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Theorem fnpsr 14153
Description: The multivariate power series constructor has a universal domain. (Contributed by Jim Kingdon, 16-Jun-2025.)
Assertion
Ref Expression
fnpsr  |- mPwSer  Fn  ( _V  X.  _V )

Proof of Theorem fnpsr
Dummy variables  b  d  f  g  h  i  k  r  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-psr 14150 . 2  |- mPwSer  =  ( i  e.  _V , 
r  e.  _V  |->  [_ { h  e.  ( NN0  ^m  i )  |  ( `' h " NN )  e.  Fin }  /  d ]_ [_ (
( Base `  r )  ^m  d )  /  b ]_ ( { <. ( Base `  ndx ) ,  b >. ,  <. ( +g  `  ndx ) ,  (  oF ( +g  `  r )  |`  ( b  X.  b
) ) >. ,  <. ( .r `  ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( k  e.  d  |->  ( r  gsumg  ( x  e.  { y  e.  d  |  y  oR  <_  k }  |->  ( ( f `  x ) ( .r
`  r ) ( g `  ( k  oF  -  x
) ) ) ) ) ) ) >. }  u.  { <. (Scalar ` 
ndx ) ,  r
>. ,  <. ( .s
`  ndx ) ,  ( x  e.  ( Base `  r ) ,  f  e.  b  |->  ( ( d  X.  { x } )  oF ( .r `  r
) f ) )
>. ,  <. (TopSet `  ndx ) ,  ( Xt_ `  ( d  X.  {
( TopOpen `  r ) } ) ) >. } ) )
2 fnmap 6709 . . . . 5  |-  ^m  Fn  ( _V  X.  _V )
3 nn0ex 9246 . . . . 5  |-  NN0  e.  _V
4 vex 2763 . . . . 5  |-  i  e. 
_V
5 fnovex 5951 . . . . 5  |-  ( (  ^m  Fn  ( _V 
X.  _V )  /\  NN0  e.  _V  /\  i  e. 
_V )  ->  ( NN0  ^m  i )  e. 
_V )
62, 3, 4, 5mp3an 1348 . . . 4  |-  ( NN0 
^m  i )  e. 
_V
76rabex 4173 . . 3  |-  { h  e.  ( NN0  ^m  i
)  |  ( `' h " NN )  e.  Fin }  e.  _V
8 basfn 12676 . . . . . 6  |-  Base  Fn  _V
9 vex 2763 . . . . . 6  |-  r  e. 
_V
10 funfvex 5571 . . . . . . 7  |-  ( ( Fun  Base  /\  r  e.  dom  Base )  ->  ( Base `  r )  e. 
_V )
1110funfni 5354 . . . . . 6  |-  ( (
Base  Fn  _V  /\  r  e.  _V )  ->  ( Base `  r )  e. 
_V )
128, 9, 11mp2an 426 . . . . 5  |-  ( Base `  r )  e.  _V
13 vex 2763 . . . . 5  |-  d  e. 
_V
14 fnovex 5951 . . . . 5  |-  ( (  ^m  Fn  ( _V 
X.  _V )  /\  ( Base `  r )  e. 
_V  /\  d  e.  _V )  ->  ( (
Base `  r )  ^m  d )  e.  _V )
152, 12, 13, 14mp3an 1348 . . . 4  |-  ( (
Base `  r )  ^m  d )  e.  _V
16 basendxnn 12674 . . . . . . 7  |-  ( Base `  ndx )  e.  NN
17 vex 2763 . . . . . . 7  |-  b  e. 
_V
18 opexg 4257 . . . . . . 7  |-  ( ( ( Base `  ndx )  e.  NN  /\  b  e.  _V )  ->  <. ( Base `  ndx ) ,  b >.  e.  _V )
1916, 17, 18mp2an 426 . . . . . 6  |-  <. ( Base `  ndx ) ,  b >.  e.  _V
20 plusgndxnn 12729 . . . . . . 7  |-  ( +g  ` 
ndx )  e.  NN
2117a1i 9 . . . . . . . . 9  |-  ( T. 
->  b  e.  _V )
2221, 21ofmresex 6189 . . . . . . . 8  |-  ( T. 
