . . . 6  |-  ( (
ph  /\  -.  R  e.  _V )  ->  (
I mPwSer  R )  =  (/) )
301, 29syl5eq 2513 . . . . 5  |-  ( (
ph  /\  -.  R  e.  _V )  ->  S  =  (/) )
3130fveq2d 5861 . . . 4  |-  ( (
ph  /\  -.  R  e.  _V )  ->  ( Base `  S )  =  ( Base `  (/) ) )
32 base0 14518 . . . 4  |-  (/)  =  (
Base `  (/) )
3331, 17, 323eqtr4g 2526 . . 3  |-  ( (
ph  /\  -.  R  e.  _V )  ->  B  =  (/) )
34 fvprc 5851 . . . . . 6  |-  ( -.  R  e.  _V  ->  (
Base `  R )  =  (/) )
3534adantl 466 . . . . 5  |-  ( (
ph  /\  -.  R  e.  _V )  ->  ( Base `  R )  =  (/) )
362, 35syl5eq 2513 . . . 4  |-  ( (
ph  /\  -.  R  e.  _V )  ->  K  =  (/) )
376fczpsrbag 17780 . . . . . . 7  |-  ( I  e.  V  ->  (
x  e.  I  |->  0 )  e.  D )
3812, 37syl 16 . . . . . 6  |-  ( ph  ->  ( x  e.  I  |->  0 )  e.  D
)
3938adantr 465 . . . . 5  |-  ( (
ph  /\  -.  R  e.  _V )  ->  (
x  e.  I  |->  0 )  e.  D )
40 ne0i 3784 . . . . 5  |-  ( ( x  e.  I  |->  0 )  e.  D  ->  D  =/=  (/) )
4139, 40syl 16 . . . 4  |-  ( (
ph  /\  -.  R  e.  _V )  ->  D  =/=  (/) )
42 fvex 5867 . . . . . 6  |-  ( Base `  R )  e.  _V
432, 42eqeltri 2544 . . . . 5  |-  K  e. 
_V
44 ovex 6300 . . . . . . 7  |-  ( NN0 
^m  I )  e. 
_V
4544rabex 4591 . . . . . 6  |-  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  e.  _V
466, 45eqeltri 2544 . . . . 5  |-  D  e. 
_V
4743, 46map0 7449 . . . 4  |-  ( ( K  ^m  D )  =  (/)  <->  ( K  =  (/)  /\  D  =/=  (/) ) )
4836, 41, 47sylanbrc 664 . . 3  |-  ( (
ph  /\  -.  R  e.  _V )  ->  ( K  ^m  D )  =  (/) )
4933, 48eqtr4d 2504 . 2  |-  ( (
ph  /\  -.  R  e.  _V )  ->  B  =  ( K  ^m  D ) )
5026, 49pm2.61dan 789 1  |-  ( ph  ->  B  =  ( K  ^m  D ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2655   {crab 2811   _Vcvv 3106    u. cun 3467   (/)c0 3778   {csn 4020   {ctp 4024   <.cop 4026   class class class wbr 4440    |-> cmpt 4498    X. cxp 4990   `'ccnv 4991    |` cres 4994   "cima 4995   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277    oFcof 6513    oRcofr 6514    ^m cmap 7410   Fincfn 7506   0cc0 9481   1c1 9482    <_ cle 9618    - cmin 9794   NNcn 10525   9c9 10581   NN0cn0 10784   ndxcnx 14476   Basecbs 14479   +g cplusg 14544   .rcmulr 14545  Scalarcsca 14547   .scvsca 14548  TopSetcts 14550   TopOpenctopn 14666   Xt_cpt 14683    gsumg cgsu 14685   mPwSer cmps 17764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  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 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-plusg 14557  df-mulr 14558  df-sca 14560  df-vsca 14561  df-tset 14563  df-psr 17769
This theorem is referenced by:  psrelbas  17796  psrplusg  17798  psraddcl  17800  psrmulr  17801  psrmulcllem  17804  psrsca  17806  psrvscafval  17807  psrvscacl  17810  psr0cl  17811  psrnegcl  17813  psr1cl  17819  resspsrbas  17834  resspsradd  17835  resspsrmul  17836  subrgpsr  17838  mvrf  17844  mplmon  17889  mplcoe1  17891  opsrtoslem2  17913  psr1bas  17994  psrbaspropd  18040  ply1plusgfvi  18047
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