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Theorem psrplusg 16126
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 2283 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
3 psrplusg.a . . . . 5  |-  .+  =  ( +g  `  R )
4 eqid 2283 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
5 eqid 2283 . . . . 5  |-  ( TopOpen `  R )  =  (
TopOpen `  R )
6 eqid 2283 . . . . 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 443 . . . . . 6  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  I  e.  _V )
91, 2, 6, 7, 8psrbas 16124 . . . . 5  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  B  =  ( (
Base `  R )  ^m  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin } ) )
10 eqid 2283 . . . . 5  |-  (  o F  .+  |`  ( B  X.  B ) )  =  (  o F 
.+  |`  ( B  X.  B ) )
11 eqid 2283 . . . . 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 2283 . . . . 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 2284 . . . . 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 447 . . . . 5  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  R  e.  _V )
151, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 8, 14psrval 16110 . . . 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 5529 . . 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 5539 . . . . . 6  |-  ( Base `  S )  e.  _V
197, 18eqeltri 2353 . . . . 5  |-  B  e. 
_V
2019, 19xpex 4801 . . . 4  |-  ( B  X.  B )  e. 
_V
21 ofexg 6082 . . . 4  |-  ( ( B  X.  B )  e.  _V  ->  (  o F  .+  |`  ( B  X.  B ) )  e.  _V )
22 psrvalstr 16111 . . . . 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 13243 . . . . 5  |-  +g  = Slot  ( +g  `  ndx )
24 snsstp2 3767 . . . . . 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 3338 . . . . . 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 3188 . . . . 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 13180 . . . 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 9 . . 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 2340 . 2  |-  ( ( I  e.  _V  /\  R  e.  _V )  -> 
.+b  =  (  o F  .+  |`  ( B  X.  B ) ) )
30 reldmpsr 16109 . . . . . . 7  |-  Rel  dom mPwSer
3130ovprc 5885 . . . . . 6  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPwSer  R )  =  (/) )
321, 31syl5eq 2327 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  S  =  (/) )
3332fveq2d 5529 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( +g  `  S
)  =  ( +g  `  (/) ) )
3423str0 13184 . . . 4  |-  (/)  =  ( +g  `  (/) )
3533, 17, 343eqtr4g 2340 . . 3  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  -> 
.+b  =  (/) )
3632fveq2d 5529 . . . . . . . 8  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( Base `  S
)  =  ( Base `  (/) ) )
37 base0 13185 . . . . . . . 8  |-  (/)  =  (
Base `  (/) )
3836, 7, 373eqtr4g 2340 . . . . . . 7  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  B  =  (/) )
3938xpeq2d 4713 . . . . . 6  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( B  X.  B
)  =  ( B  X.  (/) ) )
40 xp0 5098 . . . . . 6  |-  ( B  X.  (/) )  =  (/)
4139, 40syl6eq 2331 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( B  X.  B
)  =  (/) )
4241reseq2d 4955 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  (  o F  .+  |`  ( B  X.  B
) )  =  (  o F  .+  |`  (/) ) )
43 res0 4959 . . . 4  |-  (  o F  .+  |`  (/) )  =  (/)
4442, 43syl6eq 2331 . . 3  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  (  o F  .+  |`  ( B  X.  B
) )  =  (/) )
4535, 44eqtr4d 2318 . 2  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  -> 
.+b  =  (  o F  .+  |`  ( B  X.  B ) ) )
4629, 45pm2.61i 156 1  |-  .+b  =  (  o F  .+  |`  ( B  X.  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    u. cun 3150   (/)c0 3455   {csn 3640   {ctp 3642   <.cop 3643   class class class wbr 4023    e. cmpt 4077    X. cxp 4687   `'ccnv 4688    |` cres 4691   "cima 4692   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860    o Fcof 6076    o Rcofr 6077    ^m cmap 6772   Fincfn 6863   1c1 8738    <_ cle 8868    - cmin 9037   NNcn 9746   9c9 9802   NN0cn0 9965   ndxcnx 13145   Basecbs 13148   +g cplusg 13208   .rcmulr 13209  Scalarcsca 13211   .scvsca 13212  TopSetcts 13214   TopOpenctopn 13326   Xt_cpt 13343    gsumg cgsu 13401   mPwSer cmps 16087
This theorem is referenced by:  psradd  16127  psrmulr  16129  psrsca  16134  psrvscafval  16135  psrplusgpropd  16313  ply1plusgfvi  16320
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-psr 16098
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