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Theorem cdleme0e 30406
Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 13-Jun-2012.)
Hypotheses
Ref Expression
cdleme0.l  |-  .<_  =  ( le `  K )
cdleme0.j  |-  .\/  =  ( join `  K )
cdleme0.m  |-  ./\  =  ( meet `  K )
cdleme0.a  |-  A  =  ( Atoms `  K )
cdleme0.h  |-  H  =  ( LHyp `  K
)
cdleme0.u  |-  U  =  ( ( P  .\/  Q )  ./\  W )
cdleme0c.3  |-  V  =  ( ( P  .\/  R )  ./\  W )
Assertion
Ref Expression
cdleme0e  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  U  =/=  V )

Proof of Theorem cdleme0e
StepHypRef Expression
1 cdleme0.u . . . . 5  |-  U  =  ( ( P  .\/  Q )  ./\  W )
2 cdleme0c.3 . . . . 5  |-  V  =  ( ( P  .\/  R )  ./\  W )
31, 2oveq12i 5870 . . . 4  |-  ( U 
./\  V )  =  ( ( ( P 
.\/  Q )  ./\  W )  ./\  ( ( P  .\/  R )  ./\  W ) )
4 simp1l 979 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  K  e.  HL )
5 hlol 29551 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OL )
64, 5syl 15 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  K  e.  OL )
7 simp21l 1072 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  P  e.  A )
8 simp22 989 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  Q  e.  A )
9 eqid 2283 . . . . . . . 8  |-  ( Base `  K )  =  (
Base `  K )
10 cdleme0.j . . . . . . . 8  |-  .\/  =  ( join `  K )
11 cdleme0.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
129, 10, 11hlatjcl 29556 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
134, 7, 8, 12syl3anc 1182 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  -> 
( P  .\/  Q
)  e.  ( Base `  K ) )
14 simp23l 1076 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  R  e.  A )
159, 10, 11hlatjcl 29556 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  ( P  .\/  R
)  e.  ( Base `  K ) )
164, 7, 14, 15syl3anc 1182 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  -> 
( P  .\/  R
)  e.  ( Base `  K ) )
17 simp1r 980 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  W  e.  H )
18 cdleme0.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
199, 18lhpbase 30187 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2017, 19syl 15 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  W  e.  ( Base `  K ) )
21 cdleme0.m . . . . . . 7  |-  ./\  =  ( meet `  K )
229, 21latmmdir 29425 . . . . . 6  |-  ( ( K  e.  OL  /\  ( ( P  .\/  Q )  e.  ( Base `  K )  /\  ( P  .\/  R )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
) )  ->  (
( ( P  .\/  Q )  ./\  ( P  .\/  R ) )  ./\  W )  =  ( ( ( P  .\/  Q
)  ./\  W )  ./\  ( ( P  .\/  R )  ./\  W )
) )
236, 13, 16, 20, 22syl13anc 1184 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  -> 
( ( ( P 
.\/  Q )  ./\  ( P  .\/  R ) )  ./\  W )  =  ( ( ( P  .\/  Q ) 
./\  W )  ./\  ( ( P  .\/  R )  ./\  W )
) )
24 hllat 29553 . . . . . . . . 9  |-  ( K  e.  HL  ->  K  e.  Lat )
254, 24syl 15 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  K  e.  Lat )
269, 11atbase 29479 . . . . . . . . 9  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
2714, 26syl 15 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  R  e.  ( Base `  K ) )
289, 11atbase 29479 . . . . . . . . 9  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
297, 28syl 15 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  P  e.  ( Base `  K ) )
309, 11atbase 29479 . . . . . . . . 9  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
318, 30syl 15 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  Q  e.  ( Base `  K ) )
32 simp3r 984 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  -.  R  .<_  ( P 
.\/  Q ) )
33 cdleme0.l . . . . . . . . . 10  |-  .<_  =  ( le `  K )
349, 33, 10latnlej1r 14176 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( R  e.  ( Base `  K )  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  /\  -.  R  .<_  ( P  .\/  Q ) )  ->  R  =/=  Q )
