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Theorem psrplusg 16437
Description: The addition operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
psrplusg.s  |-  S  =  ( I mPwSer  R )
psrplusg.b  |-  B  =  ( Base `  S
)
psrplusg.a  |-  .+  =  ( +g  `  R )
psrplusg.p  |-  .+b  =  ( +g  `  S )
Assertion
Ref Expression
psrplusg  |-  .+b  =  (  o F  .+  |`  ( B  X.  B ) )

Proof of Theorem psrplusg
Dummy variables  f 
g  k  x  h  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 psrplusg.s . . . . 5  |-  S  =  ( I mPwSer  R )
2 eqid 2435 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
3 psrplusg.a . . . . 5  |-  .+  =  ( +g  `  R )
4 eqid 2435 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
5 eqid 2435 . . . . 5  |-  ( TopOpen `  R )  =  (
TopOpen `  R )
6 eqid 2435 . . . . 5  |-  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }
7 psrplusg.b . . . . . 6  |-  B  =  ( Base `  S
)
8 simpl 444 . . . . . 6  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  I  e.  _V )
91, 2, 6, 7, 8psrbas 16435 . . . . 5  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  B  =  ( (
Base `  R )  ^m  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin } ) )
10 eqid 2435 . . . . 5  |-  (  o F  .+  |`  ( B  X.  B ) )  =  (  o F 
.+  |`  ( B  X.  B ) )
11 eqid 2435 . . . . 5  |-  ( f  e.  B ,  g  e.  B  |->  ( k  e.  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  |->  ( R  gsumg  ( x  e.  {
y  e.  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  | 
y  o R  <_ 
k }  |->  ( ( f `  x ) ( .r `  R
) ( g `  ( k  o F  -  x ) ) ) ) ) ) )  =  ( f  e.  B ,  g  e.  B  |->  ( k  e.  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  |->  ( R  gsumg  ( x  e.  {
y  e.  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  | 
y  o R  <_ 
k }  |->  ( ( f `  x ) ( .r `  R
) ( g `  ( k  o F  -  x ) ) ) ) ) ) )
12 eqid 2435 . . . . 5  |-  ( x  e.  ( Base `  R
) ,  f  e.  B  |->  ( ( { h  e.  ( NN0 
^m  I )  |  ( `' h " NN )  e.  Fin }  X.  { x }
)  o F ( .r `  R ) f ) )  =  ( x  e.  (
Base `  R ) ,  f  e.  B  |->  ( ( { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  X.  { x } )  o F ( .r
`  R ) f ) )
13 eqidd 2436 . . . . 5  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  ( Xt_ `  ( { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }  X.  { ( TopOpen `  R ) } ) )  =  ( Xt_ `  ( { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  X.  { ( TopOpen `  R
) } ) ) )
14 simpr 448 . . . . 5  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  R  e.  _V )
151, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 8, 14psrval 16421 . . . 4  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  S  =  ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  (  o F  .+  |`  ( B  X.  B ) ) >. ,  <. ( .r `  ndx ) ,  ( f  e.  B ,  g  e.  B  |->  ( k  e.  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  |->  ( R  gsumg  ( x  e.  {
y  e.  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  | 
y  o R  <_ 
k }  |->  ( ( f `  x ) ( .r `  R
) ( g `  ( k  o F  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  (
Base `  R ) ,  f  e.  B  |->  ( ( { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  X.  { x } )  o F ( .r
`  R ) f ) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( {
h  e.  ( NN0 
^m  I )  |  ( `' h " NN )  e.  Fin }  X.  { ( TopOpen `  R ) } ) ) >. } ) )
1615fveq2d 5724 . . 3  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  ( +g  `  S
)  =  ( +g  `  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  (  o F  .+  |`  ( B  X.  B
) ) >. ,  <. ( .r `  ndx ) ,  ( f  e.  B ,  g  e.  B  |->  ( k  e. 
{ h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin } 
|->  ( R  gsumg  ( x  e.  {
y  e.  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  | 
y  o R  <_ 
k }  |->  ( ( f `  x ) ( .r `  R
) ( g `  ( k  o F  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  (
Base `  R ) ,  f  e.  B  |->  ( ( { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  X.  { x } )  o F ( .r
`  R ) f ) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( {
h  e.  ( NN0 
^m  I )  |  ( `' h " NN )  e.  Fin }  X.  { ( TopOpen `  R ) } ) ) >. } ) ) )
17 psrplusg.p . . 3  |-  .+b  =  ( +g  `  S )
18 fvex 5734 . . . . . 6  |-  ( Base `  S )  e.  _V
197, 18eqeltri 2505 . . . . 5  |-  B  e. 
