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Theorem cdleme23b 30539
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 1066 . . . . . 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 29551 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OL )
42, 3syl 15 . . . . 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 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 ) )  ->  S  e.  A )
6 simp13l 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 ) )  ->  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 29556 . . . . . 6  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  e.  B )
112, 5, 6, 10syl3anc 1182 . . . . 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 29553 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
132, 12syl 15 . . . . . 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 981 . . . . . . 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 1067 . . . . . . . 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 30187 . . . . . . . 8  |-  ( W  e.  H  ->  W  e.  B )
1815, 17syl 15 . . . . . . 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 14157 . . . . . . 7  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  e.  B )
2113, 14, 18, 20syl3anc 1182 . . . . . 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 14156 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( S  .\/  T )  e.  B  /\  ( X  ./\  W )  e.  B )  ->  (
( S  .\/  T
)  .\/  ( X  ./\ 
W ) )  e.  B )
2313, 11, 21, 22syl3anc 1182 . . . . 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 29419 . . . . 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 1184 . . . 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 14166 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( S  .\/  T )  e.  B  /\  ( X  ./\  W )  e.  B )  ->  ( S  .\/  T )  .<_  ( ( S  .\/  T )  .\/  ( X 
./\  W ) ) )
2813, 11, 21, 27syl3anc 1182 . . . . . 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 14185 . . . . . . 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 1182 . . . . . 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 201 . . . . 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 5873 . . . 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 29479 . . . . . . . . 9  |-  ( S  e.  A  ->  S  e.  B )
345, 33syl 15 . . . . . . . 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 29479 . . . . . . . . 9  |-  ( T  e.  A  ->  T  e.  B )
366, 35syl 15 . . . . . . . 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 14210 . . . . . . . 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 1184 . . . . . . 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 992 . . . . . . . 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 993 . . . . . . . 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 5876 . . . . . . 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 14180 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( X  .\/  X
)  =  X )
4313, 14, 42syl2anc 642 . . . . . . 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 2319 . . . . . 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 5873 . . . . 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 5874 . . . 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 2323 . . 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 1069 . . . 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 991 . . . 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 30232 . . . 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 1198 . . 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 2358 . 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 2367 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 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   Latclat 14151   OLcol 29364   Atomscatm 29453   HLchlt 29540   LHypclh 30173
This theorem is referenced by:  cdleme28a  30559
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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