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Theorem psrvscafval 18363
Description: The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
psrvsca.s  |-  S  =  ( I mPwSer  R )
psrvsca.n  |-  .xb  =  ( .s `  S )
psrvsca.k  |-  K  =  ( Base `  R
)
psrvsca.b  |-  B  =  ( Base `  S
)
psrvsca.m  |-  .x.  =  ( .r `  R )
psrvsca.d  |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }
Assertion
Ref Expression
psrvscafval  |-  .xb  =  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  oF  .x.  f
) )
Distinct variable groups:    x, f, B    f, h, I, x   
f, K, x    D, f, x    R, f, x    .x. , f, x    .xb , f, x
Allowed substitution hints:    B( h)    D( h)    R( h)    S( x, f, h)    .xb ( h)    .x. ( h)    K( h)

Proof of Theorem psrvscafval
Dummy variables  g 
k  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 psrvsca.s . . . . 5  |-  S  =  ( I mPwSer  R )
2 psrvsca.k . . . . 5  |-  K  =  ( Base `  R
)
3 eqid 2402 . . . . 5  |-  ( +g  `  R )  =  ( +g  `  R )
4 psrvsca.m . . . . 5  |-  .x.  =  ( .r `  R )
5 eqid 2402 . . . . 5  |-  ( TopOpen `  R )  =  (
TopOpen `  R )
6 psrvsca.d . . . . 5  |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }
7 psrvsca.b . . . . . 6  |-  B  =  ( Base `  S
)
8 simpl 455 . . . . . 6  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  I  e.  _V )
91, 2, 6, 7, 8psrbas 18350 . . . . 5  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  B  =  ( K  ^m  D ) )
10 eqid 2402 . . . . . 6  |-  ( +g  `  S )  =  ( +g  `  S )
111, 7, 3, 10psrplusg 18354 . . . . 5  |-  ( +g  `  S )  =  (  oF ( +g  `  R )  |`  ( B  X.  B ) )
12 eqid 2402 . . . . . 6  |-  ( .r
`  S )  =  ( .r `  S
)
131, 7, 4, 12, 6psrmulr 18357 . . . . 5  |-  ( .r
`  S )  =  ( f  e.  B ,  g  e.  B  |->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  oR  <_ 
k }  |->  ( ( f `  x ) 
.x.  ( g `  ( k  oF  -  x ) ) ) ) ) ) )
14 eqid 2402 . . . . 5  |-  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  oF  .x.  f ) )  =  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x }
)  oF  .x.  f ) )
15 eqidd 2403 . . . . 5  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  ( Xt_ `  ( D  X.  { ( TopOpen `  R ) } ) )  =  ( Xt_ `  ( D  X.  {
( TopOpen `  R ) } ) ) )
16 simpr 459 . . . . 5  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  R  e.  _V )
171, 2, 3, 4, 5, 6, 9, 11, 13, 14, 15, 8, 16psrval 18331 . . . 4  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  S  =  ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  S ) >. ,  <. ( .r `  ndx ) ,  ( .r `  S ) >. }  u.  {
<. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  oF  .x.  f
) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } ) )
1817fveq2d 5853 . . 3  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  ( .s `  S
)  =  ( .s
`  ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  ( +g  `  S
) >. ,  <. ( .r `  ndx ) ,  ( .r `  S
) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  oF  .x.  f
) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } ) ) )
19 psrvsca.n . . 3  |-  .xb  =  ( .s `  S )
20 fvex 5859 . . . . . 6  |-  ( Base `  R )  e.  _V
212, 20eqeltri 2486 . . . . 5  |-  K  e. 
_V
22 fvex 5859 . . . . . 6  |-  ( Base `  S )  e.  _V
237, 22eqeltri 2486 . . . . 5  |-  B  e. 
_V
2421, 23mpt2ex 6861 . . . 4  |-  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  oF  .x.  f ) )  e.  _V
25 psrvalstr 18332 . . . . 5  |-  ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  S ) >. ,  <. ( .r `  ndx ) ,  ( .r `  S ) >. }  u.  {
<. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  oF  .x.  f
) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } ) Struct  <. 1 ,  9 >.
