Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdleme11g Structured version   Unicode version

Theorem cdleme11g 35061
Description: Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme11 35066. (Contributed by NM, 14-Jun-2012.)
Hypotheses
Ref Expression
cdleme11.l  |-  .<_  =  ( le `  K )
cdleme11.j  |-  .\/  =  ( join `  K )
cdleme11.m  |-  ./\  =  ( meet `  K )
cdleme11.a  |-  A  =  ( Atoms `  K )
cdleme11.h  |-  H  =  ( LHyp `  K
)
cdleme11.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme11.c  |-  C  =  ( ( P  .\/  S )  ./\  W )
cdleme11.d  |-  D  =  ( ( P  .\/  T )  ./\  W )
cdleme11.f  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
Assertion
Ref Expression
cdleme11g  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  .\/  F )  =  ( Q  .\/  C ) )

Proof of Theorem cdleme11g
StepHypRef Expression
1 cdleme11.f . . . 4  |-  F  =  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) )
21oveq2i 6293 . . 3  |-  ( Q 
.\/  F )  =  ( Q  .\/  (
( S  .\/  U
)  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) ) )
3 simp1l 1020 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  K  e.  HL )
4 simp22l 1115 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  Q  e.  A )
5 hllat 34160 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
63, 5syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  K  e.  Lat )
7 simp23 1031 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  S  e.  A )
8 eqid 2467 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
9 cdleme11.a . . . . . . 7  |-  A  =  ( Atoms `  K )
108, 9atbase 34086 . . . . . 6  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
117, 10syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  S  e.  ( Base `  K )
)
12 simp1 996 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( K  e.  HL  /\  W  e.  H ) )
13 simp21 1029 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  P  e.  A )
14 cdleme11.l . . . . . . 7  |-  .<_  =  ( le `  K )
15 cdleme11.j . . . . . . 7  |-  .\/  =  ( join `  K )
16 cdleme11.m . . . . . . 7  |-  ./\  =  ( meet `  K )
17 cdleme11.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
18 cdleme11.u . . . . . . 7  |-  U  =  ( ( P  .\/  Q )  ./\  W )
1914, 15, 16, 9, 17, 18, 8cdleme0aa 35006 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  P  e.  A  /\  Q  e.  A
)  ->  U  e.  ( Base `  K )
)
2012, 13, 4, 19syl3anc 1228 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  U  e.  ( Base `  K )
)
218, 15latjcl 15531 . . . . 5  |-  ( ( K  e.  Lat  /\  S  e.  ( Base `  K )  /\  U  e.  ( Base `  K
) )  ->  ( S  .\/  U )  e.  ( Base `  K
) )
226, 11, 20, 21syl3anc 1228 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( S  .\/  U )  e.  (
Base `  K )
)
238, 9atbase 34086 . . . . . 6  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
244, 23syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  Q  e.  ( Base `  K )
)
258, 9atbase 34086 . . . . . . . 8  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
2613, 25syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  P  e.  ( Base `  K )
)
278, 15latjcl 15531 . . . . . . 7  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  S  e.  ( Base `  K
) )  ->  ( P  .\/  S )  e.  ( Base `  K
) )
286, 26, 11, 27syl3anc 1228 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( P  .\/  S )  e.  (
Base `  K )
)
29 simp1r 1021 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  W  e.  H )
308, 17lhpbase 34794 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
3129, 30syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  W  e.  ( Base `  K )
)
328, 16latmcl 15532 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  ./\  W )  e.  ( Base `  K ) )
336, 28, 31, 32syl3anc 1228 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( ( P  .\/  S )  ./\  W )  e.  ( Base `  K ) )
348, 15latjcl 15531 . . . . 5  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  (
( P  .\/  S
)  ./\  W )  e.  ( Base `  K
) )  ->  ( Q  .\/  ( ( P 
.\/  S )  ./\  W ) )  e.  (
Base `  K )
)
356, 24, 33, 34syl3anc 1228 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
)  e.  ( Base `  K ) )
368, 14, 15latlej1 15540 . . . . 5  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  (
( P  .\/  S
)  ./\  W )  e.  ( Base `  K
) )  ->  Q  .<_  ( Q  .\/  (
( P  .\/  S
)  ./\  W )
) )
376, 24, 33, 36syl3anc 1228 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  Q  .<_  ( Q  .\/  ( ( P  .\/  S ) 
./\  W ) ) )
388, 14, 15, 16, 9atmod1i1 34653 . . . 4  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  ( S  .\/  U
)  e.  ( Base `  K )  /\  ( Q  .\/  ( ( P 
.\/  S )  ./\  W ) )  e.  (
Base `  K )
)  /\  Q  .<_  ( Q  .\/  ( ( P  .\/  S ) 
./\  W ) ) )  ->  ( Q  .\/  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) ) )  =  ( ( Q  .\/  ( S  .\/  U ) )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) ) )
393, 4, 22, 35, 37, 38syl131anc 1241 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  .\/  ( ( S  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) ) )  =  ( ( Q  .\/  ( S  .\/  U ) )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) ) )
