Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dalawlem6 Unicode version

Theorem dalawlem6 30687
Description: Lemma for dalaw 30697. First piece of dalawlem8 30689. (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 2296 . 2  |-  ( Base `  K )  =  (
Base `  K )
2 dalawlem.l . 2  |-  .<_  =  ( le `  K )
3 simp11 985 . . 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 30175 . . 3  |-  ( K  e.  HL  ->  K  e.  Lat )
53, 4syl 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 ) )  ->  K  e.  Lat )
6 simp21 988 . . . . 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 989 . . . . 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 30178 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
113, 6, 7, 10syl3anc 1182 . . . 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 992 . . . . 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 30101 . . . . 5  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
1412, 13syl 15 . . . 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 14172 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  T  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  .\/  T )  e.  ( Base `  K ) )
165, 11, 14, 15syl3anc 1182 . . 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 991 . . . 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 30101 . . . 4  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
1917, 18syl 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 ) )  ->  S  e.  ( Base `  K ) )
20 dalawlem.m . . . 4  |-  ./\  =  ( meet `  K )
211, 20latmcl 14173 . . 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 1182 . 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 990 . . . . 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 30178 . . . . 5  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
253, 7, 23, 24syl3anc 1182 . . . 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 993 . . . . 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 30101 . . . . 5  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
2826, 27syl 15 . . . 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 14173 . . . 4  |-  ( ( K  e.  Lat  /\  ( Q  .\/  R )  e.  ( Base `  K
)  /\  U  e.  ( Base `  K )
)  ->  ( ( Q  .\/  R )  ./\  U )  e.  ( Base `  K ) )
305, 25, 28, 29syl3anc 1182 . . 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 30178 . . . . 5  |-  ( ( K  e.  HL  /\  R  e.  A  /\  P  e.  A )  ->  ( R  .\/  P
)  e.  ( Base `  K ) )
323, 23, 6, 31syl3anc 1182 . . . 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 30178 . . . . 5  |-  ( ( K  e.  HL  /\  U  e.  A  /\  S  e.  A )  ->  ( U  .\/  S
)  e.  ( Base `  K ) )
343, 26, 17, 33syl3anc 1182 . . . 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 14173 . . . 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 1182 . . 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 14172 . . 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 1182 . 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 30178 . . . . 5  |-  ( ( K  e.  HL  /\  T  e.  A  /\  U  e.  A )  ->  ( T  .\/  U
)  e.  ( Base `  K ) )
403, 12, 26, 39syl3anc 1182 . . . 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 14173 . . . 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 1182 . . 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 14172 . . 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 1182 . 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 30101 . . . . . . . 8  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
466, 45syl 15 . . . . . . 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 30178 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
483, 6, 17, 47syl3anc 1182 . . . . . . . 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 30178 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
503, 7, 12, 49syl3anc 1182 . . . . . . . . 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 14173 . . . . . . . . 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 1182 . . . . . . . 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 14173 . . . . . . . 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 1182 . . . . . . 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 14172 . . . . . . 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 1182 . . . . . 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 30101 . . . . . . . . 9  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
5823, 57syl 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 ) )  ->  R  e.  ( Base `  K ) )
591, 8latjcl 14172 . . . . . . . 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 1182 . . . . . . 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 14172 . . . . . . 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 1182 . . . . . 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 14215 . . . . . . . . 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 1184 . . . . . . . 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 14173 . . . . . . . . . . 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 1182 . . . . . . . . . 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 14172 . . . . . . . . . 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 1182 . . . . . . . . 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 14172 . . . . . . . . . 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 1182 . . . . . . . . 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 30187 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  S  .<_  ( P  .\/  S ) )
723, 6, 17, 71syl3anc 1182 . . . . . . . . . . 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 14172 . . . . . . . . . . . . 13  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  ( Q  .\/  T )  e.  ( Base `  K
) )  ->  ( P  .\/  ( Q  .\/  T ) )  e.  (
Base `  K )
)
745, 46, 50, 73syl3anc 1182 . . . . . . . . . . . 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 14204 . . . . . . . . . . . 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 1184 . . . . . . . . . . 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 14 . . . . . . . . . 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 30181 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  T  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  T )  =  ( P  .\/  ( Q  .\/  T ) ) )
793, 6, 7, 12, 78syl13anc 1184 . . . . . . . . . . 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 5889 . . . . . . . . . 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 30186 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  P  .<_  ( P  .\/  S ) )
823, 6, 17, 81syl3anc 1182 . . . . . . . . . . 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 30668 . . . . . . . . . . 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 1195 . . . . . . . . . 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 4069 . . . . . . . . 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 14197 . . . . . . . . . . . . 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 1182 . . . . . . . . . . . 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 986 . . . . . . . . . . . 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 4059 . . . . . . . . . . 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 14198 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( Q  .\/  T ) )
915, 50, 48, 90syl3anc 1182 . . . . . . . . . . 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 14200 . . . . . . . . . . . 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 1184 . . . . . . . . . . 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 887 . . . . . . . . . 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 14188 . . . . . . . . . . 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 1184 . . . . . . . . . 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 14 . . . . . . . . 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 14180 . . . . . . . 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 14200 . . . . . . . . 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 1184 . . . . . . . 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 887 . . . . . . 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 30675 . . . . . . . 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 1195 . . . . . . 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 4065 . . . . . 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 987 . . . . . . . . 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 14173 . . . . . . . . . . 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 1182 . . . . . . . . . 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 30178 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  R  e.  A  /\  U  e.  A )  ->  ( R  .\/  U
)  e.  ( Base `  K ) )
1093, 23, 26, 108syl3anc 1182 . . . . . . . . . 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 14204 . . . . . . . . . 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 1184 . . . . . . . . 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 14 . . . . . . . 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 30173 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  OL )
1143, 113syl 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 ) )  ->  K  e.  OL )
1151, 20latm12 30042 . . . . . . . . 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 1184 . . . . . . . 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 30187 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  R  .<_  ( Q  .\/  R ) )
1183, 7, 23, 117syl3anc 1182 . . . . . . . . 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 30675 . . . . . . . . 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 1195 . . . . . . . 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 4069 . . . . . . 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 14188 . . . . . . . 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 1184 . . . . . . 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 14 . . . . . 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 14180 . . . . 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 14220 . . . . . 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 1184 . . . . 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 4063 . . . 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 14215 . . . . 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 1184 . . . 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 14172 . . . . . 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 1182 . . . . 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 14200 . . . . 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 1184 . . . 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 887 . . 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 14214 . . . . 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 1184 . . . 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 30669 . . . 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 1202 . . 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 4065 . 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 30187 . . . . 5  |-  ( ( K  e.  HL  /\  T  e.  A  /\  U  e.  A )  ->  U  .<_  ( T  .\/  U ) )
1423, 12, 26, 141syl3anc 1182 . . . 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 14204 . . . . 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 1184 . . . 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 14 . . 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 14187 . . . 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 1184 . . 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 14 . 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 14180 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   lecple 13231   joincjn 14094   meetcmee 14095   Latclat 14167   OLcol 29986   Atomscatm 30075   HLchlt 30162
This theorem is referenced by:  dalawlem8  30689
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-undef 6314  df-riota 6320  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-lat 14168  df-clat 14230  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-psubsp 30314  df-pmap 30315  df-padd 30607
  Copyright terms: Public domain W3C validator