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Theorem dalawlem6 33826
Description: Lemma for dalaw 33836. First piece of dalawlem8 33828. (Contributed by NM, 6-Oct-2012.)
Hypotheses
Ref Expression
dalawlem.l  |-  .<_  =  ( le `  K )
dalawlem.j  |-  .\/  =  ( join `  K )
dalawlem.m  |-  ./\  =  ( meet `  K )
dalawlem.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
dalawlem6  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 )  .<_  ( ( ( Q 
.\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) )

Proof of Theorem dalawlem6
StepHypRef Expression
1 eqid 2451 . 2  |-  ( Base `  K )  =  (
Base `  K )
2 dalawlem.l . 2  |-  .<_  =  ( le `  K )
3 simp11 1018 . . 3  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 33314 . . 3  |-  ( K  e.  HL  ->  K  e.  Lat )
53, 4syl 16 . 2  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 1021 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 1022 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 . . . . . 6  |-  .\/  =  ( join `  K )
9 dalawlem.a . . . . . 6  |-  A  =  ( Atoms `  K )
101, 8, 9hlatjcl 33317 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
113, 6, 7, 10syl3anc 1219 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 simp32 1025 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 )
131, 9atbase 33240 . . . . 5  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
1412, 13syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 ) )
151, 8latjcl 15323 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  T  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  .\/  T )  e.  ( Base `  K ) )
165, 11, 14, 15syl3anc 1219 . . 3  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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
) )
17 simp31 1024 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 )
181, 9atbase 33240 . . . 4  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
1917, 18syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 ) )
20 dalawlem.m . . . 4  |-  ./\  =  ( meet `  K )
211, 20latmcl 15324 . . 3  |-  ( ( K  e.  Lat  /\  ( ( P  .\/  Q )  .\/  T )  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
)  ->  ( (
( P  .\/  Q
)  .\/  T )  ./\  S )  e.  (
Base `  K )
)
225, 16, 19, 21syl3anc 1219 . 2  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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
) )
23 simp23 1023 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 )
241, 8, 9hlatjcl 33317 . . . . 5  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
253, 7, 23, 24syl3anc 1219 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 ) )
26 simp33 1026 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 )
271, 9atbase 33240 . . . . 5  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
2826, 27syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 ) )
291, 20latmcl 15324 . . . 4  |-  ( ( K  e.  Lat  /\  ( Q  .\/  R )  e.  ( Base `  K
)  /\  U  e.  ( Base `  K )
)  ->  ( ( Q  .\/  R )  ./\  U )  e.  ( Base `  K ) )
305, 25, 28, 29syl3anc 1219 . . 3  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 )  ./\  U )  e.  ( Base `  K
) )
311, 8, 9hlatjcl 33317 . . . . 5  |-  ( ( K  e.  HL  /\  R  e.  A  /\  P  e.  A )  ->  ( R  .\/  P
)  e.  ( Base `  K ) )
323, 23, 6, 31syl3anc 1219 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 ) )
331, 8, 9hlatjcl 33317 . . . . 5  |-  ( ( K  e.  HL  /\  U  e.  A  /\  S  e.  A )  ->  ( U  .\/  S
)  e.  ( Base `  K ) )
343, 26, 17, 33syl3anc 1219 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 ) )
351, 20latmcl 15324 . . . 4  |-  ( ( K  e.  Lat  /\  ( R  .\/  P )  e.  ( Base `  K
)  /\  ( U  .\/  S )  e.  (
Base `  K )
)  ->  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )  e.  ( Base `  K ) )
365, 32, 34, 35syl3anc 1219 . . 3  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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