26 vscaid 14976 . . . . 5  |-  .s  = Slot  ( .s `  ndx )
27 snsstp2 4124 . . . . . 6  |-  { <. ( .s `  ndx ) ,  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x }
)  oF  .x.  f ) ) >. }  C_  { <. (Scalar ` 
ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  oF  .x.  f ) )
>. ,  <. (TopSet `  ndx ) ,  ( Xt_ `  ( D  X.  {
( TopOpen `  R ) } ) ) >. }
28 ssun2 3607 . . . . . 6  |-  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  oF  .x.  f ) )
>. ,  <. (TopSet `  ndx ) ,  ( Xt_ `  ( D  X.  {
( TopOpen `  R ) } ) ) >. }  C_  ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  ( +g  `  S
) >. ,  <. ( .r `  ndx ) ,  ( .r `  S
) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  oF  .x.  f
) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } )
2927, 28sstri 3451 . . . . 5  |-  { <. ( .s `  ndx ) ,  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x }
)  oF  .x.  f ) ) >. }  C_  ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  ( +g  `  S
) >. ,  <. ( .r `  ndx ) ,  ( .r `  S
) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  oF  .x.  f
) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } )
3025, 26, 29strfv 14877 . . . 4  |-  ( ( x  e.  K , 
f  e.  B  |->  ( ( D  X.  {
x } )  oF  .x.  f ) )  e.  _V  ->  ( x  e.  K , 
f  e.  B  |->  ( ( D  X.  {
x } )  oF  .x.  f ) )  =  ( .s
`  ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  ( +g  `  S
) >. ,  <. ( .r `  ndx ) ,  ( .r `  S
) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  oF  .x.  f
) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } ) ) )
3124, 30ax-mp 5 . . 3  |-  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  oF  .x.  f ) )  =  ( .s `  ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  ( +g  `  S ) >. ,  <. ( .r `  ndx ) ,  ( .r `  S ) >. }  u.  {
<. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  oF  .x.  f
) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } ) )
3218, 19, 313eqtr4g 2468 . 2  |-  ( ( I  e.  _V  /\  R  e.  _V )  -> 
.xb  =  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  oF  .x.  f ) ) )
33 eqid 2402 . . . . . 6  |-  (/)  =  (/)
34 fn0 5681 . . . . . 6  |-  ( (/)  Fn  (/) 
<->  (/)  =  (/) )
3533, 34mpbir 209 . . . . 5  |-  (/)  Fn  (/)
36 reldmpsr 18330 . . . . . . . . . 10  |-  Rel  dom mPwSer
3736ovprc 6308 . . . . . . . . 9  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPwSer  R )  =  (/) )
381, 37syl5eq 2455 . . . . . . . 8  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  S  =  (/) )
3938fveq2d 5853 . . . . . . 7  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( .s `  S
)  =  ( .s
`  (/) ) )
40 df-vsca 14926 . . . . . . . 8  |-  .s  = Slot  6
4140str0 14881 . . . . . . 7  |-  (/)  =  ( .s `  (/) )
4239, 19, 413eqtr4g 2468 . . . . . 6  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  -> 
.xb  =  (/) )
4336, 1, 7elbasov 14891 . . . . . . . . . 10  |-  ( f  e.  B  ->  (
I  e.  _V  /\  R  e.  _V )
)
4443con3i 135 . . . . . . . . 9  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  -.  f  e.  B
)
4544eq0rdv 3774 . . . . . . . 8  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  B  =  (/) )
4645xpeq2d 4847 . . . . . . 7  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( K  X.  B
)  =  ( K  X.  (/) ) )
47 xp0 5243 . . . . . . 7  |-  ( K  X.  (/) )  =  (/)
4846, 47syl6eq 2459 . . . . . 6  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( K  X.  B
)  =  (/) )
4942, 48fneq12d 5654 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  (  .xb  Fn  ( K  X.  B )  <->  (/)  Fn  (/) ) )
5035, 49mpbiri 233 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  -> 
.xb  Fn  ( K  X.  B ) )
51 fnov 6391 . . . 4  |-  (  .xb  Fn  ( K  X.  B
)  <->  .xb  =  ( x  e.  K ,  f  e.  B  |->  ( x 
.xb  f ) ) )
5250, 51sylib 196 . . 3  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  -> 
.xb  =  ( x  e.  K ,  f  e.  B  |->  ( x 
.xb  f ) ) )
5344pm2.21d 106 . . . . . 6  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( f  e.  B  ->  ( ( D  X.  { x } )  oF  .x.  f
)  =  ( x 
.xb  f ) ) )
5453a1d 25 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( x  e.  K  ->  ( f  e.  B  ->  ( ( D  X.  { x } )  oF  .x.  f
)  =  ( x 
.xb  f ) ) ) )
55543imp 1191 . . . 4  |-  ( ( -.  ( I  e. 
_V  /\  R  e.  _V )  /\  x  e.  K  /\  f  e.  B )  ->  (
( D  X.  {
x } )  oF  .x.  f )  =  ( x  .xb  f ) )
5655mpt2eq3dva 6342 . . 3  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  oF  .x.  f
) )  =  ( x  e.  K , 
f  e.  B  |->  ( x  .xb  f )
) )
5752, 56eqtr4d 2446 . 2  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  -> 
.xb  =  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  oF  .x.  f ) ) )
5832, 57pm2.61i 164 1  |-  .xb  =  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  oF  .x.  f
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   {crab 2758   _Vcvv 3059    u. cun 3412   (/)c0 3738   {csn 3972   {ctp 3976   <.cop 3978    X. cxp 4821   `'ccnv 4822   "cima 4826    Fn wfn 5564   ` cfv 5569  (class class class)co 6278    |-> cmpt2 6280    oFcof 6519    ^m cmap 7457   Fincfn 7554   1c1 9523   NNcn 10576   6c6 10630   9c9 10633   NN0cn0 10836   ndxcnx 14838   Basecbs 14841   +g cplusg 14909   .rcmulr 14910  Scalarcsca 14912   .scvsca 14913  TopSetcts 14915   TopOpenctopn 15036   Xt_cpt 15053   mPwSer cmps 18320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-n0 10837  df-z 10906  df-uz 11128  df-fz 11727  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-plusg 14922  df-mulr 14923  df-sca 14925  df-vsca 14926  df-tset 14928  df-psr 18325
This theorem is referenced by:  psrvsca  18364
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