ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cjap Unicode version

Theorem cjap 10401
Description: Complex conjugate and apartness. (Contributed by Jim Kingdon, 14-Jun-2020.)
Assertion
Ref Expression
cjap  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( * `  A ) #  ( * `  B )  <->  A #  B
) )

Proof of Theorem cjap
Dummy variables  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnre 7545 . . 3  |-  ( A  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
21adantr 271 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y
) ) )
3 cnre 7545 . . . . . 6  |-  ( B  e.  CC  ->  E. z  e.  RR  E. w  e.  RR  B  =  ( z  +  ( _i  x.  w ) ) )
43ad3antlr 478 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  E. z  e.  RR  E. w  e.  RR  B  =  ( z  +  ( _i  x.  w ) ) )
5 simplrr 504 . . . . . . . . . . . 12  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  y  e.  RR )
65ad2antrr 473 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  y  e.  RR )
76recnd 7577 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  y  e.  CC )
8 simplrr 504 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  w  e.  RR )
98recnd 7577 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  w  e.  CC )
10 apneg 8149 . . . . . . . . . 10  |-  ( ( y  e.  CC  /\  w  e.  CC )  ->  ( y #  w  <->  -u y #  -u w ) )
117, 9, 10syl2anc 404 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  (
y #  w  <->  -u y #  -u w ) )
1211orbi2d 740 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  (
( x #  z  \/  y #  w )  <->  ( x #  z  \/  -u y #  -u w ) ) )
13 simpllr 502 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  A  =  ( x  +  ( _i  x.  y
) ) )
14 simpr 109 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  B  =  ( z  +  ( _i  x.  w
) ) )
1513, 14breq12d 3864 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  ( A #  B  <->  ( x  +  ( _i  x.  y
) ) #  ( z  +  ( _i  x.  w ) ) ) )
16 simplrl 503 . . . . . . . . . . 11  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  x  e.  RR )
1716ad2antrr 473 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  x  e.  RR )
18 simplrl 503 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  z  e.  RR )
19 apreim 8141 . . . . . . . . . 10  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  ( z  e.  RR  /\  w  e.  RR ) )  -> 
( ( x  +  ( _i  x.  y
) ) #  ( z  +  ( _i  x.  w ) )  <->  ( x #  z  \/  y #  w
) ) )
2017, 6, 18, 8, 19syl22anc 1176 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  (
( x  +  ( _i  x.  y ) ) #  ( z  +  ( _i  x.  w
) )  <->  ( x #  z  \/  y #  w
) ) )
2115, 20bitrd 187 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  ( A #  B  <->  ( x #  z  \/  y #  w ) ) )
2213fveq2d 5322 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  (
* `  A )  =  ( * `  ( x  +  (
_i  x.  y )
) ) )
23 cjreim 10398 . . . . . . . . . . . 12  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( * `  (
x  +  ( _i  x.  y ) ) )  =  ( x  -  ( _i  x.  y ) ) )
2417, 6, 23syl2anc 404 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  (
* `  ( x  +  ( _i  x.  y ) ) )  =  ( x  -  ( _i  x.  y
) ) )
2522, 24eqtrd 2121 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  (
* `  A )  =  ( x  -  ( _i  x.  y
) ) )
2614fveq2d 5322 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  (
* `  B )  =  ( * `  ( z  +  ( _i  x.  w ) ) ) )
27 cjreim 10398 . . . . . . . . . . . 12  |-  ( ( z  e.  RR  /\  w  e.  RR )  ->  ( * `  (
z  +  ( _i  x.  w ) ) )  =  ( z  -  ( _i  x.  w ) ) )
2818, 8, 27syl2anc 404 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  (
* `  ( z  +  ( _i  x.  w ) ) )  =  ( z  -  ( _i  x.  w
) ) )
2926, 28eqtrd 2121 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  (
* `  B )  =  ( z  -  ( _i  x.  w
) ) )
3025, 29breq12d 3864 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  (
( * `  A
) #  ( * `  B )  <->  ( x  -  ( _i  x.  y ) ) #  ( z  -  ( _i  x.  w ) ) ) )
3117recnd 7577 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  x  e.  CC )