) )
371, 8latjcl 15323 . . 3  |-  ( ( K  e.  Lat  /\  ( ( Q  .\/  R )  ./\  U )  e.  ( Base `  K
)  /\  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )  e.  ( Base `  K ) )  -> 
( ( ( Q 
.\/  R )  ./\  U )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) )  e.  ( Base `  K
) )
385, 30, 36, 37syl3anc 1219 . 2  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 )  ./\  U )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) )  e.  ( Base `  K
) )
391, 8, 9hlatjcl 33317 . . . . 5  |-  ( ( K  e.  HL  /\  T  e.  A  /\  U  e.  A )  ->  ( T  .\/  U
)  e.  ( Base `  K ) )
403, 12, 26, 39syl3anc 1219 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 ) )
411, 20latmcl 15324 . . . 4  |-  ( ( K  e.  Lat  /\  ( Q  .\/  R )  e.  ( Base `  K
)  /\  ( T  .\/  U )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )  e.  ( Base `  K ) )
425, 25, 40, 41syl3anc 1219 . . 3  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 ) )  e.  ( Base `  K
) )
431, 8latjcl 15323 . . 3  |-  ( ( K  e.  Lat  /\  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )  e.  ( Base `  K
)  /\  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )  e.  ( Base `  K ) )  -> 
( ( ( Q 
.\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) )  e.  ( Base `  K
) )
445, 42, 36, 43syl3anc 1219 . 2  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 ) ) )  e.  ( Base `  K
) )
451, 9atbase 33240 . . . . . . . 8  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
466, 45syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 ) )
471, 8, 9hlatjcl 33317 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
483, 6, 17, 47syl3anc 1219 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 ) )
491, 8, 9hlatjcl 33317 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
503, 7, 12, 49syl3anc 1219 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 ) )
511, 20latmcl 15324 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( Q  .\/  R )  e.  ( Base `  K
)  /\  ( Q  .\/  T )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  R )  ./\  ( Q  .\/  T ) )  e.  ( Base `  K ) )
525, 25, 50, 51syl3anc 1219 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 )  ./\  ( Q  .\/  T ) )  e.  ( Base `  K
) )
531, 20latmcl 15324 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  ( ( Q  .\/  R )  ./\  ( Q  .\/  T ) )  e.  ( Base `  K ) )  -> 
( ( P  .\/  S )  ./\  ( ( Q  .\/  R )  ./\  ( Q  .\/  T ) ) )  e.  (
Base `  K )
)
545, 48, 52, 53syl3anc 1219 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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  .\/  R )  ./\  ( Q  .\/  T ) ) )  e.  (
Base `  K )
)
551, 8latjcl 15323 . . . . . . 7  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  (
( P  .\/  S
)  ./\  ( ( Q  .\/  R )  ./\  ( Q  .\/  T ) ) )  e.  (
Base `  K )
)  ->  ( P  .\/  ( ( P  .\/  S )  ./\  ( ( Q  .\/  R )  ./\  ( Q  .\/  T ) ) ) )  e.  ( Base `  K
) )
565, 46, 54, 55syl3anc 1219 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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
)  ./\  ( ( Q  .\/  R )  ./\  ( Q  .\/  T ) ) ) )  e.  ( Base `  K
) )
571, 9atbase 33240 . . . . . . . . 9  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
5823, 57syl 16 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 ) )
591, 8latjcl 15323 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  R  e.  ( Base `  K )  /\  (
( Q  .\/  R
)  ./\  U )  e.  ( Base `  K
) )  ->  ( R  .\/  ( ( Q 
.\/  R )  ./\  U ) )  e.  (
Base `  K )
)
605, 58, 30, 59syl3anc 1219 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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
)  ./\  U )
)  e.  ( Base `  K ) )
611, 8latjcl 15323 . . . . . . 7  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  ( R  .\/  ( ( Q 
.\/  R )  ./\  U ) )  e.  (
