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Theorem dalawlem11 33371
Description: Lemma for dalaw 33376. First part of dalawlem13 33373. (Contributed by NM, 17-Sep-2012.)
Hypotheses
Ref Expression
dalawlem.l  |-  .<_  =  ( le `  K )
dalawlem.j  |-  .\/  =  ( join `  K )
dalawlem.m  |-  ./\  =  ( meet `  K )
dalawlem.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
dalawlem11  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( (
( Q  .\/  R
)  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) )

Proof of Theorem dalawlem11
StepHypRef Expression
1 eqid 2423 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
2 dalawlem.l . . . 4  |-  .<_  =  ( le `  K )
3 simp11 1036 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  K  e.  HL )
4 hllat 32854 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
53, 4syl 17 . . . 4  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  K  e.  Lat )
6 simp21 1039 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  P  e.  A )
7 simp22 1040 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  Q  e.  A )
8 dalawlem.j . . . . . . 7  |-  .\/  =  ( join `  K )
9 dalawlem.a . . . . . . 7  |-  A  =  ( Atoms `  K )
101, 8, 9hlatjcl 32857 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
113, 6, 7, 10syl3anc 1265 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( P  .\/  Q )  e.  (
Base `  K )
)
12 simp31 1042 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  S  e.  A )
13 simp32 1043 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  T  e.  A )
141, 8, 9hlatjcl 32857 . . . . . 6  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
153, 12, 13, 14syl3anc 1265 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( S  .\/  T )  e.  (
Base `  K )
)
16 dalawlem.m . . . . . 6  |-  ./\  =  ( meet `  K )
171, 16latmcl 16291 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  ( S  .\/  T )  e.  (
Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  e.  ( Base `  K ) )
185, 11, 15, 17syl3anc 1265 . . . 4  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  e.  ( Base `  K ) )
19 simp23 1041 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  R  e.  A )
201, 8, 9hlatjcl 32857 . . . . 5  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
213, 7, 19, 20syl3anc 1265 . . . 4  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( Q  .\/  R )  e.  (
Base `  K )
)
221, 2, 16latmle1 16315 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  ( S  .\/  T )  e.  (
Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( P  .\/  Q ) )
235, 11, 15, 22syl3anc 1265 . . . 4  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( P  .\/  Q ) )
24 simp12 1037 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  P  .<_  ( Q  .\/  R ) )
251, 9atbase 32780 . . . . . . 7  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
267, 25syl 17 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  Q  e.  ( Base `  K )
)
271, 9atbase 32780 . . . . . . 7  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
2819, 27syl 17 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  R  e.  ( Base `  K )
)
291, 2, 8latlej1 16299 . . . . . 6  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  Q  .<_  ( Q  .\/  R
) )
305, 26, 28, 29syl3anc 1265 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  Q  .<_  ( Q  .\/  R ) )
311, 9atbase 32780 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
326, 31syl 17 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  P  e.  ( Base `  K )
)
331, 2, 8latjle12 16301 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K )  /\  ( Q  .\/  R )  e.  ( Base `  K
) ) )  -> 
( ( P  .<_  ( Q  .\/  R )  /\  Q  .<_  ( Q 
