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Theorem psrvscafval 16374
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 } )  o F  .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 2380 . . . . 5  |-  ( +g  `  R )  =  ( +g  `  R )
4 psrvsca.m . . . . 5  |-  .x.  =  ( .r `  R )
5 eqid 2380 . . . . 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 444 . . . . . 6  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  I  e.  _V )
91, 2, 6, 7, 8psrbas 16363 . . . . 5  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  B  =  ( K  ^m  D ) )
10 eqid 2380 . . . . . 6  |-  ( +g  `  S )  =  ( +g  `  S )
111, 7, 3, 10psrplusg 16365 . . . . 5  |-  ( +g  `  S )  =  (  o F ( +g  `  R )  |`  ( B  X.  B ) )
12 eqid 2380 . . . . . 6  |-  ( .r
`  S )  =  ( .r `  S
)
131, 7, 4, 12, 6psrmulr 16368 . . . . 5  |-  ( .r
`  S )  =  ( f  e.  B ,  g  e.  B  |->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  o R  <_ 
k }  |->  ( ( f `  x ) 
.x.  ( g `  ( k  o F  -  x ) ) ) ) ) ) )
14 eqid 2380 . . . . 5  |-  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  o F 
.x.  f ) )  =  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x }
)  o F  .x.  f ) )
15 eqidd 2381 . . . . 5  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  ( Xt_ `  ( D  X.  { ( TopOpen `  R ) } ) )  =  ( Xt_ `  ( D  X.  {
( TopOpen `  R ) } ) ) )
16 simpr 448 . . . . 5  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  R  e.  _V )
171, 2, 3, 4, 5, 6, 9, 11, 13, 14, 15, 8, 16psrval 16349 . . . 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 } )  o F  .x.  f
) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } ) )
1817fveq2d 5665 . . 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 } )  o F  .x.  f
) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } ) ) )
19 psrvsca.n . . 3  |-  .xb  =  ( .s `  S )
20 fvex 5675 . . . . . 6  |-  ( Base `  R )  e.  _V
212, 20eqeltri 2450 . . . . 5  |-  K  e. 
_V
22 fvex 5675 . . . . . 6  |-  ( Base `  S )  e.  _V
237, 22eqeltri 2450 . . . . 5  |-  B  e. 
_V
2421, 23mpt2ex 6357 . . . 4  |-  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  o F 
.x.  f ) )  e.  _V
25 psrvalstr 16350 . . . . 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 } )  o F  .x.  f
) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } ) Struct  <. 1 ,  9 >.
26 vscaid 13512 . . . . 5  |-  .s  = Slot  ( .s `  ndx )
27 snsstp2 3886 . . . . . 6  |-  { <. ( .s `  ndx ) ,  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x }
)  o F  .x.  f ) ) >. }  C_  { <. (Scalar ` 
ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  o F 
.x.  f ) )
>. ,  <. (TopSet `  ndx ) ,  ( Xt_ `  ( D  X.  {
( TopOpen `  R ) } ) ) >. }
28 ssun2 3447 . . . . . 6  |-  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  o F 
.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 } )  o F  .x.  f
) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } )
2927, 28sstri 3293 . . . . 5  |-  { <. ( .s `  ndx ) ,  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x }
)  o F  .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 } )  o F  .x.  f
) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } )
3025, 26, 29strfv 13421 . . . 4  |-  ( ( x  e.  K , 
f  e.  B  |->  ( ( D  X.  {
x } )  o F  .x.  f ) )  e.  _V  ->  ( x  e.  K , 
f  e.  B  |->  ( ( D  X.  {
x } )  o F  .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 } )  o F  .x.  f
) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } ) ) )
3124, 30ax-mp 8 . . 3  |-  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  o F 
.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 } )  o F  .x.  f
) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } ) )
3218, 19, 313eqtr4g 2437 . 2  |-  ( ( I  e.  _V  /\  R  e.  _V )  -> 
