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Theorem dalawlem12 30364
Description: Lemma for dalaw 30368. Second part of dalawlem13 30365. (Contributed by NM, 17-Sep-2012.)
Hypotheses
Ref Expression
dalawlem.l  |-  .<_  =  ( le `  K )
dalawlem.j  |-  .\/  =  ( join `  K )
dalawlem.m  |-  ./\  =  ( meet `  K )
dalawlem.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
dalawlem12  |-  ( ( ( K  e.  HL  /\  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 dalawlem12
StepHypRef Expression
1 eqid 2404 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
2 dalawlem.l . . . 4  |-  .<_  =  ( le `  K )
3 simp11 987 . . . . 5  |-  ( ( ( K  e.  HL  /\  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 . . . . 5  |-  ( K  e.  HL  ->  K  e.  Lat )
53, 4syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  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 . . . . . 6  |-  ( ( ( K  e.  HL  /\  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 . . . . . 6  |-  ( ( ( K  e.  HL  /\  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 . . . . . . 7  |-  .\/  =  ( join `  K )
9 dalawlem.a . . . . . . 7  |-  A  =  ( Atoms `  K )
101, 8, 9hlatjcl 29849 . . . . . 6  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  e.  ( Base `  K ) )
113, 6, 7, 10syl3anc 1184 . . . . 5  |-  ( ( ( K  e.  HL  /\  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 . . . . . 6  |-  ( ( ( K  e.  HL  /\  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 )
13 simp32 994 . . . . . 6  |-  ( ( ( K  e.  HL  /\  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 )
141, 8, 9hlatjcl 29849 . . . . . 6  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
153, 12, 13, 14syl3anc 1184 . . . . 5  |-  ( ( ( K  e.  HL  /\  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  .\/  T )  e.  (
Base `  K )
)
16 dalawlem.m . . . . . 6  |-  ./\  =  ( meet `  K )
171, 16latmcl 14435 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  ( S  .\/  T )  e.  (
Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  e.  ( Base `  K ) )
185, 11, 15, 17syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  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 ) )
191, 9atbase 29772 . . . . . . . 8  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
2012, 19syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  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 )
)
211, 8latjcl 14434 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
)  ->  ( ( P  .\/  Q )  .\/  S )  e.  ( Base `  K ) )
225, 11, 20, 21syl3anc 1184 . . . . . 6  |-  ( ( ( K  e.  HL  /\  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 ) )
231, 9atbase 29772 . . . . . . 7  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
2413, 23syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  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 )
)
251, 16latmcl 14435 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( ( P  .\/  Q )  .\/  S )  e.  ( Base `  K
)  /\  T  e.  ( Base `  K )
)  ->  ( (
( P  .\/  Q
)  .\/  S )  ./\  T )  e.  (
Base `  K )
)
265, 22, 24, 25syl3anc 1184 . . . . 5  |-  ( ( ( K  e.  HL  /\  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 )
)
271, 8latjcl 14434 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( ( P 
.\/  Q )  .\/  S )  ./\  T )  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
)  ->  ( (
( ( P  .\/  Q )  .\/  S ) 
./\  T )  .\/  S )  e.  ( Base `  K ) )
285, 26, 20, 27syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  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 )  .\/  S )  e.  ( Base `  K ) )
291, 9atbase 29772 . . . . . . 7  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
307, 29syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  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 )
)