32 ax-icn 7501 . . . . . . . . . . . 12  |-  _i  e.  CC
3332a1i 9 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  _i  e.  CC )
34 submul2 7938 . . . . . . . . . . 11  |-  ( ( x  e.  CC  /\  _i  e.  CC  /\  y  e.  CC )  ->  (
x  -  ( _i  x.  y ) )  =  ( x  +  ( _i  x.  -u y
) ) )
3531, 33, 7, 34syl3anc 1175 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  (
x  -  ( _i  x.  y ) )  =  ( x  +  ( _i  x.  -u y
) ) )
3618recnd 7577 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  z  e.  CC )
37 submul2 7938 . . . . . . . . . . 11  |-  ( ( z  e.  CC  /\  _i  e.  CC  /\  w  e.  CC )  ->  (
z  -  ( _i  x.  w ) )  =  ( z  +  ( _i  x.  -u w
) ) )
3836, 33, 9, 37syl3anc 1175 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  (
z  -  ( _i  x.  w ) )  =  ( z  +  ( _i  x.  -u w
) ) )
3935, 38breq12d 3864 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  (
( x  -  (
_i  x.  y )
) #  ( z  -  ( _i  x.  w
) )  <->  ( x  +  ( _i  x.  -u y ) ) #  ( z  +  ( _i  x.  -u w ) ) ) )
406renegcld 7919 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  -u y  e.  RR )
418renegcld 7919 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  -u w  e.  RR )
42 apreim 8141 . . . . . . . . . 10  |-  ( ( ( x  e.  RR  /\  -u y  e.  RR )  /\  ( z  e.  RR  /\  -u w  e.  RR ) )  -> 
( ( x  +  ( _i  x.  -u y
) ) #  ( z  +  ( _i  x.  -u w ) )  <->  ( x #  z  \/  -u y #  -u w ) ) )
4317, 40, 18, 41, 42syl22anc 1176 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  (
( x  +  ( _i  x.  -u y
) ) #  ( z  +  ( _i  x.  -u w ) )  <->  ( x #  z  \/  -u y #  -u w ) ) )
4430, 39, 433bitrd 213 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  (
( * `  A
) #  ( * `  B )  <->  ( x #  z  \/  -u y #  -u w ) ) )
4512, 21, 443bitr4rd 220 . . . . . . 7  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  (
( * `  A
) #  ( * `  B )  <->  A #  B
) )
4645ex 114 . . . . . 6  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  /\  (
z  e.  RR  /\  w  e.  RR )
)  ->  ( B  =  ( z  +  ( _i  x.  w
) )  ->  (
( * `  A
) #  ( * `  B )  <->  A #  B
) ) )
4746rexlimdvva 2497 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  ( E. z  e.  RR  E. w  e.  RR  B  =  ( z  +  ( _i  x.  w
) )  ->  (
( * `  A
) #  ( * `  B )  <->  A #  B
) ) )
484, 47mpd 13 . . . 4  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  (
( * `  A
) #  ( * `  B )  <->  A #  B
) )
4948ex 114 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( A  =  ( x  +  ( _i  x.  y ) )  ->  ( ( * `
 A ) #  ( * `  B )  <-> 
A #  B ) ) )
5049rexlimdvva 2497 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) )  ->  ( ( * `
 A ) #  ( * `  B )  <-> 
A #  B ) ) )
512, 50mpd 13 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( * `  A ) #  ( * `  B )  <->  A #  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 665    = wceq 1290    e. wcel 1439   E.wrex 2361   class class class wbr 3851   ` cfv 5028  (class class class)co 5666   CCcc 7409   RRcr 7410   _ici 7413    + caddc 7414    x. cmul 7416    - cmin 7714   -ucneg 7715   # cap 8119   *ccj 10334
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-cnex 7497  ax-resscn 7498  ax-1cn 7499  ax-1re 7500  ax-icn 7501  ax-addcl 7502  ax-addrcl 7503  ax-mulcl 7504  ax-mulrcl 7505  ax-addcom 7506  ax-mulcom 7507  ax-addass 7508  ax-mulass 7509  ax-distr 7510  ax-i2m1 7511  ax-0lt1 7512  ax-1rid 7513  ax-0id 7514  ax-rnegex 7515  ax-precex 7516  ax-cnre 7517  ax-pre-ltirr 7518  ax-pre-ltwlin 7519  ax-pre-lttrn 7520  ax-pre-apti 7521  ax-pre-ltadd 7522  ax-pre-mulgt0 7523  ax-pre-mulext 7524
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-po 4132  df-iso 4133  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-fv 5036  df-riota 5622  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-pnf 7585  df-mnf 7586  df-xr 7587  df-ltxr 7588  df-le 7589  df-sub 7716  df-neg 7717  df-reap 8113  df-ap 8120  df-div 8201  df-2 8542  df-cj 10337  df-re 10338  df-im 10339
This theorem is referenced by:  cjap0  10402
  Copyright terms: Public domain W3C validator