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Theorem dalawlem9 30677
Description: Lemma for dalaw 30684. Special case to eliminate the requirement  -.  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( R 
.\/  P ) in dalawlem1 30669. (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
dalawlem9  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( 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 dalawlem9
StepHypRef Expression
1 simp11 988 . . 3  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( 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 )
2 hllat 30162 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
31, 2syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( 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 )
4 simp22 992 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( 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 )
5 simp32 995 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( 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 )
6 eqid 2437 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
7 dalawlem.j . . . . . . 7  |-  .\/  =  ( join `  K )
8 dalawlem.a . . . . . . 7  |-  A  =  ( Atoms `  K )
96, 7, 8hlatjcl 30165 . . . . . 6  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
101, 4, 5, 9syl3anc 1185 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( 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 ) )
11 simp21 991 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( 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 )
12 simp31 994 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( 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 )
136, 7, 8hlatjcl 30165 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
141, 11, 12, 13syl3anc 1185 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( 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 ) )
15 dalawlem.m . . . . . 6  |-  ./\  =  ( meet `  K )
166, 15latmcom 14505 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  ( P  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  =  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) ) )
173, 10, 14, 16syl3anc 1185 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( 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 ) ) )
18 simp12 989 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( 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  .\/  P ) )
19 simp23 993 . . . . . 6  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( 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 )
207, 8hlatjcom 30166 . . . . . 6  |-  ( ( K  e.  HL  /\  R  e.  A  /\  P  e.  A )  ->  ( R  .\/  P
)  =  ( P 
.\/  R ) )
211, 19, 11, 20syl3anc 1185 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( 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
)  =  ( P 
.\/  R ) )
2218, 21breqtrd 4237 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( 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  .\/  R ) )
2317, 22eqbrtrd 4233 . . 3  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( 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 ) )
24 simp13 990 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( 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 ) )
2517, 24eqbrtrd 4233 . . 3  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( P 
.\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  -> 
( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( R  .\/  U ) )
26 simp33 996 . . 3  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( 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 )
27 dalawlem.l . . . 4  |-  .<_  =  ( le `  K )
2827, 7, 15, 8dalawlem8 30676 . . 3  |-  ( ( ( K  e.  HL  /\  ( ( Q  .\/  T )  ./\  ( P  .\/  S ) )  .<_  ( P  .\/  R )  /\  ( ( Q 
.\/  T )  ./\  ( P  .\/  S ) )  .<_  ( R  .\/  U ) )  /\  ( Q  e.  A  /\  P  e.  A  /\  R  e.  A
)  /\  ( T  e.  A  /\  S  e.  A  /\  U  e.  A ) )  -> 
( ( Q  .\/  P )  ./\  ( T  .\/  S ) )  .<_  ( ( ( P 
.\/  R )  ./\  ( S  .\/  U ) )  .\/  ( ( R  .\/  Q ) 
./\  ( U  .\/  T ) ) ) )
291, 23, 25, 4, 11, 19, 5, 12, 26, 28syl333anc 1217 . 2  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( 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  .\/  P )  ./\  ( T  .\/  S ) )  .<_  ( ( ( P 
.\/  R )  ./\  ( S  .\/  U ) )  .\/  ( ( R  .\/  Q ) 
./\  ( U  .\/  T ) ) ) )
307, 8hlatjcom 30166 . . . 4  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
311, 11, 4, 30syl3anc 1185 . . 3  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( P 
.\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  -> 
( P  .\/  Q
)  =  ( Q 
.\/  P ) )
327, 8hlatjcom 30166 . . . 4  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  =  ( T 
.\/  S ) )
331, 12, 5, 32syl3anc 1185 . . 3  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( P 
.\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  -> 
( S  .\/  T
)  =  ( T 
.\/  S ) )
3431, 33oveq12d 6100 . 2  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( 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  .\/  P )  ./\  ( T  .\/  S ) ) )
