MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frgpnabllem2 Structured version   Unicode version

Theorem frgpnabllem2 16373
Description: Lemma for frgpnabl 16374. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
frgpnabl.g  |-  G  =  (freeGrp `  I )
frgpnabl.w  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
frgpnabl.r  |-  .~  =  ( ~FG  `  I )
frgpnabl.p  |-  .+  =  ( +g  `  G )
frgpnabl.m  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
frgpnabl.t  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
frgpnabl.d  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
frgpnabl.u  |-  U  =  (varFGrp `  I )
frgpnabl.i  |-  ( ph  ->  I  e.  _V )
frgpnabl.a  |-  ( ph  ->  A  e.  I )
frgpnabl.b  |-  ( ph  ->  B  e.  I )
frgpnabl.n  |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B )
)  =  ( ( U `  B ) 
.+  ( U `  A ) ) )
Assertion
Ref Expression
frgpnabllem2  |-  ( ph  ->  A  =  B )
Distinct variable groups:    x, A    v, n, w, x, y, z, I    ph, x    x, 
.~ , y, z    x, B    n, W, v, w, x, y, z    x, G    n, M, v, w, x    x, T
Allowed substitution hints:    ph( y, z, w, v, n)    A( y, z, w, v, n)    B( y, z, w, v, n)    D( x, y, z, w, v, n)    .+ ( x, y, z, w, v, n)    .~ ( w, v, n)    T( y, z, w, v, n)    U( x, y, z, w, v, n)    G( y,
z, w, v, n)    M( y, z)

Proof of Theorem frgpnabllem2
Dummy variables  d  m  t  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgpnabl.a . 2  |-  ( ph  ->  A  e.  I )
2 0ex 4443 . . 3  |-  (/)  e.  _V
32a1i 11 . 2  |-  ( ph  -> 
(/)  e.  _V )
4 frgpnabl.d . . . . . . . 8  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
5 difss 3504 . . . . . . . 8  |-  ( W 
\  U_ x  e.  W  ran  ( T `  x
) )  C_  W
64, 5eqsstri 3407 . . . . . . 7  |-  D  C_  W
7 inss1 3591 . . . . . . . 8  |-  ( D  i^i  ( ( U `
 B )  .+  ( U `  A ) ) )  C_  D
8 frgpnabl.g . . . . . . . . 9  |-  G  =  (freeGrp `  I )
9 frgpnabl.w . . . . . . . . 9  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
10 frgpnabl.r . . . . . . . . 9  |-  .~  =  ( ~FG  `  I )
11 frgpnabl.p . . . . . . . . 9  |-  .+  =  ( +g  `  G )
12 frgpnabl.m . . . . . . . . 9  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
13 frgpnabl.t . . . . . . . . 9  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
14 frgpnabl.u . . . . . . . . 9  |-  U  =  (varFGrp `  I )
15 frgpnabl.i . . . . . . . . 9  |-  ( ph  ->  I  e.  _V )
16 frgpnabl.b . . . . . . . . 9  |-  ( ph  ->  B  e.  I )
178, 9, 10, 11, 12, 13, 4, 14, 15, 16, 1frgpnabllem1 16372 . . . . . . . 8  |-  ( ph  ->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  e.  ( D  i^i  ( ( U `
 B )  .+  ( U `  A ) ) ) )
187, 17sseldi 3375 . . . . . . 7  |-  ( ph  ->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  e.  D )
196, 18sseldi 3375 . . . . . 6  |-  ( ph  ->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  e.  W )
20 eqid 2443 . . . . . . 7  |-  ( m  e.  { t  e.  (Word  W  \  { (/)
} )  |  ( ( t `  0
)  e.  D  /\  A. k  e.  ( 1..^ ( # `  t
) ) ( t `
 k )  e. 
ran  ( T `  ( t `  (
k  -  1 ) ) ) ) } 
|->  ( m `  (
( # `  m )  -  1 ) ) )  =  ( m  e.  { t  e.  (Word  W  \  { (/)
} )  |  ( ( t `  0
)  e.  D  /\  A. k  e.  ( 1..^ ( # `  t
) ) ( t `
 k )  e. 
