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Theorem cdleme23b 30832
Description: Part of proof of Lemma E in [Crawley] p. 113, 4th paragraph, 6th line on p. 115. (Contributed by NM, 8-Dec-2012.)
Hypotheses
Ref Expression
cdleme23.b  |-  B  =  ( Base `  K
)
cdleme23.l  |-  .<_  =  ( le `  K )
cdleme23.j  |-  .\/  =  ( join `  K )
cdleme23.m  |-  ./\  =  ( meet `  K )
cdleme23.a  |-  A  =  ( Atoms `  K )
cdleme23.h  |-  H  =  ( LHyp `  K
)
cdleme23.v  |-  V  =  ( ( S  .\/  T )  ./\  ( X  ./\ 
W ) )
Assertion
Ref Expression
cdleme23b  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  V  e.  A )

Proof of Theorem cdleme23b
StepHypRef Expression
1 cdleme23.v . 2  |-  V  =  ( ( S  .\/  T )  ./\  ( X  ./\ 
W ) )
2 simp11l 1068 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  K  e.  HL )
3 hlol 29844 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OL )
42, 3syl 16 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  K  e.  OL )
5 simp12l 1070 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  S  e.  A )
6 simp13l 1072 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  T  e.  A )
7 cdleme23.b . . . . . . 7  |-  B  =  ( Base `  K
)
8 cdleme23.j . . . . . . 7  |-  .\/  =  ( join `  K )
9 cdleme23.a . . . . . . 7  |-  A  =  ( Atoms `  K )
107, 8, 9hlatjcl 29849 . . . . . 6  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  e.  B )
112, 5, 6, 10syl3anc 1184 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( S  .\/  T )  e.  B
)
12 hllat 29846 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
132, 12syl 16 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  K  e.  Lat )
14 simp2l 983 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  X  e.  B )
15 simp11r 1069 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  W  e.  H )
16 cdleme23.h . . . . . . . . 9  |-  H  =  ( LHyp `  K
)
177, 16lhpbase 30480 . . . . . . . 8  |-  ( W  e.  H  ->  W  e.  B )
1815, 17syl 16 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  W  e.  B )
19 cdleme23.m . . . . . . . 8  |-  ./\  =  ( meet `  K )
207, 19latmcl 14435 . . . . . . 7  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  e.  B )
2113, 14, 18, 20syl3anc 1184 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( X  ./\ 
W )  e.  B
)
227, 8latjcl 14434 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( S  .\/  T )  e.  B  /\  ( X  ./\  W )  e.  B )  ->  (
( S  .\/  T
)  .\/  ( X  ./\ 
W ) )  e.  B )
2313, 11, 21, 22syl3anc 1184 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( ( S  .\/  T )  .\/  ( X  ./\  W ) )  e.  B )
247, 19latmassOLD 29712 . . . . 5  |-  ( ( K  e.  OL  /\  ( ( S  .\/  T )  e.  B  /\  ( ( S  .\/  T )  .\/  ( X 
./\  W ) )  e.  B  /\  W  e.  B ) )  -> 
( ( ( S 
.\/  T )  ./\  ( ( S  .\/  T )  .\/  ( X 
./\  W ) ) )  ./\  W )  =  ( ( S 
.\/  T )  ./\  ( ( ( S 
.\/  T )  .\/  ( X  ./\  W ) )  ./\  W )
) )
254, 11, 23, 18, 24syl13anc 1186 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( (
( S  .\/  T
)  ./\  ( ( S  .\/  T )  .\/  ( X  ./\  W ) ) )  ./\  W
)  =  ( ( S  .\/  T ) 
./\  ( ( ( S  .\/  T ) 
.\/  ( X  ./\  W ) )  ./\  W
) ) )
26 cdleme23.l . . . . . . . 8  |-  .<_  =  ( le `  K )
277, 26, 8latlej1 14444 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( S  .\/  T )  e.  B  /\  ( X  ./\  W )  e.  B )  ->  ( S  .\/  T )  .<_  ( ( S  .\/  T )  .\/  ( X 
./\  W ) ) )
2813, 11, 21, 27syl3anc 1184 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( S  .\/  T )  .<_  ( ( S  .\/  T ) 
.\/  ( X  ./\  W ) ) )
297, 26, 19latleeqm1 14463 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( S  .\/  T )  e.  B  /\  (
( S  .\/  T
)  .\/  ( X  ./\ 
W ) )  e.  B )  ->  (
