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Theorem psrvscafval 18691
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 2471 . . . . 5  |-  ( +g  `  R )  =  ( +g  `  R )
4 psrvsca.m . . . . 5  |-  .x.  =  ( .r `  R )
5 eqid 2471 . . . . 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 464 . . . . . 6  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  I  e.  _V )
91, 2, 6, 7, 8psrbas 18679 . . . . 5  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  B  =  ( K  ^m  D ) )
10 eqid 2471 . . . . . 6  |-  ( +g  `  S )  =  ( +g  `  S )
111, 7, 3, 10psrplusg 18682 . . . . 5  |-  ( +g  `  S )  =  (  oF ( +g  `  R )  |`  ( B  X.  B ) )
12 eqid 2471 . . . . . 6  |-  ( .r
`  S )  =  ( .r `  S
)
131, 7, 4, 12, 6psrmulr 18685 . . . . 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 2471 . . . . 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 2472 . . . . 5  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  ( Xt_ `  ( D  X.  { ( TopOpen `  R ) } ) )  =  ( Xt_ `  ( D  X.  {
( TopOpen `  R ) } ) ) )
16 simpr 468 . . . . 5  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  R  e.  _V )
171, 2, 3, 4, 5, 6, 9, 11, 13, 14, 15, 8, 16psrval 18663 . . . 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 5883 . . 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 5889 . . . . . 6  |-  ( Base `  R )  e.  _V
212, 20eqeltri 2545 . . . . 5  |-  K  e. 
_V
22 fvex 5889 . . . . . 6  |-  ( Base `  S )  e.  _V
237, 22eqeltri 2545 . . . . 5  |-  B  e. 
_V
2421, 23mpt2ex 6889 . . . 4  |-  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  oF  .x.  f ) )  e.  _V
25 psrvalstr 18664 . . . . 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 15338 . . . . 5  |-  .s  = Slot  ( .s `  ndx )
27 snsstp2 4115 . . . . . 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 3589 . . . . . 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 3427 . . . . 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 15235 . . . 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 2530 . 2  |-  ( ( I  e.  _V  /\  R  e.  _V )  -> 
.xb  =  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  oF  .x.  f ) ) )
33 eqid 2471 . . . . . 6  |-  (/)  =  (/)
34 fn0 5705 . . . . . 6  |-  ( (/)  Fn  (/) 
<->  (/)  =  (/) )
3533, 34mpbir 214 . . . . 5  |-  (/)  Fn  (/)
36 reldmpsr 18662 . . . . . . . . . 10  |-  Rel  dom mPwSer
3736ovprc 6338 . . . . . . . . 9  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPwSer  R )  =  (/) )
381, 37syl5eq 2517 . . . . . . . 8  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  S  =  (/) )
3938fveq2d 5883 . . . . . . 7  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( .s `  S
)  =  ( .s
`  (/) ) )
40 df-vsca 15285 . . . . . . . 8  |-  .s  = Slot  6
4140str0 15239 . . . . . . 7  |-  (/)  =  ( .s `  (/) )
4239, 19, 413eqtr4g 2530 . . . . . 6  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  -> 
.xb  =  (/) )
4336, 1, 7elbasov 15249 . . . . . . . . . 10  |-  ( f  e.  B  ->  (
I  e.  _V  /\  R  e.  _V )
)
4443con3i 142 . . . . . . . . 9  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  -.  f  e.  B
)
4544eq0rdv 3773 . . . . . . . 8  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  B  =  (/) )
4645xpeq2d 4863 . . . . . . 7  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( K  X.  B
)  =  ( K  X.  (/) ) )
47 xp0 5261 . . . . . . 7  |-  ( K  X.  (/) )  =  (/)
4846, 47syl6eq 2521 . . . . . 6  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( K  X.  B
)  =  (/) )
4942, 48fneq12d 5678 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  (  .xb  Fn  ( K  X.  B )  <->  (/)  Fn  (/) ) )
5035, 49mpbiri 241 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  -> 
.xb  Fn  ( K  X.  B ) )
51 fnov 6423 . . . 4  |-  (  .xb  Fn  ( K  X.  B
)  <->  .xb  =  ( x  e.  K ,  f  e.  B  |->  ( x 
.xb  f ) ) )
5250, 51sylib 201 . . 3  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  -> 
.xb  =  ( x  e.  K ,  f  e.  B  |->  ( x 
.xb  f ) ) )
5344pm2.21d 109 . . . . . 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 1224 . . . 4  |-  ( ( -.  ( I  e. 
_V  /\  R  e.  _V )  /\  x  e.  K  /\  f  e.  B )  ->  (
( D  X.  {
x } )  oF  .x.  f )  =  ( x  .xb  f ) )
5655mpt2eq3dva 6374 . . 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 2508 . 2  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  -> 
.xb  =  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  oF  .x.  f ) ) )
5832, 57pm2.61i 169 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 376    = wceq 1452    e. wcel 1904   {crab 2760   _Vcvv 3031    u. cun 3388   (/)c0 3722   {csn 3959   {ctp 3963   <.cop 3965    X. cxp 4837   `'ccnv 4838   "cima 4842    Fn wfn 5584   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310    oFcof 6548    ^m cmap 7490   Fincfn 7587   1c1 9558   NNcn 10631   6c6 10685   9c9 10688   NN0cn0 10893   ndxcnx 15196   Basecbs 15199   +g cplusg 15268   .rcmulr 15269  Scalarcsca 15271   .scvsca 15272  TopSetcts 15274   TopOpenctopn 15398   Xt_cpt 15415   mPwSer cmps 18652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-plusg 15281  df-mulr 15282  df-sca 15284  df-vsca 15285  df-tset 15287  df-psr 18657
This theorem is referenced by:  psrvsca  18692
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