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Theorem cdleme22f 29439
Description: Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 6th and 7th lines on p. 115.  F,  N represent f(t), ft(s) respectively. If s  <_ t  \/ v, then ft(s)  <_ f(t)  \/ v. (Contributed by NM, 6-Dec-2012.)
Hypotheses
Ref Expression
cdleme22.l  |-  .<_  =  ( le `  K )
cdleme22.j  |-  .\/  =  ( join `  K )
cdleme22.m  |-  ./\  =  ( meet `  K )
cdleme22.a  |-  A  =  ( Atoms `  K )
cdleme22.h  |-  H  =  ( LHyp `  K
)
cdleme22f.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme22f.f  |-  F  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
) )
cdleme22f.n  |-  N  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( S  .\/  T )  ./\  W )
) )
Assertion
Ref Expression
cdleme22f  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  N  .<_  ( F  .\/  V
) )

Proof of Theorem cdleme22f
StepHypRef Expression
1 cdleme22f.n . 2  |-  N  =  ( ( P  .\/  Q )  ./\  ( F  .\/  ( ( S  .\/  T )  ./\  W )
) )
2 simp11l 1071 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  K  e.  HL )
3 hllat 28457 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
42, 3syl 17 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  K  e.  Lat )
5 simp12l 1073 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  P  e.  A )
6 simp13l 1075 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  Q  e.  A )
7 eqid 2253 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
8 cdleme22.j . . . . . 6  |-  .\/  =  ( join `  K )
9 cdleme22.a . . . . . 6  |-  A  =  ( Atoms `  K )
107, 8, 9hlatjcl 28460 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
112, 5, 6, 10syl3anc 1187 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
12 simp11r 1072 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  W  e.  H )
13 simp22 994 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  T  e.  A )
14 cdleme22.l . . . . . . 7  |-  .<_  =  ( le `  K )
15 cdleme22.m . . . . . . 7  |-  ./\  =  ( meet `  K )
16 cdleme22.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
17 cdleme22f.u . . . . . . 7  |-  U  =  ( ( P  .\/  Q )  ./\  W )
18 cdleme22f.f . . . . . . 7  |-  F  =  ( ( T  .\/  U )  ./\  ( Q  .\/  ( ( P  .\/  T )  ./\  W )
) )
1914, 8, 15, 9, 16, 17, 18, 7cdleme1b 29319 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A  /\  T  e.  A ) )  ->  F  e.  ( Base `  K ) )
202, 12, 5, 6, 13, 19syl23anc 1194 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  F  e.  ( Base `  K
) )
21 simp21l 1077 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  S  e.  A )
227, 8, 9hlatjcl 28460 . . . . . . 7  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
232, 21, 13, 22syl3anc 1187 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  ( S  .\/  T )  e.  ( Base `  K
) )
247, 16lhpbase 29091 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2512, 24syl 17 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  W  e.  ( Base `  K
) )
267, 15latmcl 14001 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( S  .\/  T )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( S  .\/  T )  ./\  W )  e.  ( Base `  K ) )
274, 23, 25, 26syl3anc 1187 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  (
( S  .\/  T
)  ./\  W )  e.  ( Base `  K
) )
287, 8latjcl 14000 . . . . 5  |-  ( ( K  e.  Lat  /\  F  e.  ( Base `  K )  /\  (
( S  .\/  T
)  ./\  W )  e.  ( Base `  K
) )  ->  ( F  .\/  ( ( S 
.\/  T )  ./\  W ) )  e.  (
Base `  K )
)
294, 20, 27, 28syl3anc 1187 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  ( F  .\/  ( ( S 
