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Theorem dalawlem3 30844
Description: Lemma for dalaw 30857. First piece of dalawlem5 30846. (Contributed by NM, 4-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
dalawlem3  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) )

Proof of Theorem dalawlem3
StepHypRef Expression
1 eqid 2443 . 2  |-  ( Base `  K )  =  (
Base `  K )
2 dalawlem.l . 2  |-  .<_  =  ( le `  K )
3 simp11 988 . . 3  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 30335 . . 3  |-  ( K  e.  HL  ->  K  e.  Lat )
53, 4syl 16 . 2  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 simp22 992 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 )
7 simp32 995 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 )
8 dalawlem.j . . . . . 6  |-  .\/  =  ( join `  K )
9 dalawlem.a . . . . . 6  |-  A  =  ( Atoms `  K )
101, 8, 9hlatjcl 30338 . . . . 5  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
113, 6, 7, 10syl3anc 1185 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 ) )
12 simp21 991 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 )
131, 9atbase 30261 . . . . 5  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
1412, 13syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 ) )
151, 8latjcl 14517 . . . 4  |-  ( ( K  e.  Lat  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  P  e.  ( Base `  K )
)  ->  ( ( Q  .\/  T )  .\/  P )  e.  ( Base `  K ) )
165, 11, 14, 15syl3anc 1185 . . 3  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 )  e.  ( Base `  K
) )
17 simp31 994 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 30261 . . . 4  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
1917, 18syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 14518 . . 3  |-  ( ( K  e.  Lat  /\  ( ( Q  .\/  T )  .\/  P )  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
)  ->  ( (
( Q  .\/  T
)  .\/  P )  ./\  S )  e.  (
Base `  K )
)
225, 16, 19, 21syl3anc 1185 . 2  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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
) )
23 simp23 993 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 30338 . . . . 5  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
253, 6, 23, 24syl3anc 1185 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 996 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 30261 . . . . 5  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
2826, 27syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 14518 . . . 4  |-  ( ( K  e.  Lat  /\  ( Q  .\/  R )  e.  ( Base `  K
)  /\  U  e.  ( Base `  K )
)  ->  ( ( Q  .\/  R )  ./\  U )  e.  ( Base `  K ) )
305, 25, 28, 29syl3anc 1185 . . 3  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 30338 . . . . 5  |-  ( ( K  e.  HL  /\  R  e.  A  /\  P  e.  A )  ->  ( R  .\/  P
)  e.  ( Base `  K ) )
323, 23, 12, 31syl3anc 1185 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 30338 . . . . 5  |-  ( ( K  e.  HL  /\  U  e.  A  /\  S  e.  A )  ->  ( U  .\/  S
)  e.  ( Base `  K ) )
343, 26, 17, 33syl3anc 1185 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 14518 . . . 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 1185 . . 3  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 14517 . . 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 1185 . 2  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 30338 . . . . 5  |-  ( ( K  e.  HL  /\  T  e.  A  /\  U  e.  A )  ->  ( T  .\/  U
)  e.  ( Base `  K ) )
403, 7, 26, 39syl3anc 1185 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 14518 . . . 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 1185 . . 3  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 14517 . . 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 1185 . 2  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 30261 . . . . . . . . . 10  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
466, 45syl 16 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( P 
