Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dalawlem7 Unicode version

Theorem dalawlem7 30359
Description: Lemma for dalaw 30368. Second piece of dalawlem8 30360. (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
dalawlem7  |-  ( ( ( 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 )  .\/  S )  ./\  T )  .<_  ( ( ( Q 
.\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) )

Proof of Theorem dalawlem7
StepHypRef Expression
1 eqid 2404 . 2  |-  ( Base `  K )  =  (
Base `  K )
2 dalawlem.l . 2  |-  .<_  =  ( le `  K )
3 simp11 987 . . 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 29846 . . 3  |-  ( K  e.  HL  ->  K  e.  Lat )
53, 4syl 16 . 2  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( Q  .\/  R )  /\  ( ( P 
.\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  ->  K  e.  Lat )
6 simp21 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 ) )  ->  P  e.  A )
7 simp22 991 . . . . 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 29849 . . . . 5  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
113, 6, 7, 10syl3anc 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
)  e.  ( Base `  K ) )
12 simp31 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 ) )  ->  S  e.  A )
131, 9atbase 29772 . . . . 5  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
1412, 13syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( Q  .\/  R )  /\  ( ( P 
.\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  ->  S  e.  ( Base `  K ) )
151, 8latjcl 14434 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  .\/  S )  e.  ( Base `  K ) )
165, 11, 14, 15syl3anc 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 ) )  -> 
( ( P  .\/  Q )  .\/  S )  e.  ( Base `  K
) )
17 simp32 994 . . . 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.  A )
181, 9atbase 29772 . . . 4  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
1917, 18syl 16 . . 3  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( Q  .\/  R )  /\  ( ( P 
.\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  ->  T  e.  ( Base `  K ) )
20 dalawlem.m . . . 4  |-  ./\  =  ( meet `  K )
211, 20latmcl 14435 . . 3  |-  ( ( K  e.  Lat  /\  ( ( P  .\/  Q )  .\/  S )  e.  ( Base `  K
)  /\  T  e.  ( Base `  K )
)  ->  ( (
( P  .\/  Q
)  .\/  S )  ./\  T )  e.  (
Base `  K )
)
225, 16, 19, 21syl3anc 1184 . 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 )  .\/  S )  ./\  T )  e.  ( Base `  K
) )
23 simp23 992 . . . 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  e.  A )
241, 8, 9hlatjcl 29849 . . . 4  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
253, 7, 23, 24syl3anc 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
)  e.  ( Base `  K ) )
26 simp33 995 . . . 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.  A )
271, 8, 9hlatjcl 29849 . . . 4  |-  ( ( K  e.  HL  /\  T  e.  A  /\  U  e.  A )  ->  ( T  .\/  U
)  e.  ( Base `  K ) )
283, 17, 26, 27syl3anc 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 ) )  -> 
( T  .\/  U
)  e.  ( Base `  K ) )
291, 20latmcl 14435 . . 3  |-  ( ( K  e.  Lat  /\  ( Q  .\/  R )  e.  ( Base `  K
)  /\  ( T  .\/  U )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )  e.  ( Base `  K ) )
305, 25, 28, 29syl3anc 1184 . 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 ) )  e.  ( Base `  K
) )
311, 8, 9hlatjcl 29849 . . . . 5  |-  ( ( K  e.  HL  /\  R  e.  A  /\  P  e.  A )  ->  ( R  .\/  P
)  e.  ( Base `  K ) )
323, 23, 6, 31syl3anc 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 ) )  -> 
( R  .\/  P
)  e.  ( Base `  K ) )
331, 8, 9hlatjcl 29849 . . . . 5  |-  ( ( K  e.  HL  /\  U  e.  A  /\  S  e.  A )  ->  ( U  .\/  S
)  e.  ( Base `  K ) )
343, 26, 12, 33syl3anc 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  .\/  S
)  e.  ( Base `  K ) )
351, 20latmcl 14435 . . . 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 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 ) )  -> 
( ( R  .\/  P )  ./\  ( U  .\/  S ) )  e.  ( Base `  K
) )
371, 8latjcl 14434 . . 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
) )
385, 30, 36, 37syl3anc 1184 . 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
) )
39 hlol 29844 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OL )
403, 39syl 16 . . . . 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 ) )  ->  K  e.  OL )