->  (  oF
( +g  `  r )  |`  ( b  X.  b
) )  e.  _V )
2322mptru 1373 . . . . . . 7  |-  (  oF ( +g  `  r
)  |`  ( b  X.  b ) )  e. 
_V
24 opexg 4257 . . . . . . 7  |-  ( ( ( +g  `  ndx )  e.  NN  /\  (  oF ( +g  `  r )  |`  (
b  X.  b ) )  e.  _V )  -> 
<. ( +g  `  ndx ) ,  (  oF ( +g  `  r
)  |`  ( b  X.  b ) ) >.  e.  _V )
2520, 23, 24mp2an 426 . . . . . 6  |-  <. ( +g  `  ndx ) ,  (  oF ( +g  `  r )  |`  ( b  X.  b
) ) >.  e.  _V
26 mulrslid 12749 . . . . . . . . 9  |-  ( .r  = Slot  ( .r `  ndx )  /\  ( .r `  ndx )  e.  NN )
2726simpri 113 . . . . . . . 8  |-  ( .r
`  ndx )  e.  NN
2827elexi 2772 . . . . . . 7  |-  ( .r
`  ndx )  e.  _V
2917, 17mpoex 6267 . . . . . . 7  |-  ( f  e.  b ,  g  e.  b  |->  ( k  e.  d  |->  ( r 
gsumg  ( x  e.  { y  e.  d  |  y  oR  <_  k }  |->  ( ( f `
 x ) ( .r `  r ) ( g `  (
k  oF  -  x ) ) ) ) ) ) )  e.  _V
3028, 29opex 4258 . . . . . 6  |-  <. ( .r `  ndx ) ,  ( f  e.  b ,  g  e.  b 
|->  ( k  e.  d 
|->  ( r  gsumg  ( x  e.  {
y  e.  d  |  y  oR  <_ 
k }  |->  ( ( f `  x ) ( .r `  r
) ( g `  ( k  oF  -  x ) ) ) ) ) ) ) >.  e.  _V
31 tpexg 4475 . . . . . 6  |-  ( (
<. ( Base `  ndx ) ,  b >.  e. 
_V  /\  <. ( +g  ` 
ndx ) ,  (  oF ( +g  `  r )  |`  (
b  X.  b ) ) >.  e.  _V  /\ 
<. ( .r `  ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( k  e.  d  |->  ( r  gsumg  ( x  e.  { y  e.  d  |  y  oR  <_  k }  |->  ( ( f `  x ) ( .r
`  r ) ( g `  ( k  oF  -  x
) ) ) ) ) ) ) >.  e.  _V )  ->  { <. (
Base `  ndx ) ,  b >. ,  <. ( +g  `  ndx ) ,  (  oF ( +g  `  r )  |`  ( b  X.  b
) ) >. ,  <. ( .r `  ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( k  e.  d  |->  ( r  gsumg  ( x  e.  { y  e.  d  |  y  oR  <_  k }  |->  ( ( f `  x ) ( .r
`  r ) ( g `  ( k  oF  -  x
) ) ) ) ) ) ) >. }  e.  _V )
3219, 25, 30, 31mp3an 1348 . . . . 5  |-  { <. (
Base `  ndx ) ,  b >. ,  <. ( +g  `  ndx ) ,  (  oF ( +g  `  r )  |`  ( b  X.  b
) ) >. ,  <. ( .r `  ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( k  e.  d  |->  ( r  gsumg  ( x  e.  { y  e.  d  |  y  oR  <_  k }  |->  ( ( f `  x ) ( .r
`  r ) ( g `  ( k  oF  -  x
) ) ) ) ) ) ) >. }  e.  _V
33 scaslid 12770 . . . . . . . . 9  |-  (Scalar  = Slot  (Scalar `  ndx )  /\  (Scalar `  ndx )  e.  NN )
3433simpri 113 . . . . . . . 8  |-  (Scalar `  ndx )  e.  NN
3534elexi 2772 . . . . . . 7  |-  (Scalar `  ndx )  e.  _V
3635, 9opex 4258 . . . . . 6  |-  <. (Scalar ` 
ndx ) ,  r
>.  e.  _V
37 vscaslid 12780 . . . . . . . . 9  |-  ( .s  = Slot  ( .s `  ndx )  /\  ( .s `  ndx )  e.  NN )
3837simpri 113 . . . . . . . 8  |-  ( .s
`  ndx )  e.  NN
3938elexi 2772 . . . . . . 7  |-  ( .s
`  ndx )  e.  _V
4012, 17mpoex 6267 . . . . . . 7  |-  ( x  e.  ( Base `  r
) ,  f  e.  b  |->  ( ( d  X.  { x }
)  oF ( .r `  r ) f ) )  e. 