3534necomd 2529 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( R  e.  ( Base `  K )  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  /\  -.  R  .<_  ( P  .\/  Q ) )  ->  Q  =/=  R )
3625, 27, 29, 31, 32, 35syl131anc 1195 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  Q  =/=  R )
37 simp3 957 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  -> 
( P  =/=  Q  /\  -.  R  .<_  ( P 
.\/  Q ) ) )
3833, 10, 11hlatcon3 29640 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q
) ) )  ->  -.  P  .<_  ( Q 
.\/  R ) )
394, 7, 8, 14, 37, 38syl131anc 1195 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  -.  P  .<_  ( Q 
.\/  R ) )
4033, 10, 21, 112llnma2 29978 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  P  e.  A
)  /\  ( Q  =/=  R  /\  -.  P  .<_  ( Q  .\/  R
) ) )  -> 
( ( P  .\/  Q )  ./\  ( P  .\/  R ) )  =  P )
414, 8, 14, 7, 36, 39, 40syl132anc 1200 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  -> 
( ( P  .\/  Q )  ./\  ( P  .\/  R ) )  =  P )
4241oveq1d 5873 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  -> 
( ( ( P 
.\/  Q )  ./\  ( P  .\/  R ) )  ./\  W )  =  ( P  ./\  W ) )
4323, 42eqtr3d 2317 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  -> 
( ( ( P 
.\/  Q )  ./\  W )  ./\  ( ( P  .\/  R )  ./\  W ) )  =  ( P  ./\  W )
)
443, 43syl5eq 2327 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  -> 
( U  ./\  V
)  =  ( P 
./\  W ) )
45 simp1 955 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
46 simp21 988 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
47 eqid 2283 . . . . 5  |-  ( 0.
`  K )  =  ( 0. `  K
)
4833, 21, 47, 11, 18lhpmat 30219 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  -> 
( P  ./\  W
)  =  ( 0.
`  K ) )
4945, 46, 48syl2anc 642 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  -> 
( P  ./\  W
)  =  ( 0.
`  K ) )
5044, 49eqtrd 2315 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  -> 
( U  ./\  V
)  =  ( 0.
`  K ) )
51 hlatl 29550 . . . 4  |-  ( K  e.  HL  ->  K  e.  AtLat )
524, 51syl 15 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  K  e.  AtLat )
53 simp3l 983 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  P  =/=  Q )
5433, 10, 21, 11, 18, 1lhpat2 30234 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  P  =/=  Q ) )  ->  U  e.  A
)
5545, 46, 8, 53, 54syl112anc 1186 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  U  e.  A )
569, 33, 10latnlej1l 14175 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( R  e.  ( Base `  K )  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  /\  -.  R  .<_  ( P  .\/  Q ) )  ->  R  =/=  P )
5756necomd 2529 . . . . 5  |-  ( ( K  e.  Lat  /\  ( R  e.  ( Base `  K )  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  /\  -.  R  .<_  ( P  .\/  Q ) )  ->  P  =/=  R )
5825, 27, 29, 31, 32, 57syl131anc 1195 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  P  =/=  R )
5933, 10, 21, 11, 18, 2lhpat2 30234 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( R  e.  A  /\  P  =/=  R ) )  ->  V  e.  A
)
6045, 46, 14, 58, 59syl112anc 1186 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  V  e.  A )
6121, 47, 11atnem0 29508 . . 3  |-  ( ( K  e.  AtLat  /\  U  e.  A  /\  V  e.  A )  ->  ( U  =/=  V  <->  ( U  ./\ 
V )  =  ( 0. `  K ) ) )
6252, 55, 60, 61syl3anc 1182 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  -> 
( U  =/=  V  <->  ( U  ./\  V )  =  ( 0. `  K ) ) )
6350, 62mpbird 223 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A  /\  ( R  e.  A  /\  -.  R  .<_  W ) )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  ->  U  =/=  V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   0.cp0 14143   Latclat 14151   OLcol 29364   Atomscatm 29453   AtLatcal 29454   HLchlt 29540   LHypclh 30173
This theorem is referenced by:  cdleme3fN  30422  cdleme3g  30423  cdleme11e  30452
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-lhyp 30177
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