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Theorem cdleme22d 31140
Description: Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 9th line on p. 115. (Contributed by NM, 4-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
)
Assertion
Ref Expression
cdleme22d  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  V  =  ( ( S  .\/  T )  ./\  W ) )

Proof of Theorem cdleme22d
StepHypRef Expression
1 simp3r 986 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  S  .<_  ( T  .\/  V ) )
2 simp1l 981 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  K  e.  HL )
3 simp22l 1076 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  T  e.  A )
4 simp23l 1078 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  V  e.  A )
5 cdleme22.l . . . . . . . 8  |-  .<_  =  ( le `  K )
6 cdleme22.j . . . . . . . 8  |-  .\/  =  ( join `  K )
7 cdleme22.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
85, 6, 7hlatlej1 30172 . . . . . . 7  |-  ( ( K  e.  HL  /\  T  e.  A  /\  V  e.  A )  ->  T  .<_  ( T  .\/  V ) )
92, 3, 4, 8syl3anc 1184 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  T  .<_  ( T  .\/  V ) )
10 hllat 30161 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
112, 10syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  K  e.  Lat )
12 simp21l 1074 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  S  e.  A )
13 eqid 2436 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
1413, 7atbase 30087 . . . . . . . 8  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
1512, 14syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  S  e.  ( Base `  K ) )
1613, 7atbase 30087 . . . . . . . 8  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
173, 16syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  T  e.  ( Base `  K ) )
1813, 6, 7hlatjcl 30164 . . . . . . . 8  |-  ( ( K  e.  HL  /\  T  e.  A  /\  V  e.  A )  ->  ( T  .\/  V
)  e.  ( Base `  K ) )
192, 3, 4, 18syl3anc 1184 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( T  .\/  V
)  e.  ( Base `  K ) )
2013, 5, 6latjle12 14491 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( S  e.  ( Base `  K )  /\  T  e.  ( Base `  K )  /\  ( T  .\/  V )  e.  ( Base `  K
) ) )  -> 
( ( S  .<_  ( T  .\/  V )  /\  T  .<_  ( T 
.\/  V ) )  <-> 
( S  .\/  T
)  .<_  ( T  .\/  V ) ) )
2111, 15, 17, 19, 20syl13anc 1186 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( ( S  .<_  ( T  .\/  V )  /\  T  .<_  ( T 
.\/  V ) )  <-> 
( S  .\/  T
)  .<_  ( T  .\/  V ) ) )
221, 9, 21mpbi2and 888 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( S  .\/  T
)  .<_  ( T  .\/  V ) )
2313, 6, 7hlatjcl 30164 . . . . . . 7  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
242, 12, 3, 23syl3anc 1184 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( S  .\/  T
)  e.  ( Base `  K ) )
25 simp1r 982 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  W  e.  H )
26 cdleme22.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
2713, 26lhpbase 30795 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2825, 27syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  W  e.  ( Base `  K ) )
29 cdleme22.m . . . . . . 7  |-  ./\  =  ( meet `  K )
3013, 5, 29latmlem1 14510 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( ( S  .\/  T )  e.  ( Base `  K )  /\  ( T  .\/  V )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
) )  ->  (
( S  .\/  T
)  .<_  ( T  .\/  V )  ->  ( ( S  .\/  T )  ./\  W )  .<_  ( ( T  .\/  V )  ./\  W ) ) )
3111, 24, 19, 28, 30syl13anc 1186 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( ( S  .\/  T )  .<_  ( T  .\/  V )  ->  (
( S  .\/  T
)  ./\  W )  .<_  ( ( T  .\/  V )  ./\  W )
) )
3222, 31mpd 15 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( ( S  .\/  T )  ./\  W )  .<_  ( ( T  .\/  V )  ./\  W )
)
33 simp1 957 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
34 simp22 991 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( T  e.  A  /\  -.  T  .<_  W ) )
35 eqid 2436 . . . . . . . 8  |-  ( 0.
`  K )  =  ( 0. `  K
)
365, 29, 35, 7, 26lhpmat 30827 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  -> 
( T  ./\  W
)  =  ( 0.
`  K ) )
3733, 34, 36syl2anc 643 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( T  ./\  W
)  =  ( 0.
`  K ) )
3837oveq1d 6096 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( ( T  ./\  W )  .\/  V )  =  ( ( 0.
`  K )  .\/  V ) )
39 simp23r 1079 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  V  .<_  W )
4013, 5, 6, 29, 7atmod4i1 30663 . . . . . 6  |-  ( ( K  e.  HL  /\  ( V  e.  A  /\  T  e.  ( Base `  K )  /\  W  e.  ( Base `  K ) )  /\  V  .<_  W )  -> 
( ( T  ./\  W )  .\/  V )  =  ( ( T 
.\/  V )  ./\  W ) )
412, 4, 17, 28, 39, 40syl131anc 1197 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( ( T  ./\  W )  .\/  V )  =  ( ( T 
.\/  V )  ./\  W ) )
42 hlol 30159 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OL )
432, 42syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  K  e.  OL )
4413, 7atbase 30087 . . . . . . 7  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
454, 44syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  V  e.  ( Base `  K ) )
4613, 6, 35olj02 30024 . . . . . 6  |-  ( ( K  e.  OL  /\  V  e.  ( Base `  K ) )  -> 
( ( 0. `  K )  .\/  V
)  =  V )
4743, 45, 46syl2anc 643 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( ( 0. `  K )  .\/  V
)  =  V )
4838, 41, 473eqtr3d 2476 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( ( T  .\/  V )  ./\  W )  =  V )
4932, 48breqtrd 4236 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( ( S  .\/  T )  ./\  W )  .<_  V )
50 hlatl 30158 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
512, 50syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  K  e.  AtLat )
52 simp21r 1075 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  -.  S  .<_  W )
53 simp3l 985 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  S  =/=  T )
545, 6, 29, 7, 26lhpat 30840 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  S  =/=  T ) )  ->  ( ( S 
.\/  T )  ./\  W )  e.  A )
552, 25, 12, 52, 3, 53, 54syl222anc 1200 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( ( S  .\/  T )  ./\  W )  e.  A )
565, 7atcmp 30109 . . . 4  |-  ( ( K  e.  AtLat  /\  (
( S  .\/  T
)  ./\  W )  e.  A  /\  V  e.  A )  ->  (
( ( S  .\/  T )  ./\  W )  .<_  V  <->  ( ( S 
.\/  T )  ./\  W )  =  V ) )
5751, 55, 4, 56syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( ( ( S 
.\/  T )  ./\  W )  .<_  V  <->  ( ( S  .\/  T )  ./\  W )  =  V ) )
5849, 57mpbid 202 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( ( S  .\/  T )  ./\  W )  =  V )
5958eqcomd 2441 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  V  =  ( ( S  .\/  T )  ./\  W ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   joincjn 14401   meetcmee 14402   0.cp0 14466   Latclat 14474   OLcol 29972   Atomscatm 30061   AtLatcal 30062   HLchlt 30148   LHypclh 30781
This theorem is referenced by:  cdleme22g  31145
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-p1 14469  df-lat 14475  df-clat 14537  df-oposet 29974  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-psubsp 30300  df-pmap 30301  df-padd 30593  df-lhyp 30785
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