.\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  ->  Q  e.  ( Base `  K ) )
471, 20latmcl 14518 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  U  e.  ( Base `  K
) )  ->  ( Q  ./\  U )  e.  ( Base `  K
) )
485, 46, 28, 47syl3anc 1185 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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  ./\  U
)  e.  ( Base `  K ) )
491, 8, 9hlatjcl 30338 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
503, 12, 17, 49syl3anc 1185 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 ) )
511, 20latmcl 14518 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  Q  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  ./\  Q )  e.  ( Base `  K ) )
525, 50, 46, 51syl3anc 1185 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 )  e.  ( Base `  K
) )
531, 8latjcl 14517 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( Q  ./\  U )  e.  ( Base `  K
)  /\  ( ( P  .\/  S )  ./\  Q )  e.  ( Base `  K ) )  -> 
( ( Q  ./\  U )  .\/  ( ( P  .\/  S ) 
./\  Q ) )  e.  ( Base `  K
) )
545, 48, 52, 53syl3anc 1185 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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  ./\  U )  .\/  ( ( P  .\/  S ) 
./\  Q ) )  e.  ( Base `  K
) )
551, 8latjcl 14517 . . . . . . 7  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  (
( Q  ./\  U
)  .\/  ( ( P  .\/  S )  ./\  Q ) )  e.  (
Base `  K )
)  ->  ( P  .\/  ( ( Q  ./\  U )  .\/  ( ( P  .\/  S ) 
./\  Q ) ) )  e.  ( Base `  K ) )
565, 14, 54, 55syl3anc 1185 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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  ./\  U
)  .\/  ( ( P  .\/  S )  ./\  Q ) ) )  e.  ( Base `  K
) )
571, 9atbase 30261 . . . . . . . . 9  |-  ( R  e.  A  ->  R  e.  ( Base `  K
) )
5823, 57syl 16 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 14517 . . . . . . . 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 1185 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 14517 . . . . . . 7  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  ( R  .\/  ( ( Q 
.\/  R )  ./\  U ) )  e.  (
Base `  K )
)  ->  ( P  .\/  ( R  .\/  (
( Q  .\/  R
)  ./\  U )
) )  e.  (
Base `  K )
)
625, 14, 60, 61syl3anc 1185 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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, 8latjcl 14517 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  ( Q  ./\  U )  e.  ( Base `  K
)  /\  P  e.  ( Base `  K )
)  ->  ( ( Q  ./\  U )  .\/  P )  e.  ( Base `  K ) )
645, 48, 14, 63syl3anc 1185 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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  ./\  U )  .\/  P )  e.  ( Base `  K
) )
651, 2, 8, 20latmlej22 14560 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( S  e.  ( Base `  K )  /\  ( ( Q  .\/  T )  .\/  P )  e.  ( Base `  K
)  /\  ( ( Q  ./\  U )  .\/  P )  e.  ( Base `  K ) ) )  ->  ( ( ( Q  .\/  T ) 
.\/  P )  ./\  S )  .<_  ( (
( Q  ./\  U
)  .\/  P )  .\/  S ) )
665, 19, 16, 64, 65syl13anc 1187 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 
./\  U )  .\/  P )  .\/  S ) )
671, 8latjass 14562 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( ( Q  ./\  U )  e.  ( Base `  K )  /\  P  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
) )  ->  (
( ( Q  ./\  U )  .\/  P ) 
.\/  S )  =  ( ( Q  ./\  U )  .\/  ( P 
.\/  S ) ) )
685, 48, 14, 19, 67syl13anc 1187 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 
./\  U )  .\/  P )  .\/  S )  =  ( ( Q 
./\  U )  .\/  ( P  .\/  S ) ) )
6966, 68breqtrd 4267 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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  ./\  U )  .\/  ( P 
.\/  S ) ) )
701, 20latmcl 14518 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  e.  ( Base `  K ) )
715, 11, 50, 70syl3anc 1185 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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
) )
721, 8latjcl 14517 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  e.  ( Base `  K
)  /\  P  e.  ( Base `  K )