402, 39syl5eq 2520 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  .\/  F )  =  ( ( Q  .\/  ( S  .\/  U ) ) 
./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) ) )
41 simp22 1030 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
4214, 15, 16, 9, 17, 18cdleme0cq 35011 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) ) )  ->  ( Q  .\/  U )  =  ( P  .\/  Q ) )
4312, 13, 41, 42syl12anc 1226 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  .\/  U )  =  ( P  .\/  Q ) )
4443oveq2d 6298 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( S  .\/  ( Q  .\/  U
) )  =  ( S  .\/  ( P 
.\/  Q ) ) )
458, 15latj12 15576 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Q  e.  ( Base `  K )  /\  S  e.  ( Base `  K )  /\  U  e.  ( Base `  K
) ) )  -> 
( Q  .\/  ( S  .\/  U ) )  =  ( S  .\/  ( Q  .\/  U ) ) )
466, 24, 11, 20, 45syl13anc 1230 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  .\/  ( S  .\/  U
) )  =  ( S  .\/  ( Q 
.\/  U ) ) )
478, 15latj13 15578 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Q  e.  ( Base `  K )  /\  P  e.  ( Base `  K )  /\  S  e.  ( Base `  K
) ) )  -> 
( Q  .\/  ( P  .\/  S ) )  =  ( S  .\/  ( P  .\/  Q ) ) )
486, 24, 26, 11, 47syl13anc 1230 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  .\/  ( P  .\/  S
) )  =  ( S  .\/  ( P 
.\/  Q ) ) )
4944, 46, 483eqtr4d 2518 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  .\/  ( S  .\/  U
) )  =  ( Q  .\/  ( P 
.\/  S ) ) )
5049oveq1d 6297 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( ( Q  .\/  ( S  .\/  U ) )  ./\  ( Q  .\/  ( ( P 
.\/  S )  ./\  W ) ) )  =  ( ( Q  .\/  ( P  .\/  S ) )  ./\  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
) ) )
518, 14, 16latmle1 15556 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  ./\  W )  .<_  ( P  .\/  S ) )
526, 28, 31, 51syl3anc 1228 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( ( P  .\/  S )  ./\  W )  .<_  ( P  .\/  S ) )
538, 14, 15latjlej2 15546 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( ( ( P 
.\/  S )  ./\  W )  e.  ( Base `  K )  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  Q  e.  ( Base `  K )
) )  ->  (
( ( P  .\/  S )  ./\  W )  .<_  ( P  .\/  S
)  ->  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
)  .<_  ( Q  .\/  ( P  .\/  S ) ) ) )
546, 33, 28, 24, 53syl13anc 1230 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( (
( P  .\/  S
)  ./\  W )  .<_  ( P  .\/  S
)  ->  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
)  .<_  ( Q  .\/  ( P  .\/  S ) ) ) )
5552, 54mpd 15 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  .\/  ( ( P  .\/  S )  ./\  W )
)  .<_  ( Q  .\/  ( P  .\/  S ) ) )
568, 15latjcl 15531 . . . . . 6  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  ( P  .\/  S )  e.  ( Base `  K
) )  ->  ( Q  .\/  ( P  .\/  S ) )  e.  (
Base `  K )
)
576, 24, 28, 56syl3anc 1228 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  .\/  ( P  .\/  S
) )  e.  (
Base `  K )
)
588, 14, 16latleeqm2 15560 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Q  .\/  ( ( P  .\/  S ) 
./\  W ) )  e.  ( Base `  K
)  /\  ( Q  .\/  ( P  .\/  S
) )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  ( ( P 
.\/  S )  ./\  W ) )  .<_  ( Q 
.\/  ( P  .\/  S ) )  <->  ( ( Q  .\/  ( P  .\/  S ) )  ./\  ( Q  .\/  ( ( P 
.\/  S )  ./\  W ) ) )  =  ( Q  .\/  (
( P  .\/  S
)  ./\  W )
) ) )
596, 35, 57, 58syl3anc 1228 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( ( Q  .\/  ( ( P 
.\/  S )  ./\  W ) )  .<_  ( Q 
.\/  ( P  .\/  S ) )  <->  ( ( Q  .\/  ( P  .\/  S ) )  ./\  ( Q  .\/  ( ( P 
.\/  S )  ./\  W ) ) )  =  ( Q  .\/  (
( P  .\/  S
)  ./\  W )
) ) )
6055, 59mpbid 210 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( ( Q  .\/  ( P  .\/  S ) )  ./\  ( Q  .\/  ( ( P 
.\/  S )  ./\  W ) ) )  =  ( Q  .\/  (
( P  .\/  S
)  ./\  W )
) )
61 cdleme11.c . . . 4  |-  C  =  ( ( P  .\/  S )  ./\  W )
6261oveq2i 6293 . . 3  |-  ( Q 
.\/  C )  =  ( Q  .\/  (
( P  .\/  S
)  ./\  W )
)
6360, 62syl6eqr 2526 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( ( Q  .\/  ( P  .\/  S ) )  ./\  ( Q  .\/  ( ( P 
.\/  S )  ./\  W ) ) )  =  ( Q  .\/  C
) )
6440, 50, 633eqtrd 2512 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  S  e.  A )  /\  P  =/=  Q
)  ->  ( Q  .\/  F )  =  ( Q  .\/  C ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   Basecbs 14483   lecple 14555   joincjn 15424   meetcmee 15425   Latclat 15525   Atomscatm 34060   HLchlt 34147   LHypclh 34780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-poset 15426  df-plt 15438  df-lub 15454  df-glb 15455  df-join 15456  df-meet 15457  df-p0 15519  df-p1 15520  df-lat 15526  df-clat 15588  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-psubsp 34299  df-pmap 34300  df-padd 34592  df-lhyp 34784
This theorem is referenced by:  cdleme11h  35062  cdleme11j  35063  cdleme15a  35070
  Copyright terms: Public domain W3C validator