Base `  K )
)  ->  ( P  .\/  ( R  .\/  (
( Q  .\/  R
)  ./\  U )
) )  e.  (
Base `  K )
)
625, 46, 60, 61syl3anc 1219 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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  .\/  ( ( Q 
.\/  R )  ./\  U ) ) )  e.  ( Base `  K
) )
631, 2, 8, 20latmlej22 15365 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( S  e.  ( Base `  K )  /\  ( ( P  .\/  Q )  .\/  T )  e.  ( Base `  K
)  /\  P  e.  ( Base `  K )
) )  ->  (
( ( P  .\/  Q )  .\/  T ) 
./\  S )  .<_  ( P  .\/  S ) )
645, 19, 16, 46, 63syl13anc 1221 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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  .\/  S
) )
651, 20latmcl 15324 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  e.  ( Base `  K ) )
665, 50, 48, 65syl3anc 1219 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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
) )
671, 8latjcl 15323 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  (
( Q  .\/  T
)  ./\  ( P  .\/  S ) )  e.  ( Base `  K
) )  ->  ( P  .\/  ( ( Q 
.\/  T )  ./\  ( P  .\/  S ) ) )  e.  (
Base `  K )
)
685, 46, 66, 67syl3anc 1219 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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
) )
691, 8latjcl 15323 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  (
( Q  .\/  R
)  ./\  ( Q  .\/  T ) )  e.  ( Base `  K
) )  ->  ( P  .\/  ( ( Q 
.\/  R )  ./\  ( Q  .\/  T ) ) )  e.  (
Base `  K )
)
705, 46, 52, 69syl3anc 1219 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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  .\/  T ) ) )  e.  ( Base `  K
) )
712, 8, 9hlatlej2 33326 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  S  .<_  ( P  .\/  S ) )
723, 6, 17, 71syl3anc 1219 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 ) )
731, 8latjcl 15323 . . . . . . . . . . . . 13  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  ( Q  .\/  T )  e.  ( Base `  K
) )  ->  ( P  .\/  ( Q  .\/  T ) )  e.  (
Base `  K )
)
745, 46, 50, 73syl3anc 1219 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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
) )
751, 2, 20latmlem2 15354 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  ( S  e.  ( Base `  K )  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  ( P  .\/  ( Q  .\/  T
) )  e.  (
Base `  K )
) )  ->  ( S  .<_  ( P  .\/  S )  ->  ( ( P  .\/  ( Q  .\/  T ) )  ./\  S
)  .<_  ( ( P 
.\/  ( Q  .\/  T ) )  ./\  ( P  .\/  S ) ) ) )
765, 19, 48, 74, 75syl13anc 1221 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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  .\/  ( Q  .\/  T ) )  ./\  S )  .<_  ( ( P  .\/  ( Q  .\/  T ) )  ./\  ( P  .\/  S ) ) ) )
7772, 76mpd 15 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 ) ) )
788, 9hlatjass 33320 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  T  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  T )  =  ( P  .\/  ( Q  .\/  T ) ) )
793, 6, 7, 12, 78syl13anc 1221 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 ) ) )
8079oveq1d 6205 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 ) )  ./\  S
) )
812, 8, 9hlatlej1 33325 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  P  .<_  ( P  .\/  S ) )
823, 6, 17, 81syl3anc 1219 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 ) )
831, 2, 8, 20, 9atmod1i1 33807 . . . . . . . . . . 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 ) ) )
843, 6, 50, 48, 82, 83syl131anc 1232 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 ) ) )
8577, 80, 843brtr4d 4420 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 ) ) ) )
861, 20latmcom 15347 . . . . . . . . . . . . 13  |-  ( ( K  e.  Lat  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  =  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) ) )