_V
2019, 19xpex 4982 . . . 4  |-  ( B  X.  B )  e. 
_V
21 ofexg 6301 . . . 4  |-  ( ( B  X.  B )  e.  _V  ->  (  o F  .+  |`  ( B  X.  B ) )  e.  _V )
22 psrvalstr 16422 . . . . 5  |-  ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  (  o F  .+  |`  ( B  X.  B ) ) >. ,  <. ( .r `  ndx ) ,  ( f  e.  B ,  g  e.  B  |->  ( k  e.  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  |->  ( R  gsumg  ( x  e.  {
y  e.  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  | 
y  o R  <_ 
k }  |->  ( ( f `  x ) ( .r `  R
) ( g `  ( k  o F  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  (
Base `  R ) ,  f  e.  B  |->  ( ( { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  X.  { x } )  o F ( .r
`  R ) f ) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( {
h  e.  ( NN0 
^m  I )  |  ( `' h " NN )  e.  Fin }  X.  { ( TopOpen `  R ) } ) ) >. } ) Struct  <. 1 ,  9 >.
23 plusgid 13556 . . . . 5  |-  +g  = Slot  ( +g  `  ndx )
24 snsstp2 3942 . . . . . 6  |-  { <. ( +g  `  ndx ) ,  (  o F  .+  |`  ( B  X.  B ) ) >. }  C_  { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  (  o F  .+  |`  ( B  X.  B
) ) >. ,  <. ( .r `  ndx ) ,  ( f  e.  B ,  g  e.  B  |->  ( k  e. 
{ h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin } 
|->  ( R  gsumg  ( x  e.  {
y  e.  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  | 
y  o R  <_ 
k }  |->  ( ( f `  x ) ( .r `  R
) ( g `  ( k  o F  -  x ) ) ) ) ) ) ) >. }
25 ssun1 3502 . . . . . 6  |-  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  (  o F  .+  |`  ( B  X.  B
) ) >. ,  <. ( .r `  ndx ) ,  ( f  e.  B ,  g  e.  B  |->  ( k  e. 
{ h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin } 
|->  ( R  gsumg  ( x  e.  {
y  e.  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  | 
y  o R  <_ 
k }  |->  ( ( f `  x ) ( .r `  R
) ( g `  ( k  o F  -  x ) ) ) ) ) ) ) >. }  C_  ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  (  o F  .+  |`  ( B  X.  B ) ) >. ,  <. ( .r `  ndx ) ,  ( f  e.  B ,  g  e.  B  |->  ( k  e.  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  |->  ( R  gsumg  ( x  e.  {
y  e.  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  | 
y  o R  <_ 
k }  |->  ( ( f `  x ) ( .r `  R
) ( g `  ( k  o F  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  (
Base `  R ) ,  f  e.  B  |->  ( ( { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  X.  { x } )  o F ( .r
`  R ) f ) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( {
h  e.  ( NN0 
^m  I )  |  ( `' h " NN )  e.  Fin }  X.  { ( TopOpen `  R ) } ) ) >. } )
2624, 25sstri 3349 . . . . 5  |-  { <. ( +g  `  ndx ) ,  (  o F  .+  |`  ( B  X.  B ) ) >. }  C_  ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  (  o F  .+  |`  ( B  X.  B
) ) >. ,  <. ( .r `  ndx ) ,  ( f  e.  B ,  g  e.  B  |->  ( k  e. 