)  ->  ( (
( Q  .\/  T
)  ./\  ( P  .\/  S ) )  .\/  P )  e.  ( Base `  K ) )
735, 71, 14, 72syl3anc 1185 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 )  e.  ( Base `  K
) )
741, 8, 9hlatjcl 30338 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
753, 12, 6, 74syl3anc 1185 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 ) )
762, 8, 9hlatlej2 30347 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  S  .<_  ( P  .\/  S ) )
773, 12, 17, 76syl3anc 1185 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 ) )
781, 2, 20latmlem2 14549 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  ( S  e.  ( Base `  K )  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  ( ( Q  .\/  T )  .\/  P )  e.  ( Base `  K ) ) )  ->  ( S  .<_  ( P  .\/  S )  ->  ( ( ( Q  .\/  T ) 
.\/  P )  ./\  S )  .<_  ( (
( Q  .\/  T
)  .\/  P )  ./\  ( P  .\/  S
) ) ) )
795, 19, 50, 16, 78syl13anc 1187 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 )  -> 
( ( ( Q 
.\/  T )  .\/  P )  ./\  S )  .<_  ( ( ( Q 
.\/  T )  .\/  P )  ./\  ( P  .\/  S ) ) ) )
8077, 79mpd 15 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 )  .\/  P )  ./\  ( P  .\/  S ) ) )
812, 8, 9hlatlej1 30346 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  P  .<_  ( P  .\/  S ) )
823, 12, 17, 81syl3anc 1185 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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, 9atmod4i1 30837 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  ( Q  .\/  T
)  e.  ( Base `  K )  /\  ( P  .\/  S )  e.  ( Base `  K
) )  /\  P  .<_  ( P  .\/  S
) )  ->  (
( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .\/  P )  =  ( ( ( Q  .\/  T
)  .\/  P )  ./\  ( P  .\/  S
) ) )
843, 12, 11, 50, 82, 83syl131anc 1198 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 )  =  ( ( ( Q  .\/  T ) 
.\/  P )  ./\  ( P  .\/  S ) ) )
8580, 84breqtrrd 4269 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 )  ./\  ( P  .\/  S ) )  .\/  P ) )
861, 20latmcom 14542 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  =  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) ) )
875, 11, 50, 86syl3anc 1185 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 989 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 ) )  .<_  ( P  .\/  Q ) )
8987, 88eqbrtrd 4263 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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  .\/  Q ) )
902, 8, 9hlatlej1 30346 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  P  .<_  ( P  .\/  Q ) )
913, 12, 6, 90syl3anc 1185 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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  .\/  Q ) )
921, 2, 8latjle12 14529 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  ( ( ( Q 
.\/  T )  ./\  ( P  .\/  S ) )  e.  ( Base `  K )  /\  P  e.  ( Base `  K
)  /\  ( P  .\/  Q )  e.  (
Base `  K )
) )  ->  (
( ( ( Q 
.\/  T )  ./\  ( P  .\/  S ) )  .<_  ( P  .\/  Q )  /\  P  .<_  ( P  .\/  Q
) )  <->  ( (
( Q  .\/  T
)  ./\  ( P  .\/  S ) )  .\/  P )  .<_  ( P  .\/  Q ) ) )
935, 71, 14, 75, 92syl13anc 1187 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 
.\/  Q )  /\  P  .<_  ( P  .\/  Q ) )  <->  ( (
( Q  .\/  T
)  ./\  ( P  .\/  S ) )  .\/  P )  .<_  ( P  .\/  Q ) ) )
9489, 91, 93mpbi2and 889 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 ) 
.<_  ( P  .\/  Q
) )
951, 2, 5, 22, 73, 75, 85, 94lattrd 14525 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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  .\/  Q
) )
961, 8latjcl 14517 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( Q  ./\  U )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  ./\  U )  .\/  ( P  .\/  S ) )  e.  ( Base `  K ) )
975, 48, 50, 96syl3anc 1185 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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  ./\  U )  .\/  ( P 
.\/  S ) )  e.  ( Base `  K
) )
981, 2, 20latlem12 14545 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( ( ( ( Q  .\/  T ) 
