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Theorem psrplusg 17576
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  =  (  oF  .+  |`  ( 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 2454 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
3 psrplusg.a . . . . 5  |-  .+  =  ( +g  `  R )
4 eqid 2454 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
5 eqid 2454 . . . . 5  |-  ( TopOpen `  R )  =  (
TopOpen `  R )
6 eqid 2454 . . . . 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 457 . . . . . 6  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  I  e.  _V )
91, 2, 6, 7, 8psrbas 17572 . . . . 5  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  B  =  ( (
Base `  R )  ^m  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin } ) )
10 eqid 2454 . . . . 5  |-  (  oF  .+  |`  ( B  X.  B ) )  =  (  oF  .+  |`  ( B  X.  B ) )
11 eqid 2454 . . . . 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  oR  <_ 
k }  |->  ( ( f `  x ) ( .r `  R
) ( g `  ( k  oF  -  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  oR  <_ 
k }  |->  ( ( f `  x ) ( .r `  R
) ( g `  ( k  oF  -  x ) ) ) ) ) ) )
12 eqid 2454 . . . . 5  |-  ( x  e.  ( Base `  R
) ,  f  e.  B  |->  ( ( { h  e.  ( NN0 
^m  I )  |  ( `' h " NN )  e.  Fin }  X.  { x }
)  oF ( .r `  R ) f ) )  =  ( x  e.  (
Base `  R ) ,  f  e.  B  |->  ( ( { h  e.  ( NN0  ^m  I
)  |  ( `' h " NN )  e.  Fin }  X.  { x } )  oF ( .r
`  R ) f ) )
13 eqidd 2455 . . . . 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 461 . . . . 5  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  R  e.  _V )
151, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 8, 14psrval 17553 . . . 4  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  S  =  ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  (  oF  .+  |`  ( 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  oR  <_ 
k }  |->  ( ( f `  x ) ( .r `  R
) ( g `  ( k  oF  -  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 } )  oF ( .r
`  R ) f ) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( {
h  e.  ( NN0 
^m  I )  |  ( `' h " NN )  e.  Fin }  X.  { ( TopOpen `  R ) } ) ) >. } ) )
1615fveq2d 5804 . . 3  |-  ( ( I  e.  _V  /\  R  e.  _V )  ->  ( +g  `  S
)  =  ( +g  `  ( { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  (  oF  .+  |`  ( 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  oR  <_ 
k }  |->  ( ( f `  x ) ( .r `  R
) ( g `  ( k  oF  -  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 } )  oF ( .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 5810 . . . . . 6  |-  ( Base `  S )  e.  _V
197, 18eqeltri 2538 . . . . 5  |-  B  e. 
_V
2019, 19xpex 6619 . . . 4  |-  ( B  X.  B )  e. 
_V
21 ofexg 6435 . . . 4  |-  ( ( B  X.  B )  e.  _V  ->  (  oF  .+  |`  ( B  X.  B ) )  e.  _V )
22 psrvalstr 17554 . . . . 5  |-  ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  (  oF  .+  |`  ( 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  oR  <_ 
k }  |->  ( ( f `  x ) ( .r `  R
) ( g `  ( k  oF  -  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 } )  oF ( .r
`  R ) f ) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( {
h  e.  ( NN0 
^m  I )  |  ( `' h " NN )  e.  Fin }  X.  { ( TopOpen `  R ) } ) ) >. } ) Struct  <. 1 ,  9 >.
23 plusgid 14393 . . . . 5  |-  +g  = Slot  ( +g  `  ndx )
24 snsstp2 4134 . . . . . 6  |-  { <. ( +g  `  ndx ) ,  (  oF  .+  |`  ( B  X.  B ) ) >. }  C_  { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  (  oF  .+  |`  ( 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  oR  <_ 
k }  |->  ( ( f `  x ) ( .r `  R
) ( g `  ( k  oF  -  x ) ) ) ) ) ) ) >. }
25 ssun1 3628 . . . . . 6  |-  { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  (  oF  .+  |`  ( 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  oR  <_ 
k }  |->  ( ( f `  x ) ( .r `  R
) ( g `  ( k  oF  -  x ) ) ) ) ) ) ) >. }  C_  ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  (  oF  .+  |`  ( 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  oR  <_ 
k }  |->  ( ( f `  x ) ( .r `  R
) ( g `  ( k  oF  -  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 } )  oF ( .r
`  R ) f ) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( {
h  e.  ( NN0 
^m  I )  |  ( `' h " NN )  e.  Fin }  X.  { ( TopOpen `  R ) } ) ) >. } )
2624, 25sstri 3474 . . . . 5  |-  { <. ( +g  `  ndx ) ,  (  oF  .+  |`  ( B  X.  B ) ) >. }  C_  ( { <. (
Base `  ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  (  oF  .+  |`  ( 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  oR  <_ 