.\/  R ) )  <-> 
( P  .\/  Q
)  .<_  ( Q  .\/  R ) ) )
345, 32, 26, 21, 33syl13anc 1267 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .<_  ( Q  .\/  R )  /\  Q  .<_  ( Q  .\/  R ) )  <->  ( P  .\/  Q )  .<_  ( Q  .\/  R ) ) )
3524, 30, 34mpbi2and 930 . . . 4  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( P  .\/  Q )  .<_  ( Q 
.\/  R ) )
361, 2, 5, 18, 11, 21, 23, 35lattrd 16297 . . 3  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( Q  .\/  R ) )
371, 9atbase 32780 . . . . . . . 8  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
3813, 37syl 17 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  T  e.  ( Base `  K )
)
391, 8latjcl 16290 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  T  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  .\/  T )  e.  ( Base `  K ) )
405, 11, 38, 39syl3anc 1265 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .\/  Q )  .\/  T )  e.  ( Base `  K ) )
411, 16latmcl 16291 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( ( P  .\/  Q )  .\/  T )  e.  ( Base `  K
)  /\  ( S  .\/  T )  e.  (
Base `  K )
)  ->  ( (
( P  .\/  Q
)  .\/  T )  ./\  ( S  .\/  T
) )  e.  (
Base `  K )
)
425, 40, 15, 41syl3anc 1265 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( (
( P  .\/  Q
)  .\/  T )  ./\  ( S  .\/  T
) )  e.  (
Base `  K )
)
431, 8, 9hlatjcl 32857 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  R  e.  A  /\  P  e.  A )  ->  ( R  .\/  P
)  e.  ( Base `  K ) )
443, 19, 6, 43syl3anc 1265 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( R  .\/  P )  e.  (
Base `  K )
)
45 simp33 1044 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  U  e.  A )
461, 8, 9hlatjcl 32857 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  U  e.  A  /\  S  e.  A )  ->  ( U  .\/  S
)  e.  ( Base `  K ) )
473, 45, 12, 46syl3anc 1265 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( U  .\/  S )  e.  (
Base `  K )
)
481, 16latmcl 16291 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( R  .\/  P )  e.  ( Base `  K
)  /\  ( U  .\/  S )  e.  (
Base `  K )
)  ->  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )  e.  ( Base `  K ) )
495, 44, 47, 48syl3anc 1265 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )  e.  ( Base `  K ) )
501, 9atbase 32780 . . . . . . . 8  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
5145, 50syl 17 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  U  e.  ( Base `  K )
)
521, 8latjcl 16290 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )  e.  ( Base `  K
)  /\  U  e.  ( Base `  K )
)  ->  ( (
( R  .\/  P
)  ./\  ( U  .\/  S ) )  .\/  U )  e.  ( Base `  K ) )
535, 49, 51, 52syl3anc 1265 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( (
( R  .\/  P
)  ./\  ( U  .\/  S ) )  .\/  U )  e.  ( Base `  K ) )
541, 8latjcl 16290 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( ( ( R 
.\/  P )  ./\  ( U  .\/  S ) )  .\/  U )  e.  ( Base `  K
)  /\  T  e.  ( Base `  K )
)  ->  ( (
( ( R  .\/  P )  ./\  ( U  .\/  S ) )  .\/  U )  .\/  T )  e.  ( Base `  K
) )
555, 53, 38, 54syl3anc 1265 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( (
( ( R  .\/  P )  ./\  ( U  .\/  S ) )  .\/  U )  .\/  T )  e.  ( Base `  K
) )
561, 2, 8latlej1 16299 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  T  e.  ( Base `  K )
)  ->  ( P  .\/  Q )  .<_  ( ( P  .\/  Q ) 
.\/  T ) )
575, 11, 38, 56syl3anc 1265 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( P  .\/  Q )  .<_  ( ( P  .\/  Q ) 
.\/  T ) )
581, 2, 16latmlem1 16320 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( ( P  .\/  Q )  e.  ( Base `  K )  /\  (
( P  .\/  Q
)  .\/  T )  e.  ( Base `  K