.\/  P )  ./\  S )  e.  ( Base `  K )  /\  (
( Q  ./\  U
)  .\/  ( P  .\/  S ) )  e.  ( Base `  K
)  /\  ( P  .\/  Q )  e.  (
Base `  K )
) )  ->  (
( ( ( ( Q  .\/  T ) 
.\/  P )  ./\  S )  .<_  ( ( Q  ./\  U )  .\/  ( P  .\/  S ) )  /\  ( ( ( Q  .\/  T
)  .\/  P )  ./\  S )  .<_  ( P 
.\/  Q ) )  <-> 
( ( ( Q 
.\/  T )  .\/  P )  ./\  S )  .<_  ( ( ( Q 
./\  U )  .\/  ( P  .\/  S ) )  ./\  ( P  .\/  Q ) ) ) )
995, 22, 97, 75, 98syl13anc 1187 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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  ./\  U )  .\/  ( P  .\/  S
) )  /\  (
( ( Q  .\/  T )  .\/  P ) 
./\  S )  .<_  ( P  .\/  Q ) )  <->  ( ( ( Q  .\/  T ) 
.\/  P )  ./\  S )  .<_  ( (
( Q  ./\  U
)  .\/  ( P  .\/  S ) )  ./\  ( P  .\/  Q ) ) ) )
10069, 95, 99mpbi2and 889 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 
./\  U )  .\/  ( P  .\/  S ) )  ./\  ( P  .\/  Q ) ) )
1011, 2, 8, 20, 9atmod3i1 30835 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  ( P  .\/  S
)  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  /\  P  .<_  ( P  .\/  S
) )  ->  ( P  .\/  ( ( P 
.\/  S )  ./\  Q ) )  =  ( ( P  .\/  S
)  ./\  ( P  .\/  Q ) ) )
1023, 12, 50, 46, 82, 101syl131anc 1198 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 )
)  =  ( ( P  .\/  S ) 
./\  ( P  .\/  Q ) ) )
103102oveq2d 6133 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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  ./\  U )  .\/  ( P 
.\/  ( ( P 
.\/  S )  ./\  Q ) ) )  =  ( ( Q  ./\  U )  .\/  ( ( P  .\/  S ) 
./\  ( P  .\/  Q ) ) ) )
1041, 8latj12 14563 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( ( Q  ./\  U )  e.  ( Base `  K )  /\  P  e.  ( Base `  K
)  /\  ( ( P  .\/  S )  ./\  Q )  e.  ( Base `  K ) ) )  ->  ( ( Q 
./\  U )  .\/  ( P  .\/  ( ( P  .\/  S ) 
./\  Q ) ) )  =  ( P 
.\/  ( ( Q 
./\  U )  .\/  ( ( P  .\/  S )  ./\  Q )
) ) )
1055, 48, 14, 52, 104syl13anc 1187 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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  ./\  U )  .\/  ( P 
.\/  ( ( P 
.\/  S )  ./\  Q ) ) )  =  ( P  .\/  (
( Q  ./\  U
)  .\/  ( ( P  .\/  S )  ./\  Q ) ) ) )
1061, 2, 8, 20latmlej12 14558 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( Q  e.  ( Base `  K )  /\  U  e.  ( Base `  K )  /\  P  e.  ( Base `  K
) ) )  -> 
( Q  ./\  U
)  .<_  ( P  .\/  Q ) )
1075, 46, 28, 14, 106syl13anc 1187 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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  ./\  U
)  .<_  ( P  .\/  Q ) )
1081, 2, 8, 20, 9atmod1i1m 30829 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  U  e.  A )  /\  ( Q  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )  /\  ( P  .\/  Q
)  e.  ( Base `  K ) )  /\  ( Q  ./\  U ) 
.<_  ( P  .\/  Q
) )  ->  (
( Q  ./\  U
)  .\/  ( ( P  .\/  S )  ./\  ( P  .\/  Q ) ) )  =  ( ( ( Q  ./\  U )  .\/  ( P 
.\/  S ) ) 
./\  ( P  .\/  Q ) ) )
1093, 26, 46, 50, 75, 107, 108syl231anc 1205 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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  ./\  U )  .\/  ( ( P  .\/  S ) 
./\  ( P  .\/  Q ) ) )  =  ( ( ( Q 
./\  U )  .\/  ( P  .\/  S ) )  ./\  ( P  .\/  Q ) ) )
110103, 105, 1093eqtr3rd 2484 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 
./\  U )  .\/  ( P  .\/  S ) )  ./\  ( P  .\/  Q ) )  =  ( P  .\/  (
( Q  ./\  U
)  .\/  ( ( P  .\/  S )  ./\  Q ) ) ) )
111100, 110breqtrd 4267 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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  .\/  (
( Q  ./\  U
)  .\/  ( ( P  .\/  S )  ./\  Q ) ) ) )
1122, 8, 9hlatlej1 30346 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  Q  .<_  ( Q  .\/  R ) )
1133, 6, 23, 112syl3anc 1185 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( P 
.\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  ->  Q  .<_  ( Q  .\/  R ) )
1142, 8, 9hlatlej2 30347 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  R  e.  A  /\  U  e.  A )  ->  U  .<_  ( R  .\/  U ) )
