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Theorem psrsca 16484
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 16461 . . . 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 13621 . . . 4  |- Scalar  = Slot  (Scalar ` 
ndx )
4 snsstp1 3973 . . . . 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 3497 . . . . 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 3343 . . . 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 13532 . . 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 16 . 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 2442 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
11 eqid 2442 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
12 eqid 2442 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
13 eqid 2442 . . . 4  |-  ( TopOpen `  R )  =  (
TopOpen `  R )
14 eqid 2442 . . . 4  |-  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }
15 eqid 2442 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
16 psrsca.i . . . . 5  |-  ( ph  ->  I  e.  V )
179, 10, 14, 15, 16psrbas 16474 . . . 4  |-  ( ph  ->  ( Base `  S
)  =  ( (
Base `  R )  ^m  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin } ) )
18 eqid 2442 . . . . 5  |-  ( +g  `  S )  =  ( +g  `  S )
199, 15, 11, 18psrplusg 16476 . . . 4  |-  ( +g  `  S )  =  (  o F ( +g  `  R )  |`  (
( Base `  S )  X.  ( Base `  S
) ) )
20 eqid 2442 . . . . 5  |-  ( .r
`  S )  =  ( .r `  S
)
219, 15, 12, 20, 14psrmulr 16479 . . . 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 2442 . . . 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 2443 . . . 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 16460 . . 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 5761 . 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 2477 1  |-  ( ph  ->  R  =  (Scalar `  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1727   {crab 2715    u. cun 3304   {csn 3838   {ctp 3840   <.cop 3841    X. cxp 4905   `'ccnv 4906   "cima 4910   ` cfv 5483  (class class class)co 6110    e. cmpt2 6112    o Fcof 6332    ^m cmap 7047   Fincfn 7138   1c1 9022   NNcn 10031   9c9 10087   NN0cn0 10252   ndxcnx 13497   Basecbs 13500   +g cplusg 13560   .rcmulr 13561  Scalarcsca 13563   .scvsca 13564  TopSetcts 13566   TopOpenctopn 13680   Xt_cpt 13697   mPwSer cmps 16437
This theorem is referenced by:  psrlmod  16496  psrassa  16508  mpllsslem  16530  mplsca  16539  opsrsca  16574  opsrassa  16580  ply1lss  16625  opsrlmod  16671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-of 6334  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-1o 6753  df-oadd 6757  df-er 6934  df-map 7049  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-nn 10032  df-2 10089  df-3 10090  df-4 10091  df-5 10092  df-6 10093  df-7 10094  df-8 10095  df-9 10096  df-n0 10253  df-z 10314  df-uz 10520  df-fz 11075  df-struct 13502  df-ndx 13503  df-slot 13504  df-base 13505  df-plusg 13573  df-mulr 13574  df-sca 13576  df-vsca 13577  df-tset 13579  df-psr 16448
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