)  /\  ( S  .\/  T )  e.  (
Base `  K )
) )  ->  (
( P  .\/  Q
)  .<_  ( ( P 
.\/  Q )  .\/  T )  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( (
( P  .\/  Q
)  .\/  T )  ./\  ( S  .\/  T
) ) ) )
595, 11, 40, 15, 58syl13anc 1267 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .\/  Q )  .<_  ( ( P  .\/  Q )  .\/  T )  ->  ( ( P 
.\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( (
( P  .\/  Q
)  .\/  T )  ./\  ( S  .\/  T
) ) ) )
6057, 59mpd 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( (
( P  .\/  Q
)  .\/  T )  ./\  ( S  .\/  T
) ) )
611, 2, 8latlej2 16300 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  T  e.  ( Base `  K )
)  ->  T  .<_  ( ( P  .\/  Q
)  .\/  T )
)
625, 11, 38, 61syl3anc 1265 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  T  .<_  ( ( P  .\/  Q
)  .\/  T )
)
631, 2, 8, 16, 9atmod2i2 33352 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( S  e.  A  /\  ( ( P  .\/  Q )  .\/  T )  e.  ( Base `  K
)  /\  T  e.  ( Base `  K )
)  /\  T  .<_  ( ( P  .\/  Q
)  .\/  T )
)  ->  ( (
( ( P  .\/  Q )  .\/  T ) 
./\  S )  .\/  T )  =  ( ( ( P  .\/  Q
)  .\/  T )  ./\  ( S  .\/  T
) ) )
643, 12, 40, 38, 62, 63syl131anc 1278 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( (
( ( P  .\/  Q )  .\/  T ) 
./\  S )  .\/  T )  =  ( ( ( P  .\/  Q
)  .\/  T )  ./\  ( S  .\/  T
) ) )
651, 8, 9hlatjcl 32857 . . . . . . . . . . . . . 14  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
663, 7, 13, 65syl3anc 1265 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( Q  .\/  T )  e.  (
Base `  K )
)
671, 8, 9hlatjcl 32857 . . . . . . . . . . . . . 14  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
683, 6, 12, 67syl3anc 1265 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( P  .\/  S )  e.  (
Base `  K )
)
691, 16latmcom 16314 . . . . . . . . . . . . 13  |-  ( ( K  e.  Lat  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  =  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) ) )
705, 66, 68, 69syl3anc 1265 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  =  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) ) )
71 simp13 1038 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )
7270, 71eqbrtrd 4442 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( R  .\/  U ) )
731, 16latmcl 16291 . . . . . . . . . . . . 13  |-  ( ( K  e.  Lat  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  e.  ( Base `  K ) )
745, 66, 68, 73syl3anc 1265 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  e.  ( Base `  K ) )
751, 8, 9hlatjcl 32857 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  R  e.  A  /\  U  e.  A )  ->  ( R  .\/  U
)  e.  ( Base `  K ) )
763, 19, 45, 75syl3anc 1265 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( R  .\/  U )  e.  (
Base `  K )
)
771, 2, 8latjlej2 16305 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  ( ( ( Q 
.\/  T )  ./\  ( P  .\/  S ) )  e.  ( Base `  K )  /\  ( R  .\/  U )  e.  ( Base `  K
)  /\  P  e.  ( Base `  K )
) )  ->  (
( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( R  .\/  U )  ->  ( P  .\/  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) ) ) 
.<_  ( P  .\/  ( R  .\/  U ) ) ) )
785, 74, 76, 32, 77syl13anc 1267 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( (
( Q  .\/  T
)  ./\  ( P  .\/  S ) )  .<_  ( R  .\/  U )  ->  ( P  .\/  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) ) ) 
.<_  ( P  .\/  ( R  .\/  U ) ) ) )
7972, 78mpd 15 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( P  .\/  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) ) ) 
.<_  ( P  .\/  ( R  .\/  U ) ) )
801, 9atbase 32780 . . . . . . . . . . . . 13  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
8112, 80syl 17 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  S  e.  ( Base `  K )
)
821, 2, 8latlej1 16299 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  S  e.  ( Base `  K
) )  ->  P  .<_  ( P  .\/  S
) )
835, 32, 81, 82syl3anc 1265 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  P  .<_  ( P  .\/  S ) )
841, 2, 8, 16, 9atmod1i1 33347 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  ( Q  .\/  T
)  e.  ( Base `  K )  /\  ( P  .\/  S )  e.  ( Base `  K
) )  /\  P  .<_  ( P  .\/  S
) )  ->  ( P  .\/  ( ( Q 
.\/  T )  ./\  ( P  .\/  S ) ) )  =  ( ( P  .\/  ( Q  .\/  T ) ) 
./\  ( P  .\/  S ) ) )
853, 6, 66, 68, 83, 84syl131anc 1278 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( P  .\/  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) ) )  =  ( ( P 
.\/  ( Q  .\/  T ) )  ./\  ( P  .\/  S ) ) )
868, 9hlatjass 32860 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  R  e.  A  /\  U  e.  A
) )  ->  (
( P  .\/  R
)  .\/  U )  =  ( P  .\/  ( R  .\/  U ) ) )
873, 6, 19, 45, 86syl13anc 1267 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .\/  R )  .\/  U )  =  ( P 
.\/  ( R  .\/  U ) ) )
888, 9hlatjcom 32858 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  ( P  .\/  R
)  =  ( R 
.\/  P ) )
893, 6, 19, 88syl3anc 1265 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( P  .\/  R )  =  ( R  .\/  P ) )
9089oveq1d 6318 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .\/  R )  .\/  U )  =  ( ( R  .\/  P ) 
.\/  U ) )
9187, 90eqtr3d 2466 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( P  .\/  ( R  .\/  U
) )  =  ( ( R  .\/  P
)  .\/  U )
)
9279, 85, 913brtr3d 4451 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .\/  ( Q  .\/  T ) )  ./\  ( P  .\/  S ) ) 
.<_  ( ( R  .\/  P )  .\/  U ) )
931, 2, 8latlej2 16300 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  U  e.  ( Base `  K )  /\  S  e.  ( Base `  K
) )  ->  S  .<_  ( U  .\/  S
) )
945, 51, 81, 93syl3anc 1265 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  S  .<_  ( U  .\/  S ) )
951, 8latjcl 16290 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  ( Q  .\/  T )  e.  ( Base `  K
) )  ->  ( P  .\/  ( Q  .\/  T ) )  e.  (
Base `  K )
)
965, 32, 66, 95syl3anc 1265 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( P  .\/  ( Q  .\/  T
) )  e.  (
Base `  K )
)
971, 16latmcl 16291 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  ( P  .\/  ( Q 
.\/  T ) )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( P  .\/  ( Q  .\/  T ) )  ./\  ( P  .\/  S ) )  e.  ( Base `  K
) )
985, 96, 68, 97syl3anc 1265 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .\/  ( Q  .\/  T ) )  ./\  ( P  .\/  S ) )  e.  ( Base `  K
) )
991, 8latjcl 16290 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  ( R  .\/  P )  e.  ( Base `  K
)  /\  U  e.  ( Base `  K )
)  ->  ( ( R  .\/  P )  .\/  U )  e.  ( Base `  K ) )
1005, 44, 51, 99syl3anc 1265 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( R  .\/  P )  .\/  U )  e.  ( Base `  K ) )
1011, 2, 16latmlem12 16322 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( ( ( P 
.\/  ( Q  .\/  T ) )  ./\  ( P  .\/  S ) )  e.  ( Base `  K
)  /\  ( ( R  .\/  P )  .\/  U )  e.  ( Base `  K ) )  /\  ( S  e.  ( Base `  K )  /\  ( U  .\/  S )  e.  ( Base `  K
) ) )  -> 
( ( ( ( P  .\/  ( Q 
.\/  T ) ) 
./\  ( P  .\/  S ) )  .<_  ( ( R  .\/  P ) 
.\/  U )  /\  S  .<_  ( U  .\/  S ) )  ->  (
( ( P  .\/  ( Q  .\/  T ) )  ./\  ( P  .\/  S ) )  ./\  S )  .<_  ( (
( R  .\/  P
)  .\/  U )  ./\  ( U  .\/  S
) ) ) )
1025, 98, 100, 81, 47, 101syl122anc 1274 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( (
( ( P  .\/  ( Q  .\/  T ) )  ./\  ( P  .\/  S ) )  .<_  ( ( R  .\/  P )  .\/  U )  /\  S  .<_  ( U 
.\/  S ) )  ->  ( ( ( P  .\/  ( Q 
.\/  T ) ) 
./\  ( P  .\/  S ) )  ./\  S
)  .<_  ( ( ( R  .\/  P ) 
.\/  U )  ./\  ( U  .\/  S ) ) ) )
10392, 94, 102mp2and 684 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( (
( P  .\/  ( Q  .\/  T ) ) 
./\  ( P  .\/  S ) )  ./\  S
)  .<_  ( ( ( R  .\/  P ) 
.\/  U )  ./\  ( U  .\/  S ) ) )
104 hlol 32852 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  OL )
1053, 104syl 17 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  K  e.  OL )
1061, 16latmassOLD 32720 . . . . . . . . . 10  |-  ( ( K  e.  OL  /\  ( ( P  .\/  ( Q  .\/  T ) )  e.  ( Base `  K )  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
) )  ->  (
( ( P  .\/  ( Q  .\/  T ) )  ./\  ( P  .\/  S ) )  ./\  S )  =  ( ( P  .\/  ( Q 
.\/  T ) ) 
./\  ( ( P 
.\/  S )  ./\  S ) ) )
107105, 96, 68, 81, 106syl13anc 1267 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( (
( P  .\/  ( Q  .\/  T ) ) 
./\  ( P  .\/  S ) )  ./\  S
)  =  ( ( P  .\/  ( Q 
.\/  T ) ) 
./\  ( ( P 
.\/  S )  ./\  S ) ) )
1088, 9hlatjass 32860 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  T  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  T )  =  ( P  .\/  ( Q  .\/  T ) ) )
1093, 6, 7, 13, 108syl13anc 1267 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .\/  Q )  .\/  T )  =  ( P 
.\/  ( Q  .\/  T ) ) )
110109eqcomd 2431 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( P  .\/  ( Q  .\/  T
) )  =  ( ( P  .\/  Q
)  .\/  T )
)
1111, 2, 8latlej2 16300 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  S  e.  ( Base `  K
) )  ->  S  .<_  ( P  .\/  S
) )
1125, 32, 81, 111syl3anc 1265 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  S  .<_  ( P  .\/  S ) )
1131, 2, 16latleeqm2 16319 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  S  e.  ( Base `  K )  /\  ( P  .\/  S )  e.  ( Base `  K
) )  ->  ( S  .<_  ( P  .\/  S )  <->  ( ( P 
.\/  S )  ./\  S )  =  S ) )
1145, 81, 68, 113syl3anc 1265 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( S  .<_  ( P  .\/  S
)  <->  ( ( P 
.\/  S )  ./\  S )  =  S ) )
115112, 114mpbid 214 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .\/  S )  ./\  S )  =  S )
116110, 115oveq12d 6321 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .\/  ( Q  .\/  T ) )  ./\  (
( P  .\/  S
)  ./\  S )
)  =  ( ( ( P  .\/  Q
)  .\/  T )  ./\  S ) )
117107, 116eqtr2d 2465 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( (
( P  .\/  Q
)  .\/  T )  ./\  S )  =  ( ( ( P  .\/  ( Q  .\/  T ) )  ./\  ( P  .\/  S ) )  ./\  S ) )
1181, 2, 8latlej1 16299 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  U  e.  ( Base `  K )  /\  S  e.  ( Base `  K
) )  ->  U  .<_  ( U  .\/  S
) )
1195, 51, 81, 118syl3anc 1265 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  U  .<_  ( U  .\/  S ) )
1201, 2, 8, 16, 9atmod4i1 33356 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( U  e.  A  /\  ( R  .\/  P
)  e.  ( Base `  K )  /\  ( U  .\/  S )  e.  ( Base `  K
) )  /\  U  .<_  ( U  .\/  S
) )  ->  (
( ( R  .\/  P )  ./\  ( U  .\/  S ) )  .\/  U )  =  ( ( ( R  .\/  P
)  .\/  U )  ./\  ( U  .\/  S
) ) )
1213, 45, 44, 47, 119, 120syl131anc 1278 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( (
( R  .\/  P
)  ./\  ( U  .\/  S ) )  .\/  U )  =  ( ( ( R  .\/  P
)  .\/  U )  ./\  ( U  .\/  S
) ) )
122103, 117, 1213brtr4d 4452 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( (
( P  .\/  Q
)  .\/  T )  ./\  S )  .<_  ( ( ( R  .\/  P
)  ./\  ( U  .\/  S ) )  .\/  U ) )
1231, 16latmcl 16291 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( ( P  .\/  Q )  .\/  T )  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