875, 50, 48, 86syl3anc 1219 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 ) ) )
88 simp12 1019 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 ) )  .<_  ( Q  .\/  R ) )
8987, 88eqbrtrd 4410 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 ) )  .<_  ( Q  .\/  R ) )
901, 2, 20latmle1 15348 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( Q  .\/  T ) )
915, 50, 48, 90syl3anc 1219 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 ) )  .<_  ( Q  .\/  T ) )
921, 2, 20latlem12 15350 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  ( ( ( Q 
.\/  T )  ./\  ( P  .\/  S ) )  e.  ( Base `  K )  /\  ( Q  .\/  R )  e.  ( Base `  K
)  /\  ( Q  .\/  T )  e.  (
Base `  K )
) )  ->  (
( ( ( Q 
.\/  T )  ./\  ( P  .\/  S ) )  .<_  ( Q  .\/  R )  /\  (
( Q  .\/  T
)  ./\  ( P  .\/  S ) )  .<_  ( Q  .\/  T ) )  <->  ( ( Q 
.\/  T )  ./\  ( P  .\/  S ) )  .<_  ( ( Q  .\/  R )  ./\  ( Q  .\/  T ) ) ) )
935, 66, 25, 50, 92syl13anc 1221 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 ) )  .<_  ( Q 
.\/  R )  /\  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( Q  .\/  T ) )  <->  ( ( Q 
.\/  T )  ./\  ( P  .\/  S ) )  .<_  ( ( Q  .\/  R )  ./\  ( Q  .\/  T ) ) ) )
9489, 91, 93mpbi2and 912 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 ) )  .<_  ( ( Q  .\/  R )  ./\  ( Q  .\/  T ) ) )
951, 2, 8latjlej2 15338 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  ( ( ( Q 
.\/  T )  ./\  ( P  .\/  S ) )  e.  ( Base `  K )  /\  (
( Q  .\/  R
)  ./\  ( Q  .\/  T ) )  e.  ( Base `  K
)  /\  P  e.  ( Base `  K )
) )  ->  (
( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( ( Q  .\/  R )  ./\  ( Q  .\/  T ) )  -> 
( P  .\/  (
( Q  .\/  T
)  ./\  ( P  .\/  S ) ) ) 
.<_  ( P  .\/  (
( Q  .\/  R
)  ./\  ( Q  .\/  T ) ) ) ) )
965, 66, 52, 46, 95syl13anc 1221 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 ) )  .<_  ( ( Q  .\/  R )  ./\  ( Q  .\/  T ) )  ->  ( P  .\/  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) ) ) 
.<_  ( P  .\/  (
( Q  .\/  R
)  ./\  ( Q  .\/  T ) ) ) ) )
9794, 96mpd 15 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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  .\/  R
)  ./\  ( Q  .\/  T ) ) ) )
981, 2, 5, 22, 68, 70, 85, 97lattrd 15330 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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  .\/  R
)  ./\  ( Q  .\/  T ) ) ) )
991, 2, 20latlem12 15350 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( ( ( ( P  .\/  Q ) 
.\/  T )  ./\  S )  e.  ( Base `  K )  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  ( P  .\/  ( ( Q  .\/  R )  ./\  ( Q  .\/  T ) ) )  e.  ( Base `  K
) ) )  -> 
( ( ( ( ( P  .\/  Q
)  .\/  T )  ./\  S )  .<_  ( P 
.\/  S )  /\  ( ( ( P 
.\/  Q )  .\/  T )  ./\  S )  .<_  ( P  .\/  (
( Q  .\/  R
)  ./\  ( Q  .\/  T ) ) ) )  <->  ( ( ( P  .\/  Q ) 
.\/  T )  ./\  S )  .<_  ( ( P  .\/  S )  ./\  ( P  .\/  ( ( Q  .\/  R ) 
./\  ( Q  .\/  T ) ) ) ) ) )
1005, 22, 48, 70, 99syl13anc 1221 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 
.\/  S )  /\  ( ( ( P 
.\/  Q )  .\/  T )  ./\  S )  .<_  ( P  .\/  (
( Q  .\/  R
)  ./\  ( Q  .\/  T ) ) ) )  <->  ( ( ( P  .\/  Q ) 
.\/  T )  ./\  S )  .<_  ( ( P  .\/  S )  ./\  ( P  .\/  ( ( Q  .\/  R ) 
./\  ( Q  .\/  T ) ) ) ) ) )
10164, 98, 100mpbi2and 912 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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  .\/  S )  ./\  ( P  .\/  ( ( Q  .\/  R )  ./\  ( Q  .\/  T ) ) ) ) )
1021, 2, 8, 20, 9atmod3i1 33814 . . . . . . . 8  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  ( P  .\/  S
)  e.  ( Base `  K )  /\  (
( Q  .\/  R
)  ./\  ( Q  .\/  T ) )  e.  ( Base `  K
) )  /\  P  .<_  ( P  .\/  S
) )  ->  ( P  .\/  ( ( P 
.\/  S )  ./\  ( ( Q  .\/  R )  ./\  ( Q  .\/  T ) ) ) )  =  ( ( P  .\/  S ) 
