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Theorem psrvscafval 17569
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 2451 . . . . 5  |-  ( +g  `  R )  =  ( +g  `  R )
4 psrvsca.m . . . . 5  |-  .x.  =  ( .r `  R )
5 eqid 2451 . . . . 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 457 . . . . . 6  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  I  e.  _V )
91, 2, 6, 7, 8psrbas 17556 . . . . 5  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  B  =  ( K  ^m  D ) )
10 eqid 2451 . . . . . 6  |-  ( +g  `  S )  =  ( +g  `  S )
111, 7, 3, 10psrplusg 17560 . . . . 5  |-  ( +g  `  S )  =  (  oF ( +g  `  R )  |`  ( B  X.  B ) )
12 eqid 2451 . . . . . 6  |-  ( .r
`  S )  =  ( .r `  S
)
131, 7, 4, 12, 6psrmulr 17563 . . . . 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 2451 . . . . 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 2452 . . . . 5  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  ( Xt_ `  ( D  X.  { ( TopOpen `  R ) } ) )  =  ( Xt_ `  ( D  X.  {
( TopOpen `  R ) } ) ) )
16 simpr 461 . . . . 5  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  R  e.  _V )
171, 2, 3, 4, 5, 6, 9, 11, 13, 14, 15, 8, 16psrval 17537 . . . 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 5795 . . 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 5801 . . . . . 6  |-  ( Base `  R )  e.  _V
212, 20eqeltri 2535 . . . . 5  |-  K  e. 
_V
22 fvex 5801 . . . . . 6  |-  ( Base `  S )  e.  _V
237, 22eqeltri 2535 . . . . 5  |-  B  e. 
_V
2421, 23mpt2ex 6752 . . . 4  |-  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  oF  .x.  f ) )  e.  _V
25 psrvalstr 17538 . . . . 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 14405 . . . . 5  |-  .s  = Slot  ( .s `  ndx )
27 snsstp2 4125 . . . . . 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 3620 . . . . . 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 3465 . . . . 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 14312 . . . 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 2517 . 2  |-  ( ( I  e.  _V  /\  R  e.  _V )  -> 
.xb  =  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  oF  .x.  f ) ) )
33 eqid 2451 . . . . . 6  |-  (/)  =  (/)
34 fn0 5630 . . . . . 6  |-  ( (/)  Fn  (/) 
<->  (/)  =  (/) )
3533, 34mpbir 209 . . . . 5  |-  (/)  Fn  (/)
36 reldmpsr 17536 . . . . . . . . . 10  |-  Rel  dom mPwSer
3736ovprc 6219 . . . . . . . . 9  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPwSer  R )  =  (/) )
381, 37syl5eq 2504 . . . . . . . 8  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  S  =  (/) )
3938fveq2d 5795 . . . . . . 7  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( .s `  S
)  =  ( .s
`  (/) ) )
40 df-vsca 14359 . . . . . . . 8  |-  .s  = Slot  6
4140str0 14316 . . . . . . 7  |-  (/)  =  ( .s `  (/) )
4239, 19, 413eqtr4g 2517 . . . . . 6  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  -> 
.xb  =  (/) )
4336, 1, 7elbasov 14326 . . . . . . . . . 10  |-  ( f  e.  B  ->  (
I  e.  _V  /\  R  e.  _V )
)
4443con3i 135 . . . . . . . . 9  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  -.  f  e.  B
)
4544eq0rdv 3772 . . . . . . . 8  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  B  =  (/) )
4645xpeq2d 4964 . . . . . . 7  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( K  X.  B
)  =  ( K  X.  (/) ) )
47 xp0 5356 . . . . . . 7  |-  ( K  X.  (/) )  =  (/)
4846, 47syl6eq 2508 . . . . . 6  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( K  X.  B
)  =  (/) )
4942, 48fneq12d 5603 . . . . 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 6300 . . . 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 1182 . . . 4  |-  ( ( -.  ( I  e. 
_V  /\  R  e.  _V )  /\  x  e.  K  /\  f  e.  B )  ->  (
( D  X.  {
x } )  oF  .x.  f )  =  ( x  .xb  f ) )
5655mpt2eq3dva 6251 . . 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 2495 . 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 369    = wceq 1370    e. wcel 1758   {crab 2799   _Vcvv 3070    u. cun 3426   (/)c0 3737   {csn 3977   {ctp 3981   <.cop 3983    X. cxp 4938   `'ccnv 4939   "cima 4943    Fn wfn 5513   ` cfv 5518  (class class class)co 6192    |-> cmpt2 6194    oFcof 6420    ^m cmap 7316   Fincfn 7412   1c1 9386   NNcn 10425   6c6 10478   9c9 10481   NN0cn0 10682   ndxcnx 14275   Basecbs 14278   +g cplusg 14342   .rcmulr 14343  Scalarcsca 14345   .scvsca 14346  TopSetcts 14348   TopOpenctopn 14464   Xt_cpt 14481   mPwSer cmps 17526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-of 6422  df-om 6579  df-1st 6679  df-2nd 6680  df-supp 6793  df-recs 6934  df-rdg 6968  df-1o 7022  df-oadd 7026  df-er 7203  df-map 7318  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-fsupp 7724  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-3 10484  df-4 10485  df-5 10486  df-6 10487  df-7 10488  df-8 10489  df-9 10490  df-n0 10683  df-z 10750  df-uz 10965  df-fz 11541  df-struct 14280  df-ndx 14281  df-slot 14282  df-base 14283  df-plusg 14355  df-mulr 14356  df-sca 14358  df-vsca 14359  df-tset 14361  df-psr 17531
This theorem is referenced by:  psrvsca  17570
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