356, 7, 8hlatjcl 30165 . . . . . 6  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
361, 4, 19, 35syl3anc 1185 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( 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 ) )
376, 7, 8hlatjcl 30165 . . . . . 6  |-  ( ( K  e.  HL  /\  T  e.  A  /\  U  e.  A )  ->  ( T  .\/  U
)  e.  ( Base `  K ) )
381, 5, 26, 37syl3anc 1185 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( 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 ) )
396, 15latmcl 14481 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Q  .\/  R )  e.  ( Base `  K
)  /\  ( T  .\/  U )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  R )  ./\  ( T  .\/  U ) )  e.  ( Base `  K ) )
403, 36, 38, 39syl3anc 1185 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( 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
) )
416, 7, 8hlatjcl 30165 . . . . . 6  |-  ( ( K  e.  HL  /\  R  e.  A  /\  P  e.  A )  ->  ( R  .\/  P
)  e.  ( Base `  K ) )
421, 19, 11, 41syl3anc 1185 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( 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 ) )
436, 7, 8hlatjcl 30165 . . . . . 6  |-  ( ( K  e.  HL  /\  U  e.  A  /\  S  e.  A )  ->  ( U  .\/  S
)  e.  ( Base `  K ) )
441, 26, 12, 43syl3anc 1185 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( 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 ) )
456, 15latmcl 14481 . . . . 5  |-  ( ( K  e.  Lat  /\  ( R  .\/  P )  e.  ( Base `  K
)  /\  ( U  .\/  S )  e.  (
Base `  K )
)  ->  ( ( R  .\/  P )  ./\  ( U  .\/  S ) )  e.  ( Base `  K ) )
463, 42, 44, 45syl3anc 1185 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( 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
) )
476, 7latjcom 14489 . . . 4  |-  ( ( 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 ) ) )  =  ( ( ( R 
.\/  P )  ./\  ( U  .\/  S ) )  .\/  ( ( Q  .\/  R ) 
./\  ( T  .\/  U ) ) ) )
483, 40, 46, 47syl3anc 1185 . . 3  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( 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 ) ) )  =  ( ( ( R 
.\/  P )  ./\  ( U  .\/  S ) )  .\/  ( ( Q  .\/  R ) 
./\  ( T  .\/  U ) ) ) )
497, 8hlatjcom 30166 . . . . . 6  |-  ( ( K  e.  HL  /\  U  e.  A  /\  S  e.  A )  ->  ( U  .\/  S
)  =  ( S 
.\/  U ) )
501, 26, 12, 49syl3anc 1185 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( 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
)  =  ( S 
.\/  U ) )
5121, 50oveq12d 6100 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( 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 ) )  =  ( ( P  .\/  R )  ./\  ( S  .\/  U ) ) )
527, 8hlatjcom 30166 . . . . . 6  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  =  ( R 
.\/  Q ) )
531, 4, 19, 52syl3anc 1185 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( 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
)  =  ( R 
.\/  Q ) )
547, 8hlatjcom 30166 . . . . . 6  |-  ( ( K  e.  HL  /\  T  e.  A  /\  U  e.  A )  ->  ( T  .\/  U
)  =  ( U 
.\/  T ) )
551, 5, 26, 54syl3anc 1185 . . . . 5  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( 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
)  =  ( U 
.\/  T ) )
5653, 55oveq12d 6100 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( 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  .\/  Q )  ./\  ( U  .\/  T ) ) )
5751, 56oveq12d 6100 . . 3  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( P 
.\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A
)  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A ) )  -> 
( ( ( R 
.\/  P )  ./\  ( U  .\/  S ) )  .\/  ( ( Q  .\/  R ) 
./\  ( T  .\/  U ) ) )  =  ( ( ( P 
.\/  R )  ./\  ( S  .\/  U ) )  .\/  ( ( R  .\/  Q ) 
./\  ( U  .\/  T ) ) ) )
5848, 57eqtrd 2469 . 2  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( 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 ) ) )  =  ( ( ( P 
.\/  R )  ./\  ( S  .\/  U ) )  .\/  ( ( R  .\/  Q ) 
./\  ( U  .\/  T ) ) ) )
5929, 34, 583brtr4d 4243 1  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  P )  /\  ( ( 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    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   Basecbs 13470   lecple 13537   joincjn 14402   meetcmee 14403   Latclat 14475   Atomscatm 30062   HLchlt 30149
This theorem is referenced by:  dalawlem10  30678
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
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 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-iin 4097  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-undef 6544  df-riota 6550  df-poset 14404  df-plt 14416  df-lub 14432  df-glb 14433  df-join 14434  df-meet 14435  df-p0 14469  df-lat 14476  df-clat 14538  df-oposet 29975  df-ol 29977  df-oml 29978  df-covers 30065  df-ats 30066  df-atl 30097  df-cvlat 30121  df-hlat 30150  df-psubsp 30301  df-pmap 30302  df-padd 30594
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