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Theorem psrsca 16134
Description: The scalar field of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
Hypotheses
Ref Expression
psrsca.s  |-  S  =  ( I mPwSer  R )
psrsca.i  |-  ( ph  ->  I  e.  V )
psrsca.r  |-  ( ph  ->  R  e.  W )
Assertion
Ref Expression
psrsca  |-  ( ph  ->  R  =  (Scalar `  S ) )

Proof of Theorem psrsca
Dummy variables  f  h  w  x  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 psrsca.r . . 3  |-  ( ph  ->  R  e.  W )
2 psrvalstr 16111 . . . 4  |-  ( {
<. ( Base `  ndx ) ,  ( Base `  S ) >. ,  <. ( +g  `  ndx ) ,  ( +g  `  S
) >. ,  <. ( .r `  ndx ) ,  ( .r `  S
) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  (
Base `  R ) ,  f  e.  ( Base `  S )  |->  ( ( { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  X.  { x } )  o F ( .r
`  R ) f ) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( {
h  e.  ( NN0 
^m  I )  |  ( `' h " NN )  e.  Fin }  X.  { ( TopOpen `  R ) } ) ) >. } ) Struct  <. 1 ,  9 >.
3 scaid 13269 . . . 4  |- Scalar  = Slot  (Scalar ` 
ndx )
4 snsstp1 3766 . . . . 5  |-  { <. (Scalar `  ndx ) ,  R >. }  C_  { <. (Scalar ` 
ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  ( Base `  R
) ,  f  e.  ( Base `  S
)  |->  ( ( { h  e.  ( NN0 
^m  I )  |  ( `' h " NN )  e.  Fin }  X.  { x }
)  o F ( .r `  R ) f ) ) >. ,  <. (TopSet `  ndx ) ,  ( Xt_ `  ( { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  X.  { ( TopOpen `  R
) } ) )
>. }
5 ssun2 3339 . . . . 5  |-  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  ( Base `  R
) ,  f  e.  ( Base `  S
)  |->  ( ( { h  e.  ( NN0 
^m  I )  |  ( `' h " NN )  e.  Fin }  X.  { x }
)  o F ( .r `  R ) f ) ) >. ,  <. (TopSet `  ndx ) ,  ( Xt_ `  ( { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  X.  { ( TopOpen `  R
) } ) )
>. }  C_  ( { <. ( Base `  ndx ) ,  ( Base `  S ) >. ,  <. ( +g  `  ndx ) ,  ( +g  `  S
) >. ,  <. ( .r `  ndx ) ,  ( .r `  S
) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  (
Base `  R ) ,  f  e.  ( Base `  S )  |->  ( ( { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  X.  { x } )  o F ( .r
`  R ) f ) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( {
h  e.  ( NN0 
^m  I )  |  ( `' h " NN )  e.  Fin }  X.  { ( TopOpen `  R ) } ) ) >. } )
64, 5sstri 3188 . . . 4  |-  { <. (Scalar `  ndx ) ,  R >. }  C_  ( { <. ( Base `  ndx ) ,  ( Base `  S ) >. ,  <. ( +g  `  ndx ) ,  ( +g  `  S
) >. ,  <. ( .r `  ndx ) ,  ( .r `  S
) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  (
Base `  R ) ,  f  e.  ( Base `  S )  |->  ( ( { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  X.  { x } )  o F ( .r
`  R ) f ) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( {
h  e.  ( NN0 
^m  I )  |  ( `' h " NN )  e.  Fin }  X.  { ( TopOpen `  R ) } ) ) >. } )
72, 3, 6strfv 13180 . . 3  |-  ( R  e.  W  ->  R  =  (Scalar `  ( { <. ( Base `  ndx ) ,  ( Base `  S ) >. ,  <. ( +g  `  ndx ) ,  ( +g  `  S
) >. ,  <. ( .r `  ndx ) ,  ( .r `  S
) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  (
Base `  R ) ,  f  e.  ( Base `  S )  |->  ( ( { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  X.  { x } )  o F ( .r
`  R ) f ) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( {
h  e.  ( NN0 
^m  I )  |  ( `' h " NN )  e.  Fin }  X.  { ( TopOpen `  R ) } ) ) >. } ) ) )
81, 7syl 15 . 2  |-  ( ph  ->  R  =  (Scalar `  ( { <. ( Base `  ndx ) ,  ( Base `  S ) >. ,  <. ( +g  `  ndx ) ,  ( +g  `  S
) >. ,  <. ( .r `  ndx ) ,  ( .r `  S
) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  (
Base `  R ) ,  f  e.  ( Base `  S )  |->  ( ( { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  X.  { x } )  o F ( .r
`  R ) f ) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( {
h  e.  ( NN0 