.\/  T )  ./\  W ) )  e.  (
Base `  K )
)
307, 14, 15latmle2 14027 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  ( F  .\/  ( ( S  .\/  T )  ./\  W )
)  e.  ( Base `  K ) )  -> 
( ( P  .\/  Q )  ./\  ( F  .\/  ( ( S  .\/  T )  ./\  W )
) )  .<_  ( F 
.\/  ( ( S 
.\/  T )  ./\  W ) ) )
314, 11, 29, 30syl3anc 1187 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  (
( P  .\/  Q
)  ./\  ( F  .\/  ( ( S  .\/  T )  ./\  W )
) )  .<_  ( F 
.\/  ( ( S 
.\/  T )  ./\  W ) ) )
32 simp21 993 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  ( S  e.  A  /\  -.  S  .<_  W ) )
33 simp3l 988 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  S  =/=  T )
34 simp23l 1081 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  V  e.  A )
35 simp23r 1082 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  V  .<_  W )
36 simp3r 989 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  S  .<_  ( T  .\/  V
) )
378, 9hlatjcom 28461 . . . . . . . 8  |-  ( ( K  e.  HL  /\  T  e.  A  /\  V  e.  A )  ->  ( T  .\/  V
)  =  ( V 
.\/  T ) )
382, 13, 34, 37syl3anc 1187 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  ( T  .\/  V )  =  ( V  .\/  T
) )
3936, 38breqtrd 3944 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  S  .<_  ( V  .\/  T
) )
40 hlcvl 28453 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  CvLat )
412, 40syl 17 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  K  e.  CvLat )
4214, 8, 9cvlatexch2 28431 . . . . . . 7  |-  ( ( K  e.  CvLat  /\  ( S  e.  A  /\  V  e.  A  /\  T  e.  A )  /\  S  =/=  T
)  ->  ( S  .<_  ( V  .\/  T
)  ->  V  .<_  ( S  .\/  T ) ) )
4341, 21, 34, 13, 33, 42syl131anc 1200 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  ( S  .<_  ( V  .\/  T )  ->  V  .<_  ( S  .\/  T ) ) )
4439, 43mpd 16 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  V  .<_  ( S  .\/  T
) )
45 eqid 2253 . . . . . 6  |-  ( ( S  .\/  T ) 
./\  W )  =  ( ( S  .\/  T )  ./\  W )
4614, 8, 15, 9, 16, 45cdleme22aa 29432 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  S  =/=  T )  /\  ( V  e.  A  /\  V  .<_  W  /\  V  .<_  ( S  .\/  T ) ) )  ->  V  =  ( ( S  .\/  T )  ./\  W ) )
472, 12, 32, 13, 33, 34, 35, 44, 46syl233anc 1216 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  V  =  ( ( S 
.\/  T )  ./\  W ) )
4847oveq2d 5726 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  ( F  .\/  V )  =  ( F  .\/  (
( S  .\/  T
)  ./\  W )
) )
4931, 48breqtrrd 3946 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  (
( P  .\/  Q
)  ./\  ( F  .\/  ( ( S  .\/  T )  ./\  W )
) )  .<_  ( F 
.\/  V ) )
501, 49syl5eqbr 3953 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  T  e.  A  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/= 
T  /\  S  .<_  ( T  .\/  V ) ) )  ->  N  .<_  ( F  .\/  V
) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   class class class wbr 3920   ` cfv 4592  (class class class)co 5710   Basecbs 13022   lecple 13089   joincjn 13922   meetcmee 13923   Latclat 13995   Atomscatm 28357   CvLatclc 28359   HLchlt 28444   LHypclh 29077
This theorem is referenced by:  cdleme22f2  29440  cdleme26fALTN  29455  cdleme26f  29456
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-undef 6182  df-riota 6190  df-poset 13924  df-plt 13936  df-lub 13952  df-glb 13953  df-join 13954  df-meet 13955  df-p0 13989  df-p1 13990  df-lat 13996  df-clat 14058  df-oposet 28270  df-ol 28272  df-oml 28273  df-covers 28360  df-ats 28361  df-atl 28392  df-cvlat 28416  df-hlat 28445  df-lhyp 29081
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