1153, 23, 26, 114syl3anc 1185 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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  .<_  ( R  .\/  U ) )
1161, 20latmcl 14518 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  ( Q  .\/  T )  e.  (
Base `  K )
)  ->  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  e.  ( Base `  K ) )
1175, 50, 11, 116syl3anc 1185 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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
) )
1181, 8, 9hlatjcl 30338 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  R  e.  A  /\  U  e.  A )  ->  ( R  .\/  U
)  e.  ( Base `  K ) )
1193, 23, 26, 118syl3anc 1185 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 ) )
1202, 8, 9hlatlej1 30346 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  ->  Q  .<_  ( Q  .\/  T ) )
1213, 6, 7, 120syl3anc 1185 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( P 
.\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  ->  Q  .<_  ( Q  .\/  T ) )
1221, 2, 20latmlem2 14549 . . . . . . . . . . . . 13  |-  ( ( K  e.  Lat  /\  ( Q  e.  ( Base `  K )  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
) )  ->  ( Q  .<_  ( Q  .\/  T )  ->  ( ( P  .\/  S )  ./\  Q )  .<_  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) ) ) )
1235, 46, 11, 50, 122syl13anc 1187 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( P 
.\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  -> 
( Q  .<_  ( Q 
.\/  T )  -> 
( ( P  .\/  S )  ./\  Q )  .<_  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) ) ) )
124121, 123mpd 15 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 )  .<_  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) ) )
125 simp13 990 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 ) )
1261, 2, 5, 52, 117, 119, 124, 125lattrd 14525 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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  .\/  U
) )
1271, 2, 8latjle12 14529 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  ( U  e.  ( Base `  K )  /\  ( ( P  .\/  S )  ./\  Q )  e.  ( Base `  K
)  /\  ( R  .\/  U )  e.  (
Base `  K )
) )  ->  (
( U  .<_  ( R 
.\/  U )  /\  ( ( P  .\/  S )  ./\  Q )  .<_  ( R  .\/  U
) )  <->  ( U  .\/  ( ( P  .\/  S )  ./\  Q )
)  .<_  ( R  .\/  U ) ) )
1285, 28, 52, 119, 127syl13anc 1187 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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  .<_  ( R  .\/  U )  /\  ( ( P 
.\/  S )  ./\  Q )  .<_  ( R  .\/  U ) )  <->  ( U  .\/  ( ( P  .\/  S )  ./\  Q )
)  .<_  ( R  .\/  U ) ) )
129115, 126, 128mpbi2and 889 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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  .\/  (
( P  .\/  S
)  ./\  Q )
)  .<_  ( R  .\/  U ) )
1301, 8latjcl 14517 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  U  e.  ( Base `  K )  /\  (
( P  .\/  S
)  ./\  Q )  e.  ( Base `  K
) )  ->  ( U  .\/  ( ( P 
.\/  S )  ./\  Q ) )  e.  (
Base `  K )
)
1315, 28, 52, 130syl3anc 1185 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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  .\/  (
( P  .\/  S
)  ./\  Q )
)  e.  ( Base `  K ) )
1321, 2, 20latmlem12 14550 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( Q  e.  ( Base `  K )  /\  ( Q  .\/  R )  e.  ( Base `  K
) )  /\  (
( U  .\/  (
( P  .\/  S
)  ./\  Q )
)  e.  ( Base `  K )  /\  ( R  .\/  U )  e.  ( Base `  K
) ) )  -> 
( ( Q  .<_  ( Q  .\/  R )  /\  ( U  .\/  ( ( P  .\/  S )  ./\  Q )
)  .<_  ( R  .\/  U ) )  ->  ( Q  ./\  ( U  .\/  ( ( P  .\/  S )  ./\  Q )
) )  .<_  ( ( Q  .\/  R ) 
./\  ( R  .\/  U ) ) ) )
1335, 46, 25, 131, 119, 132syl122anc 1194 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( P 
.\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  -> 
( ( Q  .<_  ( Q  .\/  R )  /\  ( U  .\/  ( ( P  .\/  S )  ./\  Q )
)  .<_  ( R  .\/  U ) )  ->  ( Q  ./\  ( U  .\/  ( ( P  .\/  S )  ./\  Q )
) )  .<_  ( ( Q  .\/  R ) 
./\  ( R  .\/  U ) ) ) )