.xb  =  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  o F 
.x.  f ) ) )
33 eqid 2380 . . . . . 6  |-  (/)  =  (/)
34 fn0 5497 . . . . . 6  |-  ( (/)  Fn  (/) 
<->  (/)  =  (/) )
3533, 34mpbir 201 . . . . 5  |-  (/)  Fn  (/)
36 reldmpsr 16348 . . . . . . . . . 10  |-  Rel  dom mPwSer
3736ovprc 6040 . . . . . . . . 9  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPwSer  R )  =  (/) )
381, 37syl5eq 2424 . . . . . . . 8  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  S  =  (/) )
3938fveq2d 5665 . . . . . . 7  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( .s `  S
)  =  ( .s
`  (/) ) )
40 df-vsca 13466 . . . . . . . 8  |-  .s  = Slot  6
4140str0 13425 . . . . . . 7  |-  (/)  =  ( .s `  (/) )
4239, 19, 413eqtr4g 2437 . . . . . 6  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  -> 
.xb  =  (/) )
4336, 1, 7elbasov 13433 . . . . . . . . . 10  |-  ( f  e.  B  ->  (
I  e.  _V  /\  R  e.  _V )
)
4443con3i 129 . . . . . . . . 9  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  -.  f  e.  B
)
4544eq0rdv 3598 . . . . . . . 8  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  B  =  (/) )
4645xpeq2d 4835 . . . . . . 7  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( K  X.  B
)  =  ( K  X.  (/) ) )
47 xp0 5224 . . . . . . 7  |-  ( K  X.  (/) )  =  (/)
4846, 47syl6eq 2428 . . . . . 6  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( K  X.  B
)  =  (/) )
4942, 48fneq12d 5471 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  (  .xb  Fn  ( K  X.  B )  <->  (/)  Fn  (/) ) )
5035, 49mpbiri 225 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  -> 
.xb  Fn  ( K  X.  B ) )
51 fnov 6110 . . . 4  |-  (  .xb  Fn  ( K  X.  B
)  <->  .xb  =  ( x  e.  K ,  f  e.  B  |->  ( x 
.xb  f ) ) )
5250, 51sylib 189 . . 3  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  -> 
.xb  =  ( x  e.  K ,  f  e.  B  |->  ( x 
.xb  f ) ) )
5344pm2.21d 100 . . . . . 6  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( f  e.  B  ->  ( ( D  X.  { x } )  o F  .x.  f
)  =  ( x 
.xb  f ) ) )
5453a1d 23 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( x  e.  K  ->  ( f  e.  B  ->  ( ( D  X.  { x } )  o F  .x.  f
)  =  ( x 
.xb  f ) ) ) )
55543imp 1147 . . . 4  |-  ( ( -.  ( I  e. 
_V  /\  R  e.  _V )  /\  x  e.  K  /\  f  e.  B )  ->  (
( D  X.  {
x } )  o F  .x.  f )  =  ( x  .xb  f ) )
5655mpt2eq3dva 6070 . . 3  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  o F  .x.  f
) )  =  ( x  e.  K , 
f  e.  B  |->  ( x  .xb  f )
) )
5752, 56eqtr4d 2415 . 2  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  -> 
.xb  =  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  o F 
.x.  f ) ) )
5832, 57pm2.61i 158 1  |-  .xb  =  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  o F  .x.  f
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   {crab 2646   _Vcvv 2892    u. cun 3254   (/)c0 3564   {csn 3750   {ctp 3752   <.cop 3753    X. cxp 4809   `'ccnv 4810   "cima 4814    Fn wfn 5382   ` cfv 5387  (class class class)co 6013    e. cmpt2 6015    o Fcof 6235    ^m cmap 6947   Fincfn 7038   1c1 8917   NNcn 9925   6c6 9978   9c9 9981   NN0cn0 10146   ndxcnx 13386   Basecbs 13389   +g cplusg 13449   .rcmulr 13450  Scalarcsca 13452   .scvsca 13453  TopSetcts 13455   TopOpenctopn 13569   Xt_cpt 13586   mPwSer cmps 16326
This theorem is referenced by:  psrvsca  16375
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-of 6237  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-3 9984  df-4 9985  df-5 9986  df-6 9987  df-7 9988  df-8 9989  df-9 9990  df-n0 10147  df-z 10208  df-uz 10414  df-fz 10969  df-struct 13391  df-ndx 13392  df-slot 13393  df-base 13394  df-plusg 13462  df-mulr 13463  df-sca 13465  df-vsca 13466  df-tset 13468  df-psr 16337
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