MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  psrbasOLD Structured version   Unicode version

Theorem psrbasOLD 17797
Description: The base set of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.) Obsolete version of psrbas 17796 as of 8-Jul-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
psrbas.s  |-  S  =  ( I mPwSer  R )
psrbas.k  |-  K  =  ( Base `  R
)
psrbas.d  |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
psrbas.b  |-  B  =  ( Base `  S
)
psrbas.i  |-  ( ph  ->  I  e.  V )
Assertion
Ref Expression
psrbasOLD  |-  ( ph  ->  B  =  ( K  ^m  D ) )
Distinct variable group:    f, I
Allowed substitution hints:    ph( f)    B( f)    D( f)    R( f)    S( f)    K( f)    V( f)

Proof of Theorem psrbasOLD
Dummy variables  g  h  k  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 psrbas.s . . . . 5  |-  S  =  ( I mPwSer  R )
2 psrbas.k . . . . 5  |-  K  =  ( Base `  R
)
3 eqid 2462 . . . . 5  |-  ( +g  `  R )  =  ( +g  `  R )
4 eqid 2462 . . . . 5  |-  ( .r
`  R )  =  ( .r `  R
)
5 eqid 2462 . . . . 5  |-  ( TopOpen `  R )  =  (
TopOpen `  R )
6 psrbas.d . . . . 5  |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
7 eqidd 2463 . . . . 5  |-  ( (
ph  /\  R  e.  _V )  ->  ( K  ^m  D )  =  ( K  ^m  D
) )
8 eqid 2462 . . . . 5  |-  (  oF ( +g  `  R
)  |`  ( ( K  ^m  D )  X.  ( K  ^m  D
) ) )  =  (  oF ( +g  `  R )  |`  ( ( K  ^m  D )  X.  ( K  ^m  D ) ) )
9 eqid 2462 . . . . 5  |-  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D
)  |->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  { y  e.  D  |  y  oR  <_  k }  |->  ( ( g `  x ) ( .r
`  R ) ( h `  ( k  oF  -  x
) ) ) ) ) ) )  =  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D ) 
|->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  oR  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  oF  -  x ) ) ) ) ) ) )
10 eqid 2462 . . . . 5  |-  ( x  e.  K ,  g  e.  ( K  ^m  D )  |->  ( ( D  X.  { x } )  oF ( .r `  R
) g ) )  =  ( x  e.  K ,  g  e.  ( K  ^m  D
)  |->  ( ( D  X.  { x }
)  oF ( .r `  R ) g ) )
11 eqidd 2463 . . . . 5  |-  ( (
ph  /\  R  e.  _V )  ->  ( Xt_ `  ( D  X.  {
( TopOpen `  R ) } ) )  =  ( Xt_ `  ( D  X.  { ( TopOpen `  R ) } ) ) )
12 psrbas.i . . . . . 6  |-  ( ph  ->  I  e.  V )
1312adantr 465 . . . . 5  |-  ( (
ph  /\  R  e.  _V )  ->  I  e.  V )
14 simpr 461 . . . . 5  |-  ( (
ph  /\  R  e.  _V )  ->  R  e. 
_V )
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14psrval 17777 . . . 4  |-  ( (
ph  /\  R  e.  _V )  ->  S  =  ( { <. ( Base `  ndx ) ,  ( K  ^m  D
) >. ,  <. ( +g  `  ndx ) ,  (  oF ( +g  `  R )  |`  ( ( K  ^m  D )  X.  ( K  ^m  D ) ) ) >. ,  <. ( .r `  ndx ) ,  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D ) 
|->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  oR  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  oF  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  g  e.  ( K  ^m  D )  |->  ( ( D  X.  {
x } )  oF ( .r `  R ) g ) ) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } ) )
1615fveq2d 5863 . . 3  |-  ( (
ph  /\  R  e.  _V )  ->  ( Base `  S )  =  (
Base `  ( { <. ( Base `  ndx ) ,  ( K  ^m  D ) >. ,  <. ( +g  `  ndx ) ,  (  oF
( +g  `  R )  |`  ( ( K  ^m  D )  X.  ( K  ^m  D ) ) ) >. ,  <. ( .r `  ndx ) ,  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D ) 
|->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  oR  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  oF  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  g  e.  ( K  ^m  D )  |->  ( ( D  X.  {
x } )  oF ( .r `  R ) g ) ) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } ) ) )
17 psrbas.b . . 3  |-  B  =  ( Base `  S
)
18 ovex 6302 . . . 4  |-  ( K  ^m  D )  e. 
