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Theorem psrsca 16344
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 16321 . . . 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 13477 . . . 4  |- Scalar  = Slot  (Scalar ` 
ndx )
4 snsstp1 3864 . . . . 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 3427 . . . . 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 3274 . . . 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 13388 . . 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 2366 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
11 eqid 2366 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
12 eqid 2366 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
13 eqid 2366 . . . 4  |-  ( TopOpen `  R )  =  (
TopOpen `  R )
14 eqid 2366 . . . 4  |-  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }
15 eqid 2366 . . . . 5  |-  ( Base `  S )  =  (
Base `  S )
16 psrsca.i . . . . 5  |-  ( ph  ->  I  e.  V )
179, 10, 14, 15, 16psrbas 16334 . . . 4  |-  ( ph  ->  ( Base `  S
)  =  ( (
Base `  R )  ^m  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin } ) )
18 eqid 2366 . . . . 5  |-  ( +g  `  S )  =  ( +g  `  S )
199, 15, 11, 18psrplusg 16336 . . . 4  |-  ( +g  `  S )  =  (  o F ( +g  `  R )  |`  (
( Base `  S )  X.  ( Base `  S
) ) )
20 eqid 2366 . . . . 5  |-  ( .r
`  S )  =  ( .r `  S
)
219, 15, 12, 20, 14psrmulr 16339 . . . 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 2366 . . . 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 2367 . . . 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 16320 . . 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 5636 . 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 2401 1  |-  ( ph  ->  R  =  (Scalar `  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1647    e. wcel 1715   {crab 2632    u. cun 3236   {csn 3729   {ctp 3731   <.cop 3732    X. cxp 4790   `'ccnv 4791   "cima 4795   ` cfv 5358  (class class class)co 5981    e. cmpt2 5983    o Fcof 6203    ^m cmap 6915   Fincfn 7006   1c1 8885   NNcn 9893   9c9 9949   NN0cn0 10114   ndxcnx 13353   Basecbs 13356   +g cplusg 13416   .rcmulr 13417  Scalarcsca 13419   .scvsca 13420  TopSetcts 13422   TopOpenctopn 13536   Xt_cpt 13553   mPwSer cmps 16297
This theorem is referenced by:  psrlmod  16356  psrassa  16368  mpllsslem  16390  mplsca  16399  opsrsca  16434  opsrassa  16440  ply1lss  16485  opsrlmod  16534
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-of 6205  df-1st 6249  df-2nd 6250  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-oadd 6625  df-er 6802  df-map 6917  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-2 9951  df-3 9952  df-4 9953  df-5 9954  df-6 9955  df-7 9956  df-8 9957  df-9 9958  df-n0 10115  df-z 10176  df-uz 10382  df-fz 10936  df-struct 13358  df-ndx 13359  df-slot 13360  df-base 13361  df-plusg 13429  df-mulr 13430  df-sca 13432  df-vsca 13433  df-tset 13435  df-psr 16308
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