ran  ( T `  ( t `  (
k  -  1 ) ) ) ) } 
|->  ( m `  (
( # `  m )  -  1 ) ) )
219, 10, 12, 13, 4, 20efgredeu 16270 . . . . . 6  |-  ( <" <. B ,  (/) >. <. A ,  (/) >. ">  e.  W  ->  E! d  e.  D  d  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> )
22 reurmo 2959 . . . . . 6  |-  ( E! d  e.  D  d  .~  <" <. B ,  (/)
>. <. A ,  (/) >. ">  ->  E* d  e.  D  d  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> )
2319, 21, 223syl 20 . . . . 5  |-  ( ph  ->  E* d  e.  D  d  .~  <" <. B ,  (/)
>. <. A ,  (/) >. "> )
24 inss1 3591 . . . . . 6  |-  ( D  i^i  ( ( U `
 A )  .+  ( U `  B ) ) )  C_  D
258, 9, 10, 11, 12, 13, 4, 14, 15, 1, 16frgpnabllem1 16372 . . . . . 6  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  ( D  i^i  ( ( U `
 A )  .+  ( U `  B ) ) ) )
2624, 25sseldi 3375 . . . . 5  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  D )
279, 10efger 16236 . . . . . . . . 9  |-  .~  Er  W
2827a1i 11 . . . . . . . 8  |-  ( ph  ->  .~  Er  W )
298frgpgrp 16280 . . . . . . . . . . 11  |-  ( I  e.  _V  ->  G  e.  Grp )
3015, 29syl 16 . . . . . . . . . 10  |-  ( ph  ->  G  e.  Grp )
31 eqid 2443 . . . . . . . . . . . . 13  |-  ( Base `  G )  =  (
Base `  G )
3210, 14, 8, 31vrgpf 16286 . . . . . . . . . . . 12  |-  ( I  e.  _V  ->  U : I --> ( Base `  G ) )
3315, 32syl 16 . . . . . . . . . . 11  |-  ( ph  ->  U : I --> ( Base `  G ) )
3433, 1ffvelrnd 5865 . . . . . . . . . 10  |-  ( ph  ->  ( U `  A
)  e.  ( Base `  G ) )
3533, 16ffvelrnd 5865 . . . . . . . . . 10  |-  ( ph  ->  ( U `  B
)  e.  ( Base `  G ) )
3631, 11grpcl 15572 . . . . . . . . . 10  |-  ( ( G  e.  Grp  /\  ( U `  A )  e.  ( Base `  G
)  /\  ( U `  B )  e.  (
Base `  G )
)  ->  ( ( U `  A )  .+  ( U `  B
) )  e.  (
Base `  G )
)
3730, 34, 35, 36syl3anc 1218 . . . . . . . . 9  |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B )
)  e.  ( Base `  G ) )
38 eqid 2443 . . . . . . . . . . . 12  |-  (freeMnd `  (
I  X.  2o ) )  =  (freeMnd `  (
I  X.  2o ) )
398, 38, 10frgpval 16276 . . . . . . . . . . 11  |-  ( I  e.  _V  ->  G  =  ( (freeMnd `  (
I  X.  2o ) )  /.s 
.~  ) )
4015, 39syl 16 . . . . . . . . . 10  |-  ( ph  ->  G  =  ( (freeMnd `  ( I  X.  2o ) )  /.s  .~  )
)
41 2on 6949 . . . . . . . . . . . . . 14  |-  2o  e.  On
42 xpexg 6528 . . . . . . . . . . . . . 14  |-  ( ( I  e.  _V  /\  2o  e.  On )  -> 
( I  X.  2o )  e.  _V )
4315, 41, 42sylancl 662 . . . . . . . . . . . . 13  |-  ( ph  ->  ( I  X.  2o )  e.  _V )
44 wrdexg 12265 . . . . . . . . . . . . 13  |-  ( ( I  X.  2o )  e.  _V  -> Word  ( I  X.  2o )  e. 
_V )
45 fvi 5769 . . . . . . . . . . . . 13  |-  (Word  (
I  X.  2o )  e.  _V  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  (
I  X.  2o ) )
4643, 44, 453syl 20 . . . . . . . . . . . 12  |-  ( ph  ->  (  _I  ` Word  ( I  X.  2o ) )  = Word  ( I  X.  2o ) )
479, 46syl5eq 2487 . . . . . . . . . . 11  |-  ( ph  ->  W  = Word  ( I  X.  2o ) )
48 eqid 2443 . . . . . . . . . . . . 13  |-  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) )
4938, 48frmdbas 15551 . . . . . . . . . . . 12  |-  ( ( I  X.  2o )  e.  _V  ->  ( Base `  (freeMnd `  (
I  X.  2o ) ) )  = Word  (
I  X.  2o ) )
5043, 49syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( Base `  (freeMnd `  ( I  X.  2o ) ) )  = Word  ( I  X.  2o ) )
5147, 50eqtr4d 2478 . . . . . . . . . 10  |-  ( ph  ->  W  =  ( Base `  (freeMnd `  ( I  X.  2o ) ) ) )
52 fvex 5722 . . . . . . . . . . . 12  |-  ( ~FG  `  I
)  e.  _V
5310, 52eqeltri 2513 . . . . . . . . . . 11  |-  .~  e.  _V
5453a1i 11 . . . . . . . . . 10  |-  ( ph  ->  .~  e.  _V )
55 fvex 5722 . . . . . . . . . . 11  |-  (freeMnd `  (
I  X.  2o ) )  e.  _V
5655a1i 11 . . . . . . . . . 10  |-  ( ph  ->  (freeMnd `  ( I  X.  2o ) )  e. 