_V
19 psrvalstr 17778 . . . . 5  |-  ( {
<. ( Base `  ndx ) ,  ( K  ^m  D ) >. ,  <. ( +g  `  ndx ) ,  (  oF
( +g  `  R )  |`  ( ( K  ^m  D )  X.  ( K  ^m  D ) ) ) >. ,  <. ( .r `  ndx ) ,  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D ) 
|->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  oR  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  oF  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  g  e.  ( K  ^m  D )  |->  ( ( D  X.  {
x } )  oF ( .r `  R ) g ) ) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } ) Struct  <. 1 ,  9 >.
20 baseid 14527 . . . . 5  |-  Base  = Slot  ( Base `  ndx )
21 snsstp1 4173 . . . . . 6  |-  { <. (
Base `  ndx ) ,  ( K  ^m  D
) >. }  C_  { <. (
Base `  ndx ) ,  ( K  ^m  D
) >. ,  <. ( +g  `  ndx ) ,  (  oF ( +g  `  R )  |`  ( ( K  ^m  D )  X.  ( K  ^m  D ) ) ) >. ,  <. ( .r `  ndx ) ,  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D ) 
|->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  oR  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  oF  -  x ) ) ) ) ) ) ) >. }
22 ssun1 3662 . . . . . 6  |-  { <. (
Base `  ndx ) ,  ( K  ^m  D
) >. ,  <. ( +g  `  ndx ) ,  (  oF ( +g  `  R )  |`  ( ( K  ^m  D )  X.  ( K  ^m  D ) ) ) >. ,  <. ( .r `  ndx ) ,  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D ) 
|->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  oR  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  oF  -  x ) ) ) ) ) ) ) >. }  C_  ( { <. ( Base `  ndx ) ,  ( K  ^m  D ) >. ,  <. ( +g  `  ndx ) ,  (  oF
( +g  `  R )  |`  ( ( K  ^m  D )  X.  ( K  ^m  D ) ) ) >. ,  <. ( .r `  ndx ) ,  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D ) 
|->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  oR  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  oF  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  g  e.  ( K  ^m  D )  |->  ( ( D  X.  {
x } )  oF ( .r `  R ) g ) ) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } )
2321, 22sstri 3508 . . . . 5  |-  { <. (
Base `  ndx ) ,  ( K  ^m  D
) >. }  C_  ( { <. ( Base `  ndx ) ,  ( K  ^m  D ) >. ,  <. ( +g  `  ndx ) ,  (  oF
( +g  `  R )  |`  ( ( K  ^m  D )  X.  ( K  ^m  D ) ) ) >. ,  <. ( .r `  ndx ) ,  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D ) 
|->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  oR  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  oF  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  g  e.  ( K  ^m  D )  |->  ( ( D  X.  {
x } )  oF ( .r `  R ) g ) ) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } )
2419, 20, 23strfv 14515 . . . 4  |-  ( ( K  ^m  D )  e.  _V  ->  ( K  ^m  D )  =  ( Base `  ( { <. ( Base `  ndx ) ,  ( K  ^m  D ) >. ,  <. ( +g  `  ndx ) ,  (  oF
( +g  `  R )  |`  ( ( K  ^m  D )  X.  ( K  ^m  D ) ) ) >. ,  <. ( .r `  ndx ) ,  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D ) 