411, 8, 9hlatjcl 29849 . . . . . . 7  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
423, 6, 12, 41syl3anc 1184 . . . . . 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  .\/  S
)  e.  ( Base `  K ) )
431, 9atbase 29772 . . . . . . 7  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
447, 43syl 16 . . . . . 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 ) )  ->  Q  e.  ( Base `  K ) )
451, 8latjcl 14434 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  Q  e.  ( Base `  K )
)  ->  ( ( P  .\/  S )  .\/  Q )  e.  ( Base `  K ) )
465, 42, 44, 45syl3anc 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  .\/  S )  .\/  Q )  e.  ( Base `  K
) )
471, 8, 9hlatjcl 29849 . . . . . 6  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
483, 7, 17, 47syl3anc 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 ) )  -> 
( Q  .\/  T
)  e.  ( Base `  K ) )
491, 20latmassOLD 29712 . . . . 5  |-  ( ( K  e.  OL  /\  ( ( ( P 
.\/  S )  .\/  Q )  e.  ( Base `  K )  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  T  e.  ( Base `  K )
) )  ->  (
( ( ( P 
.\/  S )  .\/  Q )  ./\  ( Q  .\/  T ) )  ./\  T )  =  ( ( ( P  .\/  S
)  .\/  Q )  ./\  ( ( Q  .\/  T )  ./\  T )
) )
5040, 46, 48, 19, 49syl13anc 1186 . . . 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  .\/  S ) 
.\/  Q )  ./\  ( Q  .\/  T ) )  ./\  T )  =  ( ( ( P  .\/  S ) 
.\/  Q )  ./\  ( ( Q  .\/  T )  ./\  T )
) )
518, 9hlatj32 29854 . . . . . 6  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  S  e.  A  /\  Q  e.  A
) )  ->  (
( P  .\/  S
)  .\/  Q )  =  ( ( P 
.\/  Q )  .\/  S ) )
523, 6, 12, 7, 51syl13anc 1186 . . . . 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  .\/  S )  .\/  Q )  =  ( ( P 
.\/  Q )  .\/  S ) )
532, 8, 9hlatlej2 29858 . . . . . . 7  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  ->  T  .<_  ( Q  .\/  T ) )
543, 7, 17, 53syl3anc 1184 . . . . . 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 ) )  ->  T  .<_  ( Q  .\/  T ) )
551, 2, 20latleeqm2 14464 . . . . . . 7  |-  ( ( K  e.  Lat  /\  T  e.  ( Base `  K )  /\  ( Q  .\/  T )  e.  ( Base `  K
) )  ->  ( T  .<_  ( Q  .\/  T )  <->  ( ( Q 
.\/  T )  ./\  T )  =  T ) )
565, 19, 48, 55syl3anc 1184 . . . . . 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 ) )  -> 
( T  .<_  ( Q 
.\/  T )  <->  ( ( Q  .\/  T )  ./\  T )  =  T ) )
5754, 56mpbid 202 . . . . 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  .\/  T )  ./\  T )  =  T )
5852, 57oveq12d 6058 . . . 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 
.\/  S )  .\/  Q )  ./\  ( ( Q  .\/  T )  ./\  T ) )  =  ( ( ( P  .\/  Q )  .\/  S ) 
./\  T ) )
5950, 58eqtr2d 2437 . . 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 )  .\/  S )  ./\  T )  =  ( ( ( ( P  .\/  S
)  .\/  Q )  ./\  ( Q  .\/  T
) )  ./\  T
) )
60 simp12 988 . . . . . 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  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( Q  .\/  R ) )
611, 20latmcl 14435 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  ( Q  .\/  T )  e.  (
Base `  K )
)  ->  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  e.  ( Base `  K ) )
625, 42, 48, 61syl3anc 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  .\/  T ) )  e.  ( Base `  K
) )
631, 2, 8latjlej1 14449 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( ( ( P 
.\/  S )  ./\  ( Q  .\/  T ) )  e.  ( Base `  K )  /\  ( Q  .\/  R )  e.  ( Base `  K
)  /\  Q  e.  ( Base `  K )
) )  ->  (
( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( Q  .\/  R )  ->  ( ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .\/  Q
)  .<_  ( ( Q 
.\/  R )  .\/  Q ) ) )
645, 62, 25, 44, 63syl13anc 1186 . . . . . 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 
.\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( Q  .\/  R )  ->  (
( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .\/  Q )  .<_  ( ( Q  .\/  R )  .\/  Q ) ) )
6560, 64mpd 15 . . . . 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 
.\/  S )  ./\  ( Q  .\/  T ) )  .\/  Q ) 
.<_  ( ( Q  .\/  R )  .\/  Q ) )
662, 8, 9hlatlej1 29857 . . . . . . 7  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  ->  Q  .<_  ( Q  .\/  T ) )
673, 7, 17, 66syl3anc 1184 . . . . . 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 ) )  ->  Q  .<_  ( Q  .\/  T ) )
