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Theorem psrsca 16408
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 16385 . . . 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 13545 . . . 4  |- Scalar  = Slot  (Scalar ` 
ndx )
4 snsstp1 3909 . . . . 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 3471 . . . . 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 3317 . . . 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 13456 . . 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 2404 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
11 eqid 2404 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
12 eqid 2404 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
13 eqid 2404 . . . 4  |-  ( TopOpen `  R )  =  (
TopOpen `  R )
14 eqid 2404 . . . 4  |-  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }
15 eqid 2404 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
16 psrsca.i . . . . 5  |-  ( ph  ->  I  e.  V )
179, 10, 14, 15, 16psrbas 16398 . . . 4  |-  ( ph  ->  ( Base `  S
)  =  ( (
Base `  R )  ^m  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin } ) )
18 eqid 2404 . . . . 5  |-  ( +g  `  S )  =  ( +g  `  S )
199, 15, 11, 18psrplusg 16400 . . . 4  |-  ( +g  `  S )  =  (  o F ( +g  `  R )  |`  (
( Base `  S )  X.  ( Base `  S
) ) )
20 eqid 2404 . . . . 5  |-  ( .r
`  S )  =  ( .r `  S
)
219, 15, 12, 20, 14psrmulr 16403 . . . 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 2404 . . . 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 2405 . . . 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 16384 . . 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 5691 . 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 2439 1  |-  ( ph  ->  R  =  (Scalar `  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   {crab 2670    u. cun 3278   {csn 3774   {ctp 3776   <.cop 3777    X. cxp 4835   `'ccnv 4836   "cima 4840   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042    o Fcof 6262    ^m cmap 6977   Fincfn 7068   1c1 8947   NNcn 9956   9c9 10012   NN0cn0 10177   ndxcnx 13421   Basecbs 13424   +g cplusg 13484   .rcmulr 13485  Scalarcsca 13487   .scvsca 13488  TopSetcts 13490   TopOpenctopn 13604   Xt_cpt 13621   mPwSer cmps 16361
This theorem is referenced by:  psrlmod  16420  psrassa  16432  mpllsslem  16454  mplsca  16463  opsrsca  16498  opsrassa  16504  ply1lss  16549  opsrlmod  16595
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-tset 13503  df-psr 16372
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