|->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  oR  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  oF  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  g  e.  ( K  ^m  D )  |->  ( ( D  X.  {
x } )  oF ( .r `  R ) g ) ) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } ) ) )
2518, 24ax-mp 5 . . 3  |-  ( K  ^m  D )  =  ( Base `  ( { <. ( Base `  ndx ) ,  ( K  ^m  D ) >. ,  <. ( +g  `  ndx ) ,  (  oF
( +g  `  R )  |`  ( ( K  ^m  D )  X.  ( K  ^m  D ) ) ) >. ,  <. ( .r `  ndx ) ,  ( g  e.  ( K  ^m  D ) ,  h  e.  ( K  ^m  D ) 
|->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
y  e.  D  | 
y  oR  <_ 
k }  |->  ( ( g `  x ) ( .r `  R
) ( h `  ( k  oF  -  x ) ) ) ) ) ) ) >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s `  ndx ) ,  ( x  e.  K ,  g  e.  ( K  ^m  D )  |->  ( ( D  X.  {
x } )  oF ( .r `  R ) g ) ) >. ,  <. (TopSet ` 
ndx ) ,  (
Xt_ `  ( D  X.  { ( TopOpen `  R
) } ) )
>. } ) )
2616, 17, 253eqtr4g 2528 . 2  |-  ( (
ph  /\  R  e.  _V )  ->  B  =  ( K  ^m  D
) )
27 reldmpsr 17776 . . . . . . . 8  |-  Rel  dom mPwSer
2827ovprc2 6306 . . . . . . 7  |-  ( -.  R  e.  _V  ->  ( I mPwSer  R )  =  (/) )
2928adantl 466 . . . . . 6  |-  ( (
ph  /\  -.  R  e.  _V )  ->  (
I mPwSer  R )  =  (/) )
301, 29syl5eq 2515 . . . . 5  |-  ( (
ph  /\  -.  R  e.  _V )  ->  S  =  (/) )
3130fveq2d 5863 . . . 4  |-  ( (
ph  /\  -.  R  e.  _V )  ->  ( Base `  S )  =  ( Base `  (/) ) )
32 base0 14520 . . . 4  |-  (/)  =  (
Base `  (/) )
3331, 17, 323eqtr4g 2528 . . 3  |-  ( (
ph  /\  -.  R  e.  _V )  ->  B  =  (/) )
34 fvprc 5853 . . . . . 6  |-  ( -.  R  e.  _V  ->  (
Base `  R )  =  (/) )
3534adantl 466 . . . . 5  |-  ( (
ph  /\  -.  R  e.  _V )  ->  ( Base `  R )  =  (/) )
362, 35syl5eq 2515 . . . 4  |-  ( (
ph  /\  -.  R  e.  _V )  ->  K  =  (/) )
37 0nn0 10801 . . . . . . . 8  |-  0  e.  NN0
3837a1i 11 . . . . . . 7  |-  ( ( ( ph  /\  -.  R  e.  _V )  /\  x  e.  I
)  ->  0  e.  NN0 )
39 eqid 2462 . . . . . . 7  |-  ( x  e.  I  |->  0 )  =  ( x  e.  I  |->  0 )
4038, 39fmptd 6038 . . . . . 6  |-  ( (
ph  /\  -.  R  e.  _V )  ->  (
x  e.  I  |->  0 ) : I --> NN0 )
41 0fin 7739 . . . . . . 7  |-  (/)  e.  Fin
42 nn0suppOLD 10841 . . . . . . . . 9  |-  ( ( x  e.  I  |->  0 ) : I --> NN0  ->  ( `' ( x  e.  I  |->  0 ) "
( _V  \  {
0 } ) )  =  ( `' ( x  e.  I  |->  0 ) " NN ) )
4340, 42syl 16 . . . . . . . 8  |-  ( (
ph  /\  -.  R  e.  _V )  ->  ( `' ( x  e.  I  |->  0 ) "
( _V  \  {
0 } ) )  =  ( `' ( x  e.  I  |->  0 ) " NN ) )
44 eqidd 2463 . . . . . . . . 9  |-  ( ( ( ph  /\  -.  R  e.  _V )  /\  x  e.  (
I  \  (/) ) )  ->  0  =  0 )
4544suppss2OLD 6507 . . . . . . . 8  |-  ( (
ph  /\  -.  R  e.  _V )  ->  ( `' ( x  e.  I  |->  0 ) "
( _V  \  {
0 } ) ) 
C_  (/) )
4643, 45eqsstr3d 3534 . . . . . . 7  |-  ( (
ph  /\  -.  R  e.  _V )  ->  ( `' ( x  e.  I  |->  0 ) " NN )  C_  (/) )
47 ssfi 7732 . . . . . . 7  |-  ( (
(/)  e.  Fin  /\  ( `' ( x  e.  I  |->  0 ) " NN )  C_  (/) )  -> 
( `' ( x  e.  I  |->  0 )
" NN )  e. 