)  ->  ( (
( P  .\/  Q
)  .\/  T )  ./\  S )  e.  (
Base `  K )
)
1245, 40, 81, 123syl3anc 1265 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( (
( P  .\/  Q
)  .\/  T )  ./\  S )  e.  (
Base `  K )
)
1251, 2, 8latjlej1 16304 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( ( ( ( P  .\/  Q ) 
.\/  T )  ./\  S )  e.  ( Base `  K )  /\  (
( ( R  .\/  P )  ./\  ( U  .\/  S ) )  .\/  U )  e.  ( Base `  K )  /\  T  e.  ( Base `  K
) ) )  -> 
( ( ( ( P  .\/  Q ) 
.\/  T )  ./\  S )  .<_  ( (
( R  .\/  P
)  ./\  ( U  .\/  S ) )  .\/  U )  ->  ( (
( ( P  .\/  Q )  .\/  T ) 
./\  S )  .\/  T )  .<_  ( (
( ( R  .\/  P )  ./\  ( U  .\/  S ) )  .\/  U )  .\/  T ) ) )
1265, 124, 53, 38, 125syl13anc 1267 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( (
( ( P  .\/  Q )  .\/  T ) 
./\  S )  .<_  ( ( ( R 
.\/  P )  ./\  ( U  .\/  S ) )  .\/  U )  ->  ( ( ( ( P  .\/  Q
)  .\/  T )  ./\  S )  .\/  T
)  .<_  ( ( ( ( R  .\/  P
)  ./\  ( U  .\/  S ) )  .\/  U )  .\/  T ) ) )
127122, 126mpd 15 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( (
( ( P  .\/  Q )  .\/  T ) 
./\  S )  .\/  T )  .<_  ( (
( ( R  .\/  P )  ./\  ( U  .\/  S ) )  .\/  U )  .\/  T ) )
12864, 127eqbrtrrd 4444 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( (
( P  .\/  Q
)  .\/  T )  ./\  ( S  .\/  T
) )  .<_  ( ( ( ( R  .\/  P )  ./\  ( U  .\/  S ) )  .\/  U )  .\/  T ) )
1291, 2, 5, 18, 42, 55, 60, 128lattrd 16297 . . . 4  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( (
( ( R  .\/  P )  ./\  ( U  .\/  S ) )  .\/  U )  .\/  T ) )
1301, 8latj31 16338 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( ( R 
.\/  P )  ./\  ( U  .\/  S ) )  e.  ( Base `  K )  /\  U  e.  ( Base `  K
)  /\  T  e.  ( Base `  K )
) )  ->  (
( ( ( R 
.\/  P )  ./\  ( U  .\/  S ) )  .\/  U ) 
.\/  T )  =  ( ( T  .\/  U )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) )
1315, 49, 51, 38, 130syl13anc 1267 . . . 4  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( (
( ( R  .\/  P )  ./\  ( U  .\/  S ) )  .\/  U )  .\/  T )  =  ( ( T 
.\/  U )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) )
132129, 131breqtrd 4446 . . 3  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( ( T  .\/  U )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) )
1331, 8, 9hlatjcl 32857 . . . . . 6  |-  ( ( K  e.  HL  /\  T  e.  A  /\  U  e.  A )  ->  ( T  .\/  U
)  e.  ( Base `  K ) )
1343, 13, 45, 133syl3anc 1265 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( T  .\/  U )  e.  (
Base `  K )
)
1351, 8latjcl 16290 . . . . 5  |-  ( ( K  e.  Lat  /\  ( T  .\/  U )  e.  ( Base `  K
)  /\  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )  e.  ( Base `  K ) )  -> 
( ( T  .\/  U )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) )  e.  ( Base `  K
) )
1365, 134, 49, 135syl3anc 1265 . . . 4  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( T  .\/  U )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) )  e.  ( Base `  K
) )
1371, 2, 16latlem12 16317 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( ( P 
.\/  Q )  ./\  ( S  .\/  T ) )  e.  ( Base `  K )  /\  ( Q  .\/  R )  e.  ( Base `  K
)  /\  ( ( T  .\/  U )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) )  e.  ( Base `  K
) ) )  -> 
( ( ( ( P  .\/  Q ) 
./\  ( S  .\/  T ) )  .<_  ( Q 
.\/  R )  /\  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( ( T  .\/  U )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) )  <-> 
( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( ( Q  .\/  R )  ./\  ( ( T  .\/  U )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) ) ) )