{ h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin } 
|->  ( R  gsumg  ( x  e.  {
y  e.  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  | 
y  o R  <_ 
k }  |->  ( ( f `  x ) ( .r `  R
) ( g `  ( k  o F  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  (
Base `  R ) ,  f  e.  B  |->  ( ( { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  X.  { x } )  o F ( .r
`  R ) f ) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( {
h  e.  ( NN0 
^m  I )  |  ( `' h " NN )  e.  Fin }  X.  { ( TopOpen `  R ) } ) ) >. } )
2722, 23, 26strfv 13493 . . . 4  |-  ( (  o F  .+  |`  ( B  X.  B ) )  e.  _V  ->  (  o F  .+  |`  ( B  X.  B ) )  =  ( +g  `  ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  (  o F  .+  |`  ( B  X.  B ) ) >. ,  <. ( .r `  ndx ) ,  ( f  e.  B ,  g  e.  B  |->  ( k  e.  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  |->  ( R  gsumg  ( x  e.  {
y  e.  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  | 
y  o R  <_ 
k }  |->  ( ( f `  x ) ( .r `  R
) ( g `  ( k  o F  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  (
Base `  R ) ,  f  e.  B  |->  ( ( { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  X.  { x } )  o F ( .r
`  R ) f ) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( {
h  e.  ( NN0 
^m  I )  |  ( `' h " NN )  e.  Fin }  X.  { ( TopOpen `  R ) } ) ) >. } ) ) )
2820, 21, 27mp2b 10 . . 3  |-  (  o F  .+  |`  ( B  X.  B ) )  =  ( +g  `  ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  (  o F  .+  |`  ( B  X.  B ) ) >. ,  <. ( .r `  ndx ) ,  ( f  e.  B ,  g  e.  B  |->  ( k  e.  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  |->  ( R  gsumg  ( x  e.  {
y  e.  { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  | 
y  o R  <_ 
k }  |->  ( ( f `  x ) ( .r `  R
) ( g `  ( k  o F  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  (
Base `  R ) ,  f  e.  B  |->  ( ( { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  X.  { x } )  o F ( .r
`  R ) f ) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( {
h  e.  ( NN0 
^m  I )  |  ( `' h " NN )  e.  Fin }  X.  { ( TopOpen `  R ) } ) ) >. } ) )
2916, 17, 283eqtr4g 2492 . 2  |-  ( ( I  e.  _V  /\  R  e.  _V )  -> 
.+b  =  (  o F  .+  |`  ( B  X.  B ) ) )
30 reldmpsr 16420 . . . . . . 7  |-  Rel  dom mPwSer
3130ovprc 6100 . . . . . 6  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPwSer  R )  =  (/) )
321, 31syl5eq 2479 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  S  =  (/) )
3332fveq2d 5724 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( +g  `  S
)  =  ( +g  `  (/) ) )
3423str0 13497 . . . 4  |-  (/)  =  ( +g  `  (/) )
3533, 17, 343eqtr4g 2492 . . 3  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  -> 
.+b  =  (/) )
3632fveq2d 5724 . . . . . . . 8  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( Base `  S
)  =  ( Base `  (/) ) )
37 base0 13498 . . . . . . . 8  |-  (/)  =  (
Base `  (/) )
3836, 7, 373eqtr4g 2492 . . . . . . 7  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  B  =  (/) )
3938xpeq2d 4894 . . . . . 6  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( B  X.  B
)  =  ( B  X.  (/) ) )
40 xp0 5283 . . . . . 6  |-  ( B  X.  (/) )  =  (/)
4139, 40syl6eq 2483 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( B  X.  B
)  =  (/) )
4241reseq2d 5138 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  (  o F  .+  |`  ( B  X.  B
) )  =  (  o F  .+  |`  (/) ) )
43 res0 5142 . . . 4  |-  (  o F  .+  |`  (/) )  =  (/)
4442, 43syl6eq 2483 . . 3  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  (  o F  .+  |`  ( B  X.  B
) )  =  (/) )
4535, 44eqtr4d 2470 . 2  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  -> 
.+b  =  (  o F  .+  |`  ( B  X.  B ) ) )
4629, 45pm2.61i 158 1  |-  .+b  =  (  o F  .+  |`  ( B  X.  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2701   _Vcvv 2948    u. cun 3310   (/)c0 3620   {csn 3806   {ctp 3808   <.cop 3809   class class class wbr 4204    e. cmpt 4258    X. cxp 4868   `'ccnv 4869    |` cres 4872   "cima 4873   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075    o Fcof 6295    o Rcofr 6296    ^m cmap 7010   Fincfn 7101   1c1 8983    <_ cle 9113    - cmin 9283   NNcn 9992   9c9 10048   NN0cn0 10213   ndxcnx 13458   Basecbs 13461   +g cplusg 13521   .rcmulr 13522  Scalarcsca 13524   .scvsca 13525  TopSetcts 13527   TopOpenctopn 13641   Xt_cpt 13658    gsumg cgsu 13716   mPwSer cmps 16398
This theorem is referenced by:  psradd  16438  psrmulr  16440  psrsca  16445  psrvscafval  16446  psrplusgpropd  16621  ply1plusgfvi  16628
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-plusg 13534  df-mulr 13535  df-sca 13537  df-vsca 13538  df-tset 13540  df-psr 16409
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