_V
4139, 40opex 4258 . . . . . 6  |-  <. ( .s `  ndx ) ,  ( x  e.  (
Base `  r ) ,  f  e.  b  |->  ( ( d  X. 
{ x } )  oF ( .r
`  r ) f ) ) >.  e.  _V
42 tsetndxnn 12806 . . . . . . . 8  |-  (TopSet `  ndx )  e.  NN
4342elexi 2772 . . . . . . 7  |-  (TopSet `  ndx )  e.  _V
44 topnfn 12855 . . . . . . . . . . 11  |-  TopOpen  Fn  _V
45 funfvex 5571 . . . . . . . . . . . 12  |-  ( ( Fun  TopOpen  /\  r  e.  dom 
TopOpen )  ->  ( TopOpen `  r )  e.  _V )
4645funfni 5354 . . . . . . . . . . 11  |-  ( (
TopOpen  Fn  _V  /\  r  e.  _V )  ->  ( TopOpen
`  r )  e. 
_V )
4744, 9, 46mp2an 426 . . . . . . . . . 10  |-  ( TopOpen `  r )  e.  _V
4847snex 4214 . . . . . . . . 9  |-  { (
TopOpen `  r ) }  e.  _V
4913, 48xpex 4774 . . . . . . . 8  |-  ( d  X.  { ( TopOpen `  r ) } )  e.  _V
50 ptex 12875 . . . . . . . 8  |-  ( ( d  X.  { (
TopOpen `  r ) } )  e.  _V  ->  (
Xt_ `  ( d  X.  { ( TopOpen `  r
) } ) )  e.  _V )
5149, 50ax-mp 5 . . . . . . 7  |-  ( Xt_ `  ( d  X.  {
( TopOpen `  r ) } ) )  e. 
_V
5243, 51opex 4258 . . . . . 6  |-  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( d  X.  { ( TopOpen `  r
) } ) )
>.  e.  _V
53 tpexg 4475 . . . . . 6  |-  ( (
<. (Scalar `  ndx ) ,  r >.  e.  _V  /\ 
<. ( .s `  ndx ) ,  ( x  e.  ( Base `  r
) ,  f  e.  b  |->  ( ( d  X.  { x }
)  oF ( .r `  r ) f ) ) >.  e.  _V  /\  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( d  X.  { ( TopOpen `  r
) } ) )
>.  e.  _V )  ->  { <. (Scalar `  ndx ) ,  r >. , 
<. ( .s `  ndx ) ,  ( x  e.  ( Base `  r
) ,  f  e.  b  |->  ( ( d  X.  { x }
)  oF ( .r `  r ) f ) ) >. ,  <. (TopSet `  ndx ) ,  ( Xt_ `  ( d  X.  {
( TopOpen `  r ) } ) ) >. }  e.  _V )
5436, 41, 52, 53mp3an 1348 . . . . 5  |-  { <. (Scalar `  ndx ) ,  r
>. ,  <. ( .s
`  ndx ) ,  ( x  e.  ( Base `  r ) ,  f  e.  b  |->  ( ( d  X.  { x } )  oF ( .r `  r
) f ) )
>. ,  <. (TopSet `  ndx ) ,  ( Xt_ `  ( d  X.  {
( TopOpen `  r ) } ) ) >. }  e.  _V
5532, 54unex 4472 . . . 4  |-  ( {
<. ( Base `  ndx ) ,  b >. , 
<. ( +g  `  ndx ) ,  (  oF ( +g  `  r
)  |`  ( b  X.  b ) ) >. ,  <. ( .r `  ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( k  e.  d  |->  ( r 
gsumg  ( x  e.  { y  e.  d  |  y  oR  <_  k }  |->  ( ( f `
 x ) ( .r `  r ) ( g `  (
k  oF  -  x ) ) ) ) ) ) )
>. }  u.  { <. (Scalar `  ndx ) ,  r
>. ,  <. ( .s
`  ndx ) ,  ( x  e.  ( Base `  r ) ,  f  e.  b  |->  ( ( d  X.  { x } )  oF ( .r `  r
) f ) )
>. ,  <. (TopSet `  ndx ) ,  ( Xt_ `  ( d  X.  {
( TopOpen `  r ) } ) ) >. } )  e.  _V
5615, 55csbexa 4158 . . 3  |-  [_ (
( Base `  r )  ^m  d )  /  b ]_ ( { <. ( Base `  ndx ) ,  b >. ,  <. ( +g  `  ndx ) ,  (  oF ( +g  `  r )  |`  ( b  X.  b