134113, 129, 133mp2and 662 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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  ./\  ( U  .\/  ( ( P 
.\/  S )  ./\  Q ) ) )  .<_  ( ( Q  .\/  R )  ./\  ( R  .\/  U ) ) )
1351, 2, 20latmle2 14544 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  Q  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  ./\  Q )  .<_  Q )
1365, 50, 46, 135syl3anc 1185 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 )  .<_  Q )
1371, 2, 8, 20, 9atmod2i2 30833 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( U  e.  A  /\  Q  e.  ( Base `  K )  /\  ( ( P  .\/  S )  ./\  Q )  e.  ( Base `  K
) )  /\  (
( P  .\/  S
)  ./\  Q )  .<_  Q )  ->  (
( Q  ./\  U
)  .\/  ( ( P  .\/  S )  ./\  Q ) )  =  ( Q  ./\  ( U  .\/  ( ( P  .\/  S )  ./\  Q )
) ) )
1383, 26, 46, 52, 136, 137syl131anc 1198 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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  ./\  U )  .\/  ( ( P  .\/  S ) 
./\  Q ) )  =  ( Q  ./\  ( U  .\/  ( ( P  .\/  S ) 
./\  Q ) ) ) )
1392, 8, 9hlatlej2 30347 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  R  .<_  ( Q  .\/  R ) )
1403, 6, 23, 139syl3anc 1185 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 ) )
1411, 2, 8, 20, 9atmod3i2 30836 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( U  e.  A  /\  R  e.  ( Base `  K )  /\  ( Q  .\/  R )  e.  ( Base `  K
) )  /\  R  .<_  ( Q  .\/  R
) )  ->  ( R  .\/  ( ( Q 
.\/  R )  ./\  U ) )  =  ( ( Q  .\/  R
)  ./\  ( R  .\/  U ) ) )
1423, 26, 58, 25, 140, 141syl131anc 1198 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 ) ) )
143134, 138, 1423brtr4d 4273 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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  ./\  U )  .\/  ( ( P  .\/  S ) 
./\  Q ) ) 
.<_  ( R  .\/  (
( Q  .\/  R
)  ./\  U )
) )
1441, 2, 8latjlej2 14533 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( ( ( Q 
./\  U )  .\/  ( ( P  .\/  S )  ./\  Q )
)  e.  ( Base `  K )  /\  ( R  .\/  ( ( Q 
.\/  R )  ./\  U ) )  e.  (
Base `  K )  /\  P  e.  ( Base `  K ) ) )  ->  ( (
( Q  ./\  U
)  .\/  ( ( P  .\/  S )  ./\  Q ) )  .<_  ( R 
.\/  ( ( Q 
.\/  R )  ./\  U ) )  ->  ( P  .\/  ( ( Q 
./\  U )  .\/  ( ( P  .\/  S )  ./\  Q )
) )  .<_  ( P 
.\/  ( R  .\/  ( ( Q  .\/  R )  ./\  U )
) ) ) )
1455, 54, 60, 14, 144syl13anc 1187 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 
./\  U )  .\/  ( ( P  .\/  S )  ./\  Q )
)  .<_  ( R  .\/  ( ( Q  .\/  R )  ./\  U )
)  ->  ( P  .\/  ( ( Q  ./\  U )  .\/  ( ( P  .\/  S ) 
./\  Q ) ) )  .<_  ( P  .\/  ( R  .\/  (
( Q  .\/  R
)  ./\  U )
) ) ) )
146143, 145mpd 15 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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  ./\  U
)  .\/  ( ( P  .\/  S )  ./\  Q ) ) )  .<_  ( P  .\/  ( R 
.\/  ( ( Q 
.\/  R )  ./\  U ) ) ) )
1471, 2, 5, 22, 56, 62, 111, 146lattrd 14525 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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  .\/  ( R  .\/  ( ( Q 
.\/  R )  ./\  U ) ) ) )
1481, 8latj13 14565 . . . . . 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 ) ) )
1495, 14, 58, 30, 148syl13anc 1187 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 ) ) )
150147, 149breqtrd 4267 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 )  ./\  U )  .\/  ( R 
.\/  P ) ) )
1511, 2, 8, 20latmlej22 14560 . . . . 5  |-  ( ( K  e.  Lat  /\  ( S  e.  ( Base `  K )  /\  ( ( Q  .\/  T )  .\/  P )  e.  ( Base `  K
)  /\  U  e.  ( Base `  K )
) )  ->  (
( ( Q  .\/  T )  .\/  P ) 
./\  S )  .<_  ( U  .\/  S ) )
1525, 19, 16, 28, 151syl13anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 )  .<_  ( U  .\/  S
) )
1531, 8latjcl 14517 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( ( Q  .\/  R )  ./\  U )  e.  ( Base `  K
)  /\  ( R  .\/  P )  e.  (
Base `  K )
)  ->  ( (
( Q  .\/  R
)  ./\  U )  .\/  ( R  .\/  P
) )  e.  (
Base `  K )
)
1545, 30, 32, 153syl3anc 1185 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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