./\  ( P  .\/  ( ( Q  .\/  R )  ./\  ( Q  .\/  T ) ) ) ) )
1033, 6, 48, 52, 82, 102syl131anc 1232 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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
)  ./\  ( ( Q  .\/  R )  ./\  ( Q  .\/  T ) ) ) )  =  ( ( P  .\/  S )  ./\  ( P  .\/  ( ( Q  .\/  R )  ./\  ( Q  .\/  T ) ) ) ) )
104101, 103breqtrrd 4416 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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  .\/  (
( P  .\/  S
)  ./\  ( ( Q  .\/  R )  ./\  ( Q  .\/  T ) ) ) ) )
105 simp13 1020 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 ) )
1061, 20latmcl 15324 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  ( Q  .\/  T )  e.  (
Base `  K )
)  ->  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  e.  ( Base `  K ) )
1075, 48, 50, 106syl3anc 1219 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 ) )  e.  ( Base `  K
) )
1081, 8, 9hlatjcl 33317 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  R  e.  A  /\  U  e.  A )  ->  ( R  .\/  U
)  e.  ( Base `  K ) )
1093, 23, 26, 108syl3anc 1219 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 ) )
1101, 2, 20latmlem2 15354 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( ( ( P 
.\/  S )  ./\  ( Q  .\/  T ) )  e.  ( Base `  K )  /\  ( R  .\/  U )  e.  ( Base `  K
)  /\  ( Q  .\/  R )  e.  (
Base `  K )
) )  ->  (
( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U )  ->  ( ( Q 
.\/  R )  ./\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) ) ) 
.<_  ( ( Q  .\/  R )  ./\  ( R  .\/  U ) ) ) )
1115, 107, 109, 25, 110syl13anc 1221 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 )  ->  (
( Q  .\/  R
)  ./\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) ) )  .<_  ( ( Q  .\/  R ) 
./\  ( R  .\/  U ) ) ) )
112105, 111mpd 15 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 )  ./\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) ) )  .<_  ( ( Q  .\/  R ) 
./\  ( R  .\/  U ) ) )
113 hlol 33312 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  OL )
1143, 113syl 16 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 )
1151, 20latm12 33181 . . . . . . . . 9  |-  ( ( K  e.  OL  /\  ( ( P  .\/  S )  e.  ( Base `  K )  /\  ( Q  .\/  R )  e.  ( Base `  K
)  /\  ( Q  .\/  T )  e.  (
Base `  K )
) )  ->  (
( P  .\/  S
)  ./\  ( ( Q  .\/  R )  ./\  ( Q  .\/  T ) ) )  =  ( ( Q  .\/  R
)  ./\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) ) ) )
116114, 48, 25, 50, 115syl13anc 1221 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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  .\/  R )  ./\  ( Q  .\/  T ) ) )  =  ( ( Q  .\/  R
)  ./\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) ) ) )
1172, 8, 9hlatlej2 33326 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  R  .<_  ( Q  .\/  R ) )
1183, 7, 23, 117syl3anc 1219 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 ) )
1191, 2, 8, 20, 9atmod3i1 33814 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( R  e.  A  /\  ( Q  .\/  R
)  e.  ( Base `  K )  /\  U  e.  ( Base `  K
) )  /\  R  .<_  ( Q  .\/  R
) )  ->  ( R  .\/  ( ( Q 
.\/  R )  ./\  U ) )  =  ( ( Q  .\/  R
)  ./\  ( R  .\/  U ) ) )
1203, 23, 25, 28, 118, 119syl131anc 1232 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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
)  ./\  U )
)  =  ( ( Q  .\/  R ) 
./\  ( R  .\/  U ) ) )
121112, 116, 1203brtr4d 4420 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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  .\/  R )  ./\  ( Q  .\/  T ) ) )  .<_  ( R 
.\/  ( ( Q 
.\/  R )  ./\  U ) ) )
1221, 2, 8latjlej2 15338 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( ( ( P 
.\/  S )  ./\  ( ( Q  .\/  R )  ./\  ( Q  .\/  T ) ) )  e.  ( Base `  K
)  /\  ( R  .\/  ( ( Q  .\/  R )  ./\  U )
)  e.  ( Base `  K )  /\  P  e.  ( Base `  K
) ) )  -> 
