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Theorem psrplusg 16450
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 2438 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
3 psrplusg.a . . . . 5  |-  .+  =  ( +g  `  R )
4 eqid 2438 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
5 eqid 2438 . . . . 5  |-  ( TopOpen `  R )  =  (
TopOpen `  R )
6 eqid 2438 . . . . 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 445 . . . . . 6  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  I  e.  _V )
91, 2, 6, 7, 8psrbas 16448 . . . . 5  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  B  =  ( (
Base `  R )  ^m  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin } ) )
10 eqid 2438 . . . . 5  |-  (  o F  .+  |`  ( B  X.  B ) )  =  (  o F 
.+  |`  ( B  X.  B ) )
11 eqid 2438 . . . . 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 2438 . . . . 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 2439 . . . . 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 449 . . . . 5  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  R  e.  _V )
151, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 8, 14psrval 16434 . . . 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 5735 . . 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 5745 . . . . . 6  |-  ( Base `  S )  e.  _V
197, 18eqeltri 2508 . . . . 5  |-  B  e. 
_V
2019, 19xpex 4993 . . . 4  |-  ( B  X.  B )  e. 
_V
21 ofexg 6312 . . . 4  |-  ( ( B  X.  B )  e.  _V  ->  (  o F  .+  |`  ( B  X.  B ) )  e.  _V )
22 psrvalstr 16435 . . . . 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 13569 . . . . 5  |-  +g  = Slot  ( +g  `  ndx )
24 snsstp2 3952 . . . . . 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 3512 . . . . . 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 3359 . . . . 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 13506 . . . 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 2495 . 2  |-  ( ( I  e.  _V  /\  R  e.  _V )  -> 
.+b  =  (  o F  .+  |`  ( B  X.  B ) ) )
30 reldmpsr 16433 . . . . . . 7  |-  Rel  dom mPwSer
3130ovprc 6111 . . . . . 6  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPwSer  R )  =  (/) )
321, 31syl5eq 2482 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  S  =  (/) )
3332fveq2d 5735 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( +g  `  S
)  =  ( +g  `  (/) ) )
3423str0 13510 . . . 4  |-  (/)  =  ( +g  `  (/) )
3533, 17, 343eqtr4g 2495 . . 3  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  -> 
.+b  =  (/) )
3632fveq2d 5735 . . . . . . . 8  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( Base `  S
)  =  ( Base `  (/) ) )
37 base0 13511 . . . . . . . 8  |-  (/)  =  (
Base `  (/) )
3836, 7, 373eqtr4g 2495 . . . . . . 7  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  B  =  (/) )
3938xpeq2d 4905 . . . . . 6  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( B  X.  B
)  =  ( B  X.  (/) ) )
40 xp0 5294 . . . . . 6  |-  ( B  X.  (/) )  =  (/)
4139, 40syl6eq 2486 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( B  X.  B
)  =  (/) )
4241reseq2d 5149 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  (  o F  .+  |`  ( B  X.  B
) )  =  (  o F  .+  |`  (/) ) )
43 res0 5153 . . . 4  |-  (  o F  .+  |`  (/) )  =  (/)
4442, 43syl6eq 2486 . . 3  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  (  o F  .+  |`  ( B  X.  B
) )  =  (/) )
4535, 44eqtr4d 2473 . 2  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  -> 
.+b  =  (  o F  .+  |`  ( B  X.  B ) ) )
4629, 45pm2.61i 159 1  |-  .+b  =  (  o F  .+  |`  ( B  X.  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 360    = wceq 1653    e. wcel 1726   {crab 2711   _Vcvv 2958    u. cun 3320   (/)c0 3630   {csn 3816   {ctp 3818   <.cop 3819   class class class wbr 4215    e. cmpt 4269    X. cxp 4879   `'ccnv 4880    |` cres 4883   "cima 4884   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086    o Fcof 6306    o Rcofr 6307    ^m cmap 7021   Fincfn 7112   1c1 8996    <_ cle 9126    - cmin 9296   NNcn 10005   9c9 10061   NN0cn0 10226   ndxcnx 13471   Basecbs 13474   +g cplusg 13534   .rcmulr 13535  Scalarcsca 13537   .scvsca 13538  TopSetcts 13540   TopOpenctopn 13654   Xt_cpt 13671    gsumg cgsu 13729   mPwSer cmps 16411
This theorem is referenced by:  psradd  16451  psrmulr  16453  psrsca  16458  psrvscafval  16459  psrplusgpropd  16634  ply1plusgfvi  16641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-7 10068  df-8 10069  df-9 10070  df-n0 10227  df-z 10288  df-uz 10494  df-fz 11049  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-plusg 13547  df-mulr 13548  df-sca 13550  df-vsca 13551  df-tset 13553  df-psr 16422
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