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Theorem psrvscafval 16151
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 2296 . . . . 5  |-  ( +g  `  R )  =  ( +g  `  R )
4 psrvsca.m . . . . 5  |-  .x.  =  ( .r `  R )
5 eqid 2296 . . . . 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 443 . . . . . 6  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  I  e.  _V )
91, 2, 6, 7, 8psrbas 16140 . . . . 5  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  B  =  ( K  ^m  D ) )
10 eqid 2296 . . . . . 6  |-  ( +g  `  S )  =  ( +g  `  S )
111, 7, 3, 10psrplusg 16142 . . . . 5  |-  ( +g  `  S )  =  (  o F ( +g  `  R )  |`  ( B  X.  B ) )
12 eqid 2296 . . . . . 6  |-  ( .r
`  S )  =  ( .r `  S
)
131, 7, 4, 12, 6psrmulr 16145 . . . . 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 2296 . . . . 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 2297 . . . . 5  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  ( Xt_ `  ( D  X.  { ( TopOpen `  R ) } ) )  =  ( Xt_ `  ( D  X.  {
( TopOpen `  R ) } ) ) )
16 simpr 447 . . . . 5  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  R  e.  _V )
171, 2, 3, 4, 5, 6, 9, 11, 13, 14, 15, 8, 16psrval 16126 . . . 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 5545 . . 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 5555 . . . . . 6  |-  ( Base `  R )  e.  _V
212, 20eqeltri 2366 . . . . 5  |-  K  e. 
_V
22 fvex 5555 . . . . . 6  |-  ( Base `  S )  e.  _V
237, 22eqeltri 2366 . . . . 5  |-  B  e. 
_V
2421, 23mpt2ex 6214 . . . 4  |-  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  o F 
.x.  f ) )  e.  _V
25 psrvalstr 16127 . . . . 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 13287 . . . . 5  |-  .s  = Slot  ( .s `  ndx )
27 snsstp2 3783 . . . . . 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 3352 . . . . . 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 3201 . . . . 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 13196 . . . 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 2353 . 2  |-  ( ( I  e.  _V  /\  R  e.  _V )  -> 
.xb  =  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  o F 
.x.  f ) ) )
33 eqid 2296 . . . . . 6  |-  (/)  =  (/)
34 fn0 5379 . . . . . 6  |-  ( (/)  Fn  (/) 
<->  (/)  =  (/) )
3533, 34mpbir 200 . . . . 5  |-  (/)  Fn  (/)
36 reldmpsr 16125 . . . . . . . . . 10  |-  Rel  dom mPwSer
3736ovprc 5901 . . . . . . . . 9  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPwSer  R )  =  (/) )
381, 37syl5eq 2340 . . . . . . . 8  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  S  =  (/) )
3938fveq2d 5545 . . . . . . 7  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( .s `  S
)  =  ( .s
`  (/) ) )
40 df-vsca 13241 . . . . . . . 8  |-  .s  = Slot  6
4140str0 13200 . . . . . . 7  |-  (/)  =  ( .s `  (/) )
4239, 19, 413eqtr4g 2353 . . . . . 6  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  -> 
.xb  =  (/) )
4336, 1, 7elbasov 13208 . . . . . . . . . 10  |-  ( f  e.  B  ->  (
I  e.  _V  /\  R  e.  _V )
)
4443con3i 127 . . . . . . . . 9  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  -.  f  e.  B
)
4544eq0rdv 3502 . . . . . . . 8  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  B  =  (/) )
4645xpeq2d 4729 . . . . . . 7  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( K  X.  B
)  =  ( K  X.  (/) ) )
47 xp0 5114 . . . . . . 7  |-  ( K  X.  (/) )  =  (/)
4846, 47syl6eq 2344 . . . . . 6  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( K  X.  B
)  =  (/) )
4942, 48fneq12d 5353 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  (  .xb  Fn  ( K  X.  B )  <->  (/)  Fn  (/) ) )
5035, 49mpbiri 224 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  -> 
.xb  Fn  ( K  X.  B ) )
51 fnov 5968 . . . 4  |-  (  .xb  Fn  ( K  X.  B
)  <->  .xb  =  ( x  e.  K ,  f  e.  B  |->  ( x 
.xb  f ) ) )
5250, 51sylib 188 . . 3  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  -> 
.xb  =  ( x  e.  K ,  f  e.  B  |->  ( x 
.xb  f ) ) )
5344pm2.21d 98 . . . . . 6  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( f  e.  B  ->  ( ( D  X.  { x } )  o F  .x.  f
)  =  ( x 
.xb  f ) ) )
5453a1d 22 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( x  e.  K  ->  ( f  e.  B  ->  ( ( D  X.  { x } )  o F  .x.  f
)  =  ( x 
.xb  f ) ) ) )
55543imp 1145 . . . 4  |-  ( ( -.  ( I  e. 
_V  /\  R  e.  _V )  /\  x  e.  K  /\  f  e.  B )  ->  (
( D  X.  {
x } )  o F  .x.  f )  =  ( x  .xb  f ) )
5655mpt2eq3dva 5928 . . 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 2331 . 2  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  -> 
.xb  =  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  o F 
.x.  f ) ) )
5832, 57pm2.61i 156 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 358    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801    u. cun 3163   (/)c0 3468   {csn 3653   {ctp 3655   <.cop 3656    X. cxp 4703   `'ccnv 4704   "cima 4708    Fn wfn 5266   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876    o Fcof 6092    ^m cmap 6788   Fincfn 6879   1c1 8754   NNcn 9762   6c6 9815   9c9 9818   NN0cn0 9981   ndxcnx 13161   Basecbs 13164   +g cplusg 13224   .rcmulr 13225  Scalarcsca 13227   .scvsca 13228  TopSetcts 13230   TopOpenctopn 13342   Xt_cpt 13359   mPwSer cmps 16103
This theorem is referenced by:  psrvsca  16152
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-psr 16114
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