681, 2, 8, 20, 9atmod4i1 30348 . . . . . 6  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  ( P  .\/  S
)  e.  ( Base `  K )  /\  ( Q  .\/  T )  e.  ( Base `  K
) )  /\  Q  .<_  ( Q  .\/  T
) )  ->  (
( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .\/  Q )  =  ( ( ( P  .\/  S
)  .\/  Q )  ./\  ( Q  .\/  T
) ) )
693, 7, 42, 48, 67, 68syl131anc 1197 . . . . 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 
.\/  S )  ./\  ( Q  .\/  T ) )  .\/  Q )  =  ( ( ( P  .\/  S ) 
.\/  Q )  ./\  ( Q  .\/  T ) ) )
708, 9hlatj32 29854 . . . . . . 7  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  R  e.  A  /\  Q  e.  A
) )  ->  (
( Q  .\/  R
)  .\/  Q )  =  ( ( Q 
.\/  Q )  .\/  R ) )
713, 7, 23, 7, 70syl13anc 1186 . . . . . 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 ) )  -> 
( ( Q  .\/  R )  .\/  Q )  =  ( ( Q 
.\/  Q )  .\/  R ) )
721, 8latjidm 14458 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K ) )  -> 
( Q  .\/  Q
)  =  Q )
735, 44, 72syl2anc 643 . . . . . . 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 ) )  -> 
( Q  .\/  Q
)  =  Q )
7473oveq1d 6055 . . . . . 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 ) )  -> 
( ( Q  .\/  Q )  .\/  R )  =  ( Q  .\/  R ) )
7571, 74eqtrd 2436 . . . . 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 )  .\/  Q )  =  ( Q  .\/  R ) )
7665, 69, 753brtr3d 4201 . . . 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 
.\/  S )  .\/  Q )  ./\  ( Q  .\/  T ) )  .<_  ( Q  .\/  R ) )
772, 8, 9hlatlej1 29857 . . . . 5  |-  ( ( K  e.  HL  /\  T  e.  A  /\  U  e.  A )  ->  T  .<_  ( T  .\/  U ) )
783, 17, 26, 77syl3anc 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 ) )  ->  T  .<_  ( T  .\/  U ) )
791, 20latmcl 14435 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( ( P  .\/  S )  .\/  Q )  e.  ( Base `  K
)  /\  ( Q  .\/  T )  e.  (
Base `  K )
)  ->  ( (
( P  .\/  S
)  .\/  Q )  ./\  ( Q  .\/  T
) )  e.  (
Base `  K )
)
805, 46, 48, 79syl3anc 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 
.\/  S )  .\/  Q )  ./\  ( Q  .\/  T ) )  e.  ( Base `  K
) )
811, 2, 20latmlem12 14467 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( ( ( P  .\/  S ) 
.\/  Q )  ./\  ( Q  .\/  T ) )  e.  ( Base `  K )  /\  ( Q  .\/  R )  e.  ( Base `  K
) )  /\  ( T  e.  ( Base `  K )  /\  ( T  .\/  U )  e.  ( Base `  K
) ) )  -> 
( ( ( ( ( P  .\/  S
)  .\/  Q )  ./\  ( Q  .\/  T
) )  .<_  ( Q 
.\/  R )  /\  T  .<_  ( T  .\/  U ) )  ->  (
( ( ( P 
.\/  S )  .\/  Q )  ./\  ( Q  .\/  T ) )  ./\  T )  .<_  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) ) ) )
825, 80, 25, 19, 28, 81syl122anc 1193 . . . 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  .\/  S
)  .\/  Q )  ./\  ( Q  .\/  T
) )  .<_  ( Q 
.\/  R )  /\  T  .<_  ( T  .\/  U ) )  ->  (
( ( ( P 
.\/  S )  .\/  Q )  ./\  ( Q  .\/  T ) )  ./\  T )  .<_  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) ) ) )
8376, 78, 82mp2and 661 . . 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  .\/  S ) 
.\/  Q )  ./\  ( Q  .\/  T ) )  ./\  T )  .<_  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) ) )
8459, 83eqbrtrd 4192 . 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 )  .\/  S )  ./\  T )  .<_  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) ) )
851, 2, 8latlej1 14444 . . 3  |-  ( ( K  e.  Lat  /\  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )  e.  ( Base `  K
)  /\  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )  e.  ( Base `  K ) )  -> 
( ( Q  .\/  R )  ./\  ( T  .\/  U ) )  .<_  ( ( ( Q 
.\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) )
865, 30, 36, 85syl3anc 1184 . 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 ) )  .<_  ( ( ( Q 
.\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) )
871, 2, 5, 22, 30, 38, 84, 86lattrd 14442 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 )  .\/  S )  ./\  T )  .<_  ( ( ( Q 
.\/  R )  ./\  ( T  .\/  U ) )  .\/  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   Latclat 14429   OLcol 29657   Atomscatm 29746   HLchlt 29833
This theorem is referenced by:  dalawlem8  30360
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-lat 14430  df-clat 14492  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-psubsp 29985  df-pmap 29986  df-padd 30278
  Copyright terms: Public domain W3C validator