) ) >. ,  <. ( .r `  ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( k  e.  d  |->  ( r  gsumg  ( x  e.  { y  e.  d  |  y  oR  <_  k }  |->  ( ( f `  x ) ( .r
`  r ) ( g `  ( k  oF  -  x
) ) ) ) ) ) ) >. }  u.  { <. (Scalar ` 
ndx ) ,  r
>. ,  <. ( .s
`  ndx ) ,  ( x  e.  ( Base `  r ) ,  f  e.  b  |->  ( ( d  X.  { x } )  oF ( .r `  r
) f ) )
>. ,  <. (TopSet `  ndx ) ,  ( Xt_ `  ( d  X.  {
( TopOpen `  r ) } ) ) >. } )  e.  _V
577, 56csbexa 4158 . 2  |-  [_ {
h  e.  ( NN0 
^m  i )  |  ( `' h " NN )  e.  Fin }  /  d ]_ [_ (
( Base `  r )  ^m  d )  /  b ]_ ( { <. ( Base `  ndx ) ,  b >. ,  <. ( +g  `  ndx ) ,  (  oF ( +g  `  r )  |`  ( b  X.  b
) ) >. ,  <. ( .r `  ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( k  e.  d  |->  ( r  gsumg  ( x  e.  { y  e.  d  |  y  oR  <_  k }  |->  ( ( f `  x ) ( .r
`  r ) ( g `  ( k  oF  -  x
) ) ) ) ) ) ) >. }  u.  { <. (Scalar ` 
ndx ) ,  r
>. ,  <. ( .s
`  ndx ) ,  ( x  e.  ( Base `  r ) ,  f  e.  b  |->  ( ( d  X.  { x } )  oF ( .r `  r
) f ) )
>. ,  <. (TopSet `  ndx ) ,  ( Xt_ `  ( d  X.  {
( TopOpen `  r ) } ) ) >. } )  e.  _V
581, 57fnmpoi 6257 1  |- mPwSer  Fn  ( _V  X.  _V )
Colors of variables: wff set class
Syntax hints:    = wceq 1364   T. wtru 1365    e. wcel 2164   {crab 2476   _Vcvv 2760   [_csb 3080    u. cun 3151   {csn 3618   {ctp 3620   <.cop 3621   class class class wbr 4029    |-> cmpt 4090    X. cxp 4657   `'ccnv 4658    |` cres 4661   "cima 4662    Fn wfn 5249   ` cfv 5254  (class class class)co 5918    e. cmpo 5920    oFcof 6128    oRcofr 6129    ^m cmap 6702   Fincfn 6794    <_ cle 8055    - cmin 8190   NNcn 8982   NN0cn0 9240   ndxcnx 12615  Slot cslot 12617   Basecbs 12618   +g cplusg 12695   .rcmulr 12696  Scalarcsca 12698   .scvsca 12699  TopSetcts 12701   TopOpenctopn 12851   Xt_cpt 12866    gsumg cgsu 12868   mPwSer cmps 14149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-i2m1 7977
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-tp 3626  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-of 6130  df-1st 6193  df-2nd 6194  df-map 6704  df-ixp 6753  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-5 9044  df-6 9045  df-7 9046  df-8 9047  df-9 9048  df-n0 9241  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-mulr 12709  df-sca 12711  df-vsca 12712  df-tset 12714  df-rest 12852  df-topn 12853  df-topgen 12871  df-pt 12872  df-psr 14150
This theorem is referenced by:  psrelbas  14160  psrplusgg  14162  psradd  14163  psraddcl  14164
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