^m  I )  |  ( `' h " NN )  e.  Fin }  X.  { ( TopOpen `  R ) } ) ) >. } ) ) )
9 psrsca.s . . . 4  |-  S  =  ( I mPwSer  R )
10 eqid 2283 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
11 eqid 2283 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
12 eqid 2283 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
13 eqid 2283 . . . 4  |-  ( TopOpen `  R )  =  (
TopOpen `  R )
14 eqid 2283 . . . 4  |-  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }
15 eqid 2283 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
16 psrsca.i . . . . 5  |-  ( ph  ->  I  e.  V )
179, 10, 14, 15, 16psrbas 16124 . . . 4  |-  ( ph  ->  ( Base `  S
)  =  ( (
Base `  R )  ^m  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin } ) )
18 eqid 2283 . . . . 5  |-  ( +g  `  S )  =  ( +g  `  S )
199, 15, 11, 18psrplusg 16126 . . . 4  |-  ( +g  `  S )  =  (  o F ( +g  `  R )  |`  (
( Base `  S )  X.  ( Base `  S
) ) )
20 eqid 2283 . . . . 5  |-  ( .r
`  S )  =  ( .r `  S
)
219, 15, 12, 20, 14psrmulr 16129 . . . 4  |-  ( .r
`  S )  =  ( f  e.  (
Base `  S ) ,  z  e.  ( Base `  S )  |->  ( w  e.  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  |->  ( R  gsumg  ( x  e.  {
y  e.  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  | 
y  o R  <_  w }  |->  ( ( f `  x ) ( .r `  R
) ( z `  ( w  o F  -  x ) ) ) ) ) ) )
22 eqid 2283 . . . 4  |-  ( x  e.  ( Base `  R
) ,  f  e.  ( Base `  S
)  |->  ( ( { h  e.  ( NN0 
^m  I )  |  ( `' h " NN )  e.  Fin }  X.  { x }
)  o F ( .r `  R ) f ) )  =  ( x  e.  (
Base `  R ) ,  f  e.  ( Base `  S )  |->  ( ( { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  X.  { x } )  o F ( .r
`  R ) f ) )
23 eqidd 2284 . . . 4  |-  ( ph  ->  ( Xt_ `  ( { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }  X.  { ( TopOpen `  R ) } ) )  =  ( Xt_ `  ( { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  X.  { ( TopOpen `  R
) } ) ) )
249, 10, 11, 12, 13, 14, 17, 19, 21, 22, 23, 16, 1psrval 16110 . . 3  |-  ( ph  ->  S  =  ( {
<. ( Base `  ndx ) ,  ( Base `  S ) >. ,  <. ( +g  `  ndx ) ,  ( +g  `  S
) >. ,  <. ( .r `  ndx ) ,  ( .r `  S
) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  (
Base `  R ) ,  f  e.  ( Base `  S )  |->  ( ( { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  X.  { x } )  o F ( .r
`  R ) f ) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( {
h  e.  ( NN0 
^m  I )  |  ( `' h " NN )  e.  Fin }  X.  { ( TopOpen `  R ) } ) ) >. } ) )
2524fveq2d 5529 . 2  |-  ( ph  ->  (Scalar `  S )  =  (Scalar `  ( { <. ( Base `  ndx ) ,  ( Base `  S ) >. ,  <. ( +g  `  ndx ) ,  ( +g  `  S
) >. ,  <. ( .r `  ndx ) ,  ( .r `  S
) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  (
Base `  R ) ,  f  e.  ( Base `  S )  |->  ( ( { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  X.  { x } )  o F ( .r
`  R ) f ) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( {
h  e.  ( NN0 
^m  I )  |  ( `' h " NN )  e.  Fin }  X.  { ( TopOpen `  R ) } ) ) >. } ) ) )
268, 25eqtr4d 2318 1  |-  ( ph  ->  R  =  (Scalar `  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   {crab 2547    u. cun 3150   {csn 3640   {ctp 3642   <.cop 3643    X. cxp 4687   `'ccnv 4688   "cima 4692   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860    o Fcof 6076    ^m cmap 6772   Fincfn 6863   1c1 8738   NNcn 9746   9c9 9802   NN0cn0 9965   ndxcnx 13145   Basecbs 13148   +g cplusg 13208   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212  TopSetcts 13214   TopOpenctopn 13326   Xt_cpt 13343   mPwSer cmps 16087
This theorem is referenced by:  psrlmod  16146  psrassa  16158  mpllsslem  16180  mplsca  16189  opsrsca  16224  opsrassa  16230  ply1lss  16275  opsrlmod  16324
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-psr 16098
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