Fin )
4841, 46, 47sylancr 663 . . . . . 6  |-  ( (
ph  /\  -.  R  e.  _V )  ->  ( `' ( x  e.  I  |->  0 ) " NN )  e.  Fin )
496psrbag 17779 . . . . . . . 8  |-  ( I  e.  V  ->  (
( x  e.  I  |->  0 )  e.  D  <->  ( ( x  e.  I  |->  0 ) : I --> NN0  /\  ( `' ( x  e.  I  |->  0 ) " NN )  e.  Fin )
) )
5012, 49syl 16 . . . . . . 7  |-  ( ph  ->  ( ( x  e.  I  |->  0 )  e.  D  <->  ( ( x  e.  I  |->  0 ) : I --> NN0  /\  ( `' ( x  e.  I  |->  0 ) " NN )  e.  Fin ) ) )
5150adantr 465 . . . . . 6  |-  ( (
ph  /\  -.  R  e.  _V )  ->  (
( x  e.  I  |->  0 )  e.  D  <->  ( ( x  e.  I  |->  0 ) : I --> NN0  /\  ( `' ( x  e.  I  |->  0 ) " NN )  e.  Fin )
) )
5240, 48, 51mpbir2and 915 . . . . 5  |-  ( (
ph  /\  -.  R  e.  _V )  ->  (
x  e.  I  |->  0 )  e.  D )
53 ne0i 3786 . . . . 5  |-  ( ( x  e.  I  |->  0 )  e.  D  ->  D  =/=  (/) )
5452, 53syl 16 . . . 4  |-  ( (
ph  /\  -.  R  e.  _V )  ->  D  =/=  (/) )
55 fvex 5869 . . . . . 6  |-  ( Base `  R )  e.  _V
562, 55eqeltri 2546 . . . . 5  |-  K  e. 
_V
57 ovex 6302 . . . . . . 7  |-  ( NN0 
^m  I )  e. 
_V
5857rabex 4593 . . . . . 6  |-  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  e.  _V
596, 58eqeltri 2546 . . . . 5  |-  D  e. 
_V
6056, 59map0 7451 . . . 4  |-  ( ( K  ^m  D )  =  (/)  <->  ( K  =  (/)  /\  D  =/=  (/) ) )
6136, 54, 60sylanbrc 664 . . 3  |-  ( (
ph  /\  -.  R  e.  _V )  ->  ( K  ^m  D )  =  (/) )
6233, 61eqtr4d 2506 . 2  |-  ( (
ph  /\  -.  R  e.  _V )  ->  B  =  ( K  ^m  D ) )
6326, 62pm2.61dan 789 1  |-  ( ph  ->  B  =  ( K  ^m  D ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2657   {crab 2813   _Vcvv 3108    \ cdif 3468    u. cun 3469    C_ wss 3471   (/)c0 3780   {csn 4022   {ctp 4026   <.cop 4028   class class class wbr 4442    |-> cmpt 4500    X. cxp 4992   `'ccnv 4993    |` cres 4996   "cima 4997   -->wf 5577   ` cfv 5581  (class class class)co 6277    |-> cmpt2 6279    oFcof 6515    oRcofr 6516    ^m cmap 7412   Fincfn 7508   0cc0 9483   1c1 9484    <_ cle 9620    - cmin 9796   NNcn 10527   9c9 10583   NN0cn0 10786   ndxcnx 14478   Basecbs 14481   +g cplusg 14546   .rcmulr 14547  Scalarcsca 14549   .scvsca 14550  TopSetcts 14552   TopOpenctopn 14668   Xt_cpt 14685    gsumg cgsu 14687   mPwSer cmps 17766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6517  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-en 7509  df-dom 7510  df-sdom 7511  df-fin 7512  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-nn 10528  df-2 10585  df-3 10586  df-4 10587  df-5 10588  df-6 10589  df-7 10590  df-8 10591  df-9 10592  df-n0 10787  df-z 10856  df-uz 11074  df-fz 11664  df-struct 14483  df-ndx 14484  df-slot 14485  df-base 14486  df-plusg 14559  df-mulr 14560  df-sca 14562  df-vsca 14563  df-tset 14565  df-psr 17771
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator