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Theorem cdleme23b 33999
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 1099 . . . . . 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 33011 . . . . . 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 1101 . . . . . 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 1103 . . . . . 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 33016 . . . . . 6  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  e.  B )
112, 5, 6, 10syl3anc 1218 . . . . 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 33013 . . . . . . 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 1014 . . . . . . 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 1100 . . . . . . . 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 33647 . . . . . . . 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 15227 . . . . . . 7  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  W  e.  B )  ->  ( X  ./\  W
)  e.  B )
2113, 14, 18, 20syl3anc 1218 . . . . . 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 15226 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( S  .\/  T )  e.  B  /\  ( X  ./\  W )  e.  B )  ->  (
( S  .\/  T
)  .\/  ( X  ./\ 
W ) )  e.  B )
2313, 11, 21, 22syl3anc 1218 . . . . 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 32879 . . . . 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 1220 . . . 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 15235 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( S  .\/  T )  e.  B  /\  ( X  ./\  W )  e.  B )  ->  ( S  .\/  T )  .<_  ( ( S  .\/  T )  .\/  ( X 
./\  W ) ) )
2813, 11, 21, 27syl3anc 1218 . . . . . 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 15254 . . . . . . 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 1218 . . . . . 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 210 . . . . 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 6111 . . . 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 32939 . . . . . . . . 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 32939 . . . . . . . . 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 15279 . . . . . . . 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 1220 . . . . . . 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 1025 . . . . . . . 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 1026 . . . . . . . 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 6114 . . . . . . 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 15249 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  X  e.  B )  ->  ( X  .\/  X
)  =  X )
4313, 14, 42syl2anc 661 . . . . . . 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 2479 . . . . . 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 6111 . . . . 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 6112 . . . 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 2483 . . 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 1102 . . . 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 1024 . . . 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 33692 . . . 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 1234 . . 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 2518 . 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 2527 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   Basecbs 14179   lecple 14250   joincjn 15119   meetcmee 15120   Latclat 15220   OLcol 32824   Atomscatm 32913   HLchlt 33000   LHypclh 33633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-poset 15121  df-plt 15133  df-lub 15149  df-glb 15150  df-join 15151  df-meet 15152  df-p0 15214  df-p1 15215  df-lat 15221  df-clat 15283  df-oposet 32826  df-ol 32828  df-oml 32829  df-covers 32916  df-ats 32917  df-atl 32948  df-cvlat 32972  df-hlat 33001  df-lhyp 33637
This theorem is referenced by:  cdleme28a  34019
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