) )
1551, 2, 20latlem12 14545 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( ( ( Q  .\/  T ) 
.\/  P )  ./\  S )  e.  ( Base `  K )  /\  (
( ( Q  .\/  R )  ./\  U )  .\/  ( R  .\/  P
) )  e.  (
Base `  K )  /\  ( U  .\/  S
)  e.  ( Base `  K ) ) )  ->  ( ( ( ( ( Q  .\/  T )  .\/  P ) 
./\  S )  .<_  ( ( ( Q 
.\/  R )  ./\  U )  .\/  ( R 
.\/  P ) )  /\  ( ( ( Q  .\/  T ) 
.\/  P )  ./\  S )  .<_  ( U  .\/  S ) )  <->  ( (
( Q  .\/  T
)  .\/  P )  ./\  S )  .<_  ( ( ( ( Q  .\/  R )  ./\  U )  .\/  ( R  .\/  P
) )  ./\  ( U  .\/  S ) ) ) )
1565, 22, 154, 34, 155syl13anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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
)  ./\  U )  .\/  ( R  .\/  P
) )  /\  (
( ( Q  .\/  T )  .\/  P ) 
./\  S )  .<_  ( U  .\/  S ) )  <->  ( ( ( Q  .\/  T ) 
.\/  P )  ./\  S )  .<_  ( (
( ( Q  .\/  R )  ./\  U )  .\/  ( R  .\/  P
) )  ./\  ( U  .\/  S ) ) ) )
157150, 152, 156mpbi2and 889 . . 3  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 ) 
./\  U )  .\/  ( R  .\/  P ) )  ./\  ( U  .\/  S ) ) )
1581, 2, 8, 20latmlej21 14559 . . . . 5  |-  ( ( K  e.  Lat  /\  ( U  e.  ( Base `  K )  /\  ( Q  .\/  R )  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
) )  ->  (
( Q  .\/  R
)  ./\  U )  .<_  ( U  .\/  S
) )
1595, 28, 25, 19, 158syl13anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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
) )
1601, 2, 8, 20, 9atmod1i1m 30829 . . . 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 ) ) )
1613, 26, 25, 32, 34, 159, 160syl231anc 1205 . . 3  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 ) ) )
162157, 161breqtrrd 4269 . 2  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 )  ./\  U )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) )
1632, 8, 9hlatlej2 30347 . . . . 5  |-  ( ( K  e.  HL  /\  T  e.  A  /\  U  e.  A )  ->  U  .<_  ( T  .\/  U ) )
1643, 7, 26, 163syl3anc 1185 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 ) )
1651, 2, 20latmlem2 14549 . . . . 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 ) ) ) )
1665, 28, 40, 25, 165syl13anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 ) ) ) )
167164, 166mpd 15 . . 3  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 ) ) )
1681, 2, 8latjlej1 14532 . . . 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 ) ) ) ) )
1695, 30, 42, 36, 168syl13anc 1187 . . 3  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 ) ) ) ) )
170167, 169mpd 15 . 2  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 ) ) ) )
1711, 2, 5, 22, 38, 44, 162, 170lattrd 14525 1  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( P  .\/  Q )  /\  ( ( 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 )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1654    e. wcel 1728   class class class wbr 4243   ` cfv 5489  (class class class)co 6117   Basecbs 13507   lecple 13574   joincjn 14439   meetcmee 14440   Latclat 14512   Atomscatm 30235   HLchlt 30322
This theorem is referenced by:  dalawlem4  30845  dalawlem5  30846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-rep 4351  ax-sep 4361  ax-nul 4369  ax-pow 4412  ax-pr 4438  ax-un 4736
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2717  df-rex 2718  df-reu 2719  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-op 3852  df-uni 4045  df-iun 4124  df-iin 4125  df-br 4244  df-opab 4298  df-mpt 4299  df-id 4533  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5453  df-fun 5491  df-fn 5492  df-f 5493  df-f1 5494  df-fo 5495  df-f1o 5496  df-fv 5497  df-ov 6120  df-oprab 6121  df-mpt2 6122  df-1st 6385  df-2nd 6386  df-undef 6579  df-riota 6585  df-poset 14441  df-plt 14453  df-lub 14469  df-glb 14470  df-join 14471  df-meet 14472  df-p0 14506  df-lat 14513  df-clat 14575  df-oposet 30148  df-ol 30150  df-oml 30151  df-covers 30238  df-ats 30239  df-atl 30270  df-cvlat 30294  df-hlat 30323  df-psubsp 30474  df-pmap 30475  df-padd 30767
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