1385, 18, 21, 136, 137syl13anc 1267 . . 3  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( (
( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( Q  .\/  R )  /\  ( ( P 
.\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( ( T  .\/  U )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) )  <->  ( ( P 
.\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( ( Q  .\/  R )  ./\  ( ( T  .\/  U )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) ) ) )
13936, 132, 138mpbi2and 930 . 2  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( ( Q  .\/  R )  ./\  ( ( T  .\/  U )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) ) )
1401, 2, 16latmle1 16315 . . . . 5  |-  ( ( K  e.  Lat  /\  ( R  .\/  P )  e.  ( Base `  K
)  /\  ( U  .\/  S )  e.  (
Base `  K )
)  ->  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )  .<_  ( R  .\/  P ) )
1415, 44, 47, 140syl3anc 1265 . . . 4  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )  .<_  ( R  .\/  P ) )
1421, 2, 8latlej2 16300 . . . . . 6  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  R  e.  ( Base `  K
) )  ->  R  .<_  ( Q  .\/  R
) )
1435, 26, 28, 142syl3anc 1265 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  R  .<_  ( Q  .\/  R ) )
1441, 2, 8latjle12 16301 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( R  e.  ( Base `  K )  /\  P  e.  ( Base `  K )  /\  ( Q  .\/  R )  e.  ( Base `  K
) ) )  -> 
( ( R  .<_  ( Q  .\/  R )  /\  P  .<_  ( Q 
.\/  R ) )  <-> 
( R  .\/  P
)  .<_  ( Q  .\/  R ) ) )
1455, 28, 32, 21, 144syl13anc 1267 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( R  .<_  ( Q  .\/  R )  /\  P  .<_  ( Q  .\/  R ) )  <->  ( R  .\/  P )  .<_  ( Q  .\/  R ) ) )
146143, 24, 145mpbi2and 930 . . . 4  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( R  .\/  P )  .<_  ( Q 
.\/  R ) )
1471, 2, 5, 49, 44, 21, 141, 146lattrd 16297 . . 3  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )  .<_  ( Q  .\/  R ) )
1481, 2, 8, 16, 9llnmod2i2 33353 . . 3  |-  ( ( ( K  e.  HL  /\  ( Q  .\/  R
)  e.  ( Base `  K )  /\  (
( R  .\/  P
)  ./\  ( U  .\/  S ) )  e.  ( Base `  K
) )  /\  ( T  e.  A  /\  U  e.  A )  /\  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )  .<_  ( Q  .\/  R ) )  ->  ( (
( Q  .\/  R
)  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) )  =  ( ( Q 
.\/  R )  ./\  ( ( T  .\/  U )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) ) )
1493, 21, 49, 13, 45, 147, 148syl321anc 1287 . 2  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( (
( Q  .\/  R
)  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) )  =  ( ( Q 
.\/  R )  ./\  ( ( T  .\/  U )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) ) )
150139, 149breqtrrd 4448 1  |-  ( ( ( K  e.  HL  /\  P  .<_  ( Q  .\/  R )  /\  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( (
( Q  .\/  R
)  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869   class class class wbr 4421   ` cfv 5599  (class class class)co 6303   Basecbs 15114   lecple 15190   joincjn 16182   meetcmee 16183   Latclat 16284   OLcol 32665   Atomscatm 32754   HLchlt 32841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-iun 4299  df-iin 4300  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-1st 6805  df-2nd 6806  df-preset 16166  df-poset 16184  df-plt 16197  df-lub 16213  df-glb 16214  df-join 16215  df-meet 16216  df-p0 16278  df-lat 16285  df-clat 16347  df-oposet 32667  df-ol 32669  df-oml 32670  df-covers 32757  df-ats 32758  df-atl 32789  df-cvlat 32813  df-hlat 32842  df-psubsp 32993  df-pmap 32994  df-padd 33286
This theorem is referenced by:  dalawlem13  33373
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