( ( ( P 
.\/  S )  ./\  ( ( Q  .\/  R )  ./\  ( Q  .\/  T ) ) ) 
.<_  ( R  .\/  (
( Q  .\/  R
)  ./\  U )
)  ->  ( P  .\/  ( ( P  .\/  S )  ./\  ( ( Q  .\/  R )  ./\  ( Q  .\/  T ) ) ) )  .<_  ( P  .\/  ( R 
.\/  ( ( Q 
.\/  R )  ./\  U ) ) ) ) )
1235, 54, 60, 46, 122syl13anc 1221 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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  .\/  R )  ./\  ( Q  .\/  T ) ) ) 
.<_  ( R  .\/  (
( Q  .\/  R
)  ./\  U )
)  ->  ( P  .\/  ( ( P  .\/  S )  ./\  ( ( Q  .\/  R )  ./\  ( Q  .\/  T ) ) ) )  .<_  ( P  .\/  ( R 
.\/  ( ( Q 
.\/  R )  ./\  U ) ) ) ) )
124121, 123mpd 15 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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
)  ./\  ( ( Q  .\/  R )  ./\  ( Q  .\/  T ) ) ) )  .<_  ( P  .\/  ( R 
.\/  ( ( Q 
.\/  R )  ./\  U ) ) ) )
1251, 2, 5, 22, 56, 62, 104, 124lattrd 15330 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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  .\/  ( R  .\/  ( ( Q 
.\/  R )  ./\  U ) ) ) )
1261, 8latj13 15370 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  R  e.  ( Base `  K )  /\  (
( Q  .\/  R
)  ./\  U )  e.  ( Base `  K
) ) )  -> 
( P  .\/  ( R  .\/  ( ( Q 
.\/  R )  ./\  U ) ) )  =  ( ( ( Q 
.\/  R )  ./\  U )  .\/  ( R 
.\/  P ) ) )
1275, 46, 58, 30, 126syl13anc 1221 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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  .\/  ( ( Q 
.\/  R )  ./\  U ) ) )  =  ( ( ( Q 
.\/  R )  ./\  U )  .\/  ( R 
.\/  P ) ) )
128125, 127breqtrd 4414 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 )  .<_  ( ( ( Q 
.\/  R )  ./\  U )  .\/  ( R 
.\/  P ) ) )
1291, 2, 8, 20latmlej22 15365 . . . . 5  |-  ( ( K  e.  Lat  /\  ( S  e.  ( Base `  K )  /\  ( ( P  .\/  Q )  .\/  T )  e.  ( Base `  K
)  /\  U  e.  ( Base `  K )
) )  ->  (
( ( P  .\/  Q )  .\/  T ) 
./\  S )  .<_  ( U  .\/  S ) )
1305, 19, 16, 28, 129syl13anc 1221 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 )  .<_  ( U  .\/  S
) )
1311, 8latjcl 15323 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( ( Q  .\/  R )  ./\  U )  e.  ( Base `  K
)  /\  ( R  .\/  P )  e.  (
Base `  K )
)  ->  ( (
( Q  .\/  R
)  ./\  U )  .\/  ( R  .\/  P
) )  e.  (
Base `  K )
)
1325, 30, 32, 131syl3anc 1219 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 )  ./\  U )  .\/  ( R 
.\/  P ) )  e.  ( Base `  K
) )
1331, 2, 20latlem12 15350 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( ( ( P  .\/  Q ) 
.\/  T )  ./\  S )  e.  ( Base `  K )  /\  (
( ( Q  .\/  R )  ./\  U )  .\/  ( R  .\/  P
) )  e.  (
Base `  K )  /\  ( U  .\/  S
)  e.  ( Base `  K ) ) )  ->  ( ( ( ( ( P  .\/  Q )  .\/  T ) 
./\  S )  .<_  ( ( ( Q 
.\/  R )  ./\  U )  .\/  ( R 
.\/  P ) )  /\  ( ( ( P  .\/  Q ) 
.\/  T )  ./\  S )  .<_  ( U  .\/  S ) )  <->  ( (
( P  .\/  Q
)  .\/  T )  ./\  S )  .<_  ( ( ( ( Q  .\/  R )  ./\  U )  .\/  ( R  .\/  P
) )  ./\  ( U  .\/  S ) ) ) )
1345, 22, 132, 34, 133syl13anc 1221 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 )  .<_  ( ( ( Q  .\/  R
)  ./\  U )  .\/  ( R  .\/  P
) )  /\  (
( ( P  .\/  Q )  .\/  T ) 
./\  S )  .<_  ( U  .\/  S ) )  <->  ( ( ( P  .\/  Q ) 
.\/  T )  ./\  S )  .<_  ( (
( ( Q  .\/  R )  ./\  U )  .\/  ( R  .\/  P
) )  ./\  ( U  .\/  S ) ) ) )
135128, 130, 134mpbi2and 912 . . 3  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 )  .<_  ( ( ( ( Q  .\/  R ) 
./\  U )  .\/  ( R  .\/  P ) )  ./\  ( U  .\/  S ) ) )
1361, 2, 8, 20latmlej21 15364 . . . . 5  |-  ( ( K  e.  Lat  /\  ( U  e.  ( Base `  K )  /\  ( Q  .\/  R )  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
) )  ->  (
( Q  .\/  R
)  ./\  U )  .<_  ( U  .\/  S
) )
1375, 28, 25, 19, 136syl13anc 1221 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 )  ./\  U )  .<_  ( U  .\/  S
) )