_V )
5740, 51, 54, 56divsbas 14504 . . . . . . . . 9  |-  ( ph  ->  ( W /.  .~  )  =  ( Base `  G ) )
5837, 57eleqtrrd 2520 . . . . . . . 8  |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B )
)  e.  ( W /.  .~  ) )
59 inss2 3592 . . . . . . . . 9  |-  ( D  i^i  ( ( U `
 A )  .+  ( U `  B ) ) )  C_  (
( U `  A
)  .+  ( U `  B ) )
6059, 25sseldi 3375 . . . . . . . 8  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  ( ( U `  A ) 
.+  ( U `  B ) ) )
61 qsel 7200 . . . . . . . 8  |-  ( (  .~  Er  W  /\  ( ( U `  A )  .+  ( U `  B )
)  e.  ( W /.  .~  )  /\  <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ( ( U `  A )  .+  ( U `  B )
) )  ->  (
( U `  A
)  .+  ( U `  B ) )  =  [ <" <. A ,  (/) >. <. B ,  (/) >. "> ]  .~  )
6228, 58, 60, 61syl3anc 1218 . . . . . . 7  |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B )
)  =  [ <"
<. A ,  (/) >. <. B ,  (/)
>. "> ]  .~  )
63 inss2 3592 . . . . . . . . . 10  |-  ( D  i^i  ( ( U `
 B )  .+  ( U `  A ) ) )  C_  (
( U `  B
)  .+  ( U `  A ) )
6463, 17sseldi 3375 . . . . . . . . 9  |-  ( ph  ->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  e.  ( ( U `  B ) 
.+  ( U `  A ) ) )
65 frgpnabl.n . . . . . . . . 9  |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B )
)  =  ( ( U `  B ) 
.+  ( U `  A ) ) )
6664, 65eleqtrrd 2520 . . . . . . . 8  |-  ( ph  ->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  e.  ( ( U `  A ) 
.+  ( U `  B ) ) )
67 qsel 7200 . . . . . . . 8  |-  ( (  .~  Er  W  /\  ( ( U `  A )  .+  ( U `  B )
)  e.  ( W /.  .~  )  /\  <" <. B ,  (/) >. <. A ,  (/) >. ">  e.  ( ( U `  A )  .+  ( U `  B )
) )  ->  (
( U `  A
)  .+  ( U `  B ) )  =  [ <" <. B ,  (/) >. <. A ,  (/) >. "> ]  .~  )
6828, 58, 66, 67syl3anc 1218 . . . . . . 7  |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B )
)  =  [ <"
<. B ,  (/) >. <. A ,  (/)
>. "> ]  .~  )
6962, 68eqtr3d 2477 . . . . . 6  |-  ( ph  ->  [ <" <. A ,  (/) >. <. B ,  (/) >. "> ]  .~  =  [ <" <. B ,  (/)
>. <. A ,  (/) >. "> ]  .~  )
706, 26sseldi 3375 . . . . . . 7  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  e.  W )
7128, 70erth 7166 . . . . . 6  |-  ( ph  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. ">  .~  <" <. B ,  (/) >. <. A ,  (/) >. ">  <->  [ <" <. A ,  (/) >. <. B ,  (/) >. "> ]  .~  =  [ <" <. B ,  (/)
>. <. A ,  (/) >. "> ]  .~  )
)
7269, 71mpbird 232 . . . . 5  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> )
7328, 19erref 7142 . . . . 5  |-  ( ph  ->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> )
74 breq1 4316 . . . . . 6  |-  ( d  =  <" <. A ,  (/)
>. <. B ,  (/) >. ">  ->  ( d  .~  <" <. B ,  (/)
>. <. A ,  (/) >. ">  <->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> ) )
75 breq1 4316 . . . . . 6  |-  ( d  =  <" <. B ,  (/)
>. <. A ,  (/) >. ">  ->  ( d  .~  <" <. B ,  (/)
>. <. A ,  (/) >. ">  <->  <" <. B ,  (/)
>. <. A ,  (/) >. ">  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> ) )
7674, 75rmoi 3308 . . . . 5  |-  ( ( E* d  e.  D  d  .~  <" <. B ,  (/)
>. <. A ,  (/) >. ">  /\  ( <"
<. A ,  (/) >. <. B ,  (/)
>. ">  e.  D  /\  <" <. A ,  (/)
>. <. B ,  (/) >. ">  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> )  /\  ( <" <. B ,  (/) >. <. A ,  (/) >. ">  e.  D  /\  <" <. B ,  (/) >. <. A ,  (/) >. ">  .~  <" <. B ,  (/) >. <. A ,  (/) >. "> ) )  ->  <" <. A ,  (/) >. <. B ,  (/) >. ">  =  <" <. B ,  (/)
>. <. A ,  (/) >. "> )
7723, 26, 72, 18, 73, 76syl122anc 1227 . . . 4  |-  ( ph  ->  <" <. A ,  (/)
>. <. B ,  (/) >. ">  =  <" <. B ,  (/) >. <. A ,  (/) >. "> )
7877fveq1d 5714 . . 3  |-  ( ph  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. "> `  0 )  =  ( <" <. B ,  (/) >. <. A ,  (/) >. "> `  0 )
)
79 opex 4577 . . . 4  |-  <. A ,  (/)
>.  e.  _V
80 s2fv0 12533 . . . 4  |-  ( <. A ,  (/) >.  e.  _V  ->  ( <" <. A ,  (/) >. <. B ,  (/) >. "> `  0 )  =  <. A ,  (/) >.