( S  .\/  T
)  .<_  ( ( S 
.\/  T )  .\/  ( X  ./\  W ) )  <->  ( ( S 
.\/  T )  ./\  ( ( S  .\/  T )  .\/  ( X 
./\  W ) ) )  =  ( S 
.\/  T ) ) )
3013, 11, 23, 29syl3anc 1184 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( ( S  .\/  T )  .<_  ( ( S  .\/  T )  .\/  ( X 
./\  W ) )  <-> 
( ( S  .\/  T )  ./\  ( ( S  .\/  T )  .\/  ( X  ./\  W ) ) )  =  ( S  .\/  T ) ) )
3128, 30mpbid 202 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( ( S  .\/  T )  ./\  ( ( S  .\/  T )  .\/  ( X 
./\  W ) ) )  =  ( S 
.\/  T ) )
3231oveq1d 6055 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( (
( S  .\/  T
)  ./\  ( ( S  .\/  T )  .\/  ( X  ./\  W ) ) )  ./\  W
)  =  ( ( S  .\/  T ) 
./\  W ) )
337, 9atbase 29772 . . . . . . . . 9  |-  ( S  e.  A  ->  S  e.  B )
345, 33syl 16 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  S  e.  B )
357, 9atbase 29772 . . . . . . . . 9  |-  ( T  e.  A  ->  T  e.  B )
366, 35syl 16 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  T  e.  B )
377, 8latjjdir 14488 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( S  e.  B  /\  T  e.  B  /\  ( X  ./\  W
)  e.  B ) )  ->  ( ( S  .\/  T )  .\/  ( X  ./\  W ) )  =  ( ( S  .\/  ( X 
./\  W ) ) 
.\/  ( T  .\/  ( X  ./\  W ) ) ) )
3813, 34, 36, 21, 37syl13anc 1186 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( ( S  .\/  T )  .\/  ( X  ./\  W ) )  =  ( ( S  .\/  ( X 
./\  W ) ) 
.\/  ( T  .\/  ( X  ./\  W ) ) ) )
39 simp32 994 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( S  .\/  ( X  ./\  W
) )  =  X )
40 simp33 995 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( T  .\/  ( X  ./\  W
) )  =  X )
4139, 40oveq12d 6058 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( ( S  .\/  ( X  ./\  W ) )  .\/  ( T  .\/  ( X  ./\  W ) ) )  =  ( X  .\/  X
) )
427, 8latjidm 14458 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( X  .\/  X
)  =  X )
4313, 14, 42syl2anc 643 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( X  .\/  X )  =  X )
4438, 41, 433eqtrd 2440 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( ( S  .\/  T )  .\/  ( X  ./\  W ) )  =  X )
4544oveq1d 6055 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( (
( S  .\/  T
)  .\/  ( X  ./\ 
W ) )  ./\  W )  =  ( X 
./\  W ) )
4645oveq2d 6056 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( ( S  .\/  T )  ./\  ( ( ( S 
.\/  T )  .\/  ( X  ./\  W ) )  ./\  W )
)  =  ( ( S  .\/  T ) 
./\  ( X  ./\  W ) ) )
4725, 32, 463eqtr3d 2444 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( ( S  .\/  T )  ./\  W )  =  ( ( S  .\/  T ) 
./\  ( X  ./\  W ) ) )
48 simp12r 1071 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  -.  S  .<_  W )
49 simp31 993 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  S  =/=  T )
5026, 8, 19, 9, 16lhpat 30525 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  S  =/=  T ) )  ->  ( ( S 
.\/  T )  ./\  W )  e.  A )
512, 15, 5, 48, 6, 49, 50syl222anc 1200 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( ( S  .\/  T )  ./\  W )  e.  A )
5247, 51eqeltrrd 2479 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  ( ( S  .\/  T )  ./\  ( X  ./\  W ) )  e.  A )
531, 52syl5eqel 2488 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  /\  ( X  e.  B  /\  -.  X  .<_  W )  /\  ( S  =/= 
T  /\  ( S  .\/  ( X  ./\  W
) )  =  X  /\  ( T  .\/  ( X  ./\  W ) )  =  X ) )  ->  V  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   Latclat 14429   OLcol 29657   Atomscatm 29746   HLchlt 29833   LHypclh 30466
This theorem is referenced by:  cdleme28a  30852
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-lhyp 30470
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