k }  |->  ( ( f `  x ) ( .r `  R
) ( g `  ( k  oF  -  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 } )  oF ( .r
`  R ) f ) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( {
h  e.  ( NN0 
^m  I )  |  ( `' h " NN )  e.  Fin }  X.  { ( TopOpen `  R ) } ) ) >. } )
2722, 23, 26strfv 14327 . . . 4  |-  ( (  oF  .+  |`  ( B  X.  B ) )  e.  _V  ->  (  oF  .+  |`  ( B  X.  B ) )  =  ( +g  `  ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  (  oF  .+  |`  ( 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  oR  <_ 
k }  |->  ( ( f `  x ) ( .r `  R
) ( g `  ( k  oF  -  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 } )  oF ( .r
`  R ) f ) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( {
h  e.  ( NN0 
^m  I )  |  ( `' h " NN )  e.  Fin }  X.  { ( TopOpen `  R ) } ) ) >. } ) ) )
2820, 21, 27mp2b 10 . . 3  |-  (  oF  .+  |`  ( B  X.  B ) )  =  ( +g  `  ( { <. ( Base `  ndx ) ,  B >. , 
<. ( +g  `  ndx ) ,  (  oF  .+  |`  ( 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  oR  <_ 
k }  |->  ( ( f `  x ) ( .r `  R
) ( g `  ( k  oF  -  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 } )  oF ( .r
`  R ) f ) ) >. ,  <. (TopSet `  ndx ) ,  (
Xt_ `  ( {
h  e.  ( NN0 
^m  I )  |  ( `' h " NN )  e.  Fin }  X.  { ( TopOpen `  R ) } ) ) >. } ) )
2916, 17, 283eqtr4g 2520 . 2  |-  ( ( I  e.  _V  /\  R  e.  _V )  -> 
.+b  =  (  oF  .+  |`  ( B  X.  B ) ) )
30 reldmpsr 17552 . . . . . . 7  |-  Rel  dom mPwSer
3130ovprc 6228 . . . . . 6  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( I mPwSer  R )  =  (/) )
321, 31syl5eq 2507 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  S  =  (/) )
3332fveq2d 5804 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( +g  `  S
)  =  ( +g  `  (/) ) )
3423str0 14331 . . . 4  |-  (/)  =  ( +g  `  (/) )
3533, 17, 343eqtr4g 2520 . . 3  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  -> 
.+b  =  (/) )
3632fveq2d 5804 . . . . . . . 8  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( Base `  S
)  =  ( Base `  (/) ) )
37 base0 14332 . . . . . . . 8  |-  (/)  =  (
Base `  (/) )
3836, 7, 373eqtr4g 2520 . . . . . . 7  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  B  =  (/) )
3938xpeq2d 4973 . . . . . 6  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( B  X.  B
)  =  ( B  X.  (/) ) )
40 xp0 5365 . . . . . 6  |-  ( B  X.  (/) )  =  (/)
4139, 40syl6eq 2511 . . . . 5  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  ( B  X.  B
)  =  (/) )
4241reseq2d 5219 . . . 4  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  (  oF  .+  |`  ( B  X.  B
) )  =  (  oF  .+  |`  (/) ) )
43 res0 5224 . . . 4  |-  (  oF  .+  |`  (/) )  =  (/)
4442, 43syl6eq 2511 . . 3  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  ->  (  oF  .+  |`  ( B  X.  B
) )  =  (/) )
4535, 44eqtr4d 2498 . 2  |-  ( -.  ( I  e.  _V  /\  R  e.  _V )  -> 
.+b  =  (  oF  .+  |`  ( B  X.  B ) ) )
4629, 45pm2.61i 164 1  |-  .+b  =  (  oF  .+  |`  ( B  X.  B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    = wceq 1370    e. wcel 1758   {crab 2803   _Vcvv 3078    u. cun 3435   (/)c0 3746   {csn 3986   {ctp 3990   <.cop 3992   class class class wbr 4401    |-> cmpt 4459    X. cxp 4947   `'ccnv 4948    |` cres 4951   "cima 4952   ` cfv 5527  (class class class)co 6201    |-> cmpt2 6203    oFcof 6429    oRcofr 6430    ^m cmap 7325   Fincfn 7421   1c1 9395    <_ cle 9531    - cmin 9707   NNcn 10434   9c9 10490   NN0cn0 10691   ndxcnx 14290   Basecbs 14293   +g cplusg 14358   .rcmulr 14359  Scalarcsca 14361   .scvsca 14362  TopSetcts 14364   TopOpenctopn 14480   Xt_cpt 14497    gsumg cgsu 14499   mPwSer cmps 17542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-of 6431  df-om 6588  df-1st 6688  df-2nd 6689  df-supp 6802  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-map 7327  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-fsupp 7733  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-3 10493  df-4 10494  df-5 10495  df-6 10496  df-7 10497  df-8 10498  df-9 10499  df-n0 10692  df-z 10759  df-uz 10974  df-fz 11556  df-struct 14295  df-ndx 14296  df-slot 14297  df-base 14298  df-plusg 14371  df-mulr 14372  df-sca 14374  df-vsca 14375  df-tset 14377  df-psr 17547
This theorem is referenced by:  psradd  17577  psrmulr  17579  psrsca  17584  psrvscafval  17585  psrplusgpropd  17815  ply1plusgfvi  17821
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