)
8179, 80ax-mp 5 . . 3  |-  ( <" <. A ,  (/) >. <. B ,  (/) >. "> `  0 )  =  <. A ,  (/) >.
82 opex 4577 . . . 4  |-  <. B ,  (/)
>.  e.  _V
83 s2fv0 12533 . . . 4  |-  ( <. B ,  (/) >.  e.  _V  ->  ( <" <. B ,  (/) >. <. A ,  (/) >. "> `  0 )  =  <. B ,  (/) >.
)
8482, 83ax-mp 5 . . 3  |-  ( <" <. B ,  (/) >. <. A ,  (/) >. "> `  0 )  =  <. B ,  (/) >.
8578, 81, 843eqtr3g 2498 . 2  |-  ( ph  -> 
<. A ,  (/) >.  =  <. B ,  (/) >. )
86 opthg 4588 . . 3  |-  ( ( A  e.  I  /\  (/) 
e.  _V )  ->  ( <. A ,  (/) >.  =  <. B ,  (/) >.  <->  ( A  =  B  /\  (/)  =  (/) ) ) )
8786simprbda 623 . 2  |-  ( ( ( A  e.  I  /\  (/)  e.  _V )  /\  <. A ,  (/) >.  =  <. B ,  (/) >.
)  ->  A  =  B )
881, 3, 85, 87syl21anc 1217 1  |-  ( ph  ->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2736   E!wreu 2738   E*wrmo 2739   {crab 2740   _Vcvv 2993    \ cdif 3346    i^i cin 3348   (/)c0 3658   {csn 3898   <.cop 3904   <.cotp 3906   U_ciun 4192   class class class wbr 4313    e. cmpt 4371    _I cid 4652   Oncon0 4740    X. cxp 4859   ran crn 4862   -->wf 5435   ` cfv 5439  (class class class)co 6112    e. cmpt2 6114   1oc1o 6934   2oc2o 6935    Er wer 7119   [cec 7120   /.cqs 7121   0cc0 9303   1c1 9304    - cmin 9616   ...cfz 11458  ..^cfzo 11569   #chash 12124  Word cword 12242   splice csplice 12247   <"cs2 12489   Basecbs 14195   +g cplusg 14259    /.s cqus 14464   Grpcgrp 15431  freeMndcfrmd 15546   ~FG cefg 16224  freeGrpcfrgp 16225  varFGrpcvrgp 16226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-ot 3907  df-uni 4113  df-int 4150  df-iun 4194  df-iin 4195  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-2o 6942  df-oadd 6945  df-er 7122  df-ec 7124  df-qs 7128  df-map 7237  df-pm 7238  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-sup 7712  df-card 8130  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-7 10406  df-8 10407  df-9 10408  df-10 10409  df-n0 10601  df-z 10668  df-dec 10777  df-uz 10883  df-rp 11013  df-fz 11459  df-fzo 11570  df-hash 12125  df-word 12250  df-concat 12252  df-s1 12253  df-substr 12254  df-splice 12255  df-reverse 12256  df-s2 12496  df-struct 14197  df-ndx 14198  df-slot 14199  df-base 14200  df-plusg 14272  df-mulr 14273  df-sca 14275  df-vsca 14276  df-ip 14277  df-tset 14278  df-ple 14279  df-ds 14281  df-0g 14401  df-imas 14467  df-divs 14468  df-mnd 15436  df-frmd 15548  df-grp 15566  df-efg 16227  df-frgp 16228  df-vrgp 16229
This theorem is referenced by:  frgpnabl  16374
  Copyright terms: Public domain W3C validator