1381, 2, 8, 20, 9atmod1i1m 33808 . . . 4  |-  ( ( ( K  e.  HL  /\  U  e.  A )  /\  ( ( Q 
.\/  R )  e.  ( Base `  K
)  /\  ( R  .\/  P )  e.  (
Base `  K )  /\  ( U  .\/  S
)  e.  ( Base `  K ) )  /\  ( ( Q  .\/  R )  ./\  U )  .<_  ( U  .\/  S
) )  ->  (
( ( Q  .\/  R )  ./\  U )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) )  =  ( ( ( ( Q  .\/  R
)  ./\  U )  .\/  ( R  .\/  P
) )  ./\  ( U  .\/  S ) ) )
1393, 26, 25, 32, 34, 137, 138syl231anc 1239 . . 3  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 )  ./\  U )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) )  =  ( ( ( ( Q  .\/  R ) 
./\  U )  .\/  ( R  .\/  P ) )  ./\  ( U  .\/  S ) ) )
140135, 139breqtrrd 4416 . 2  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 )  .<_  ( ( ( Q 
.\/  R )  ./\  U )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) )
1412, 8, 9hlatlej2 33326 . . . . 5  |-  ( ( K  e.  HL  /\  T  e.  A  /\  U  e.  A )  ->  U  .<_  ( T  .\/  U ) )
1423, 12, 26, 141syl3anc 1219 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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  .<_  ( T  .\/  U ) )
1431, 2, 20latmlem2 15354 . . . . 5  |-  ( ( K  e.  Lat  /\  ( U  e.  ( Base `  K )  /\  ( T  .\/  U )  e.  ( Base `  K
)  /\  ( Q  .\/  R )  e.  (
Base `  K )
) )  ->  ( U  .<_  ( T  .\/  U )  ->  ( ( Q  .\/  R )  ./\  U )  .<_  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) ) ) )
1445, 28, 40, 25, 143syl13anc 1221 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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  .<_  ( T 
.\/  U )  -> 
( ( Q  .\/  R )  ./\  U )  .<_  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) ) ) )
145142, 144mpd 15 . . 3  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 )  ./\  U )  .<_  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) ) )
1461, 2, 8latjlej1 15337 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( ( Q 
.\/  R )  ./\  U )  e.  ( Base `  K )  /\  (
( Q  .\/  R
)  ./\  ( T  .\/  U ) )  e.  ( Base `  K
)  /\  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )  e.  ( Base `  K ) ) )  ->  ( ( ( Q  .\/  R ) 
./\  U )  .<_  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )  -> 
( ( ( Q 
.\/  R )  ./\  U )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) )  .<_  ( ( ( Q 
.\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) ) )
1475, 30, 42, 36, 146syl13anc 1221 . . 3  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 )  ./\  U )  .<_  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )  ->  ( (
( Q  .\/  R
)  ./\  U )  .\/  ( ( R  .\/  P )  ./\  ( U  .\/  S ) ) ) 
.<_  ( ( ( Q 
.\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) ) )
148145, 147mpd 15 . 2  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 )  ./\  U )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) )  .<_  ( ( ( Q 
.\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) )
1491, 2, 5, 22, 38, 44, 140, 148lattrd 15330 1  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( 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 )  .<_  ( ( ( Q 
.\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   class class class wbr 4390   ` cfv 5516  (class class class)co 6190   Basecbs 14276   lecple 14347   joincjn 15216   meetcmee 15217   Latclat 15317   OLcol 33125   Atomscatm 33214   HLchlt 33301
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-iin 4272  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-1st 6677  df-2nd 6678  df-poset 15218  df-plt 15230  df-lub 15246  df-glb 15247  df-join 15248  df-meet 15249  df-p0 15311  df-lat 15318  df-clat 15380  df-oposet 33127  df-ol 33129  df-oml 33130  df-covers 33217  df-ats 33218  df-atl 33249  df-cvlat 33273  df-hlat 33302  df-psubsp 33453  df-pmap 33454  df-padd 33746
This theorem is referenced by:  dalawlem8  33828
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