31 simp33 995 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  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 )
321, 8, 9hlatjcl 29849 . . . . . . 7  |-  ( ( K  e.  HL  /\  T  e.  A  /\  U  e.  A )  ->  ( T  .\/  U
)  e.  ( Base `  K ) )
333, 13, 31, 32syl3anc 1184 . . . . . 6  |-  ( ( ( K  e.  HL  /\  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 )
)
341, 16latmcl 14435 . . . . . 6  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  ( T  .\/  U )  e.  ( Base `  K
) )  ->  ( Q  ./\  ( T  .\/  U ) )  e.  (
Base `  K )
)
355, 30, 33, 34syl3anc 1184 . . . . 5  |-  ( ( ( K  e.  HL  /\  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  .\/  U
) )  e.  (
Base `  K )
)
361, 8, 9hlatjcl 29849 . . . . . 6  |-  ( ( K  e.  HL  /\  U  e.  A  /\  S  e.  A )  ->  ( U  .\/  S
)  e.  ( Base `  K ) )
373, 31, 12, 36syl3anc 1184 . . . . 5  |-  ( ( ( K  e.  HL  /\  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 )
)
381, 8latjcl 14434 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Q  ./\  ( T 
.\/  U ) )  e.  ( Base `  K
)  /\  ( U  .\/  S )  e.  (
Base `  K )
)  ->  ( ( Q  ./\  ( T  .\/  U ) )  .\/  ( U  .\/  S ) )  e.  ( Base `  K
) )
395, 35, 37, 38syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  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  .\/  U ) )  .\/  ( U  .\/  S ) )  e.  ( Base `  K
) )
401, 2, 8latlej1 14444 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
)  ->  ( P  .\/  Q )  .<_  ( ( P  .\/  Q ) 
.\/  S ) )
415, 11, 20, 40syl3anc 1184 . . . . . 6  |-  ( ( ( K  e.  HL  /\  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 )  .<_  ( ( P  .\/  Q ) 
.\/  S ) )
421, 8, 9hlatjcl 29849 . . . . . . . 8  |-  ( ( K  e.  HL  /\  T  e.  A  /\  S  e.  A )  ->  ( T  .\/  S
)  e.  ( Base `  K ) )
433, 13, 12, 42syl3anc 1184 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  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  .\/  S )  e.  (
Base `  K )
)
441, 2, 16latmlem1 14465 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( ( P  .\/  Q )  e.  ( Base `  K )  /\  (
( P  .\/  Q
)  .\/  S )  e.  ( Base `  K
)  /\  ( T  .\/  S )  e.  (
Base `  K )
) )  ->  (
( P  .\/  Q
)  .<_  ( ( P 
.\/  Q )  .\/  S )  ->  ( ( P  .\/  Q )  ./\  ( T  .\/  S ) )  .<_  ( (
( P  .\/  Q
)  .\/  S )  ./\  ( T  .\/  S
) ) ) )
455, 11, 22, 43, 44syl13anc 1186 . . . . . 6  |-  ( ( ( K  e.  HL  /\  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 )  .<_  ( ( P  .\/  Q )  .\/  S )  ->  ( ( P 
.\/  Q )  ./\  ( T  .\/  S ) )  .<_  ( (
( P  .\/  Q
)  .\/  S )  ./\  ( T  .\/  S
) ) ) )
4641, 45mpd 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  Q  =  R  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .\/  Q )  ./\  ( T  .\/  S ) )  .<_  ( (
( P  .\/  Q
)  .\/  S )  ./\  ( T  .\/  S
) ) )
478, 9hlatjcom 29850 . . . . . . 7  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  =  ( T 
.\/  S ) )
483, 12, 13, 47syl3anc 1184 . . . . . 6  |-  ( ( ( K  e.  HL  /\  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  .\/  T )  =  ( T  .\/  S ) )
4948oveq2d 6056 . . . . 5  |-  ( ( ( K  e.  HL  /\  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  .\/  Q ) 
./\  ( T  .\/  S ) ) )
501, 2, 8latlej2 14445 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
)  ->  S  .<_  ( ( P  .\/  Q
)  .\/  S )
)
515, 11, 20, 50syl3anc 1184 . . . . . 6  |-  ( ( ( K  e.  HL  /\  Q  =  R  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  S  .<_  ( ( P  .\/  Q
)  .\/  S )
)
521, 2, 8, 16, 9atmod2i2 30344 . . . . . 6  |-  ( ( K  e.  HL  /\  ( T  e.  A  /\  ( ( P  .\/  Q )  .\/  S )  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
)  /\  S  .<_  ( ( P  .\/  Q
)  .\/  S )
)  ->  ( (
( ( P  .\/  Q )  .\/  S ) 
./\  T )  .\/  S )  =  ( ( ( P  .\/  Q
)  .\/  S )  ./\  ( T  .\/  S
) ) )
533, 13, 22, 20, 51, 52syl131anc 1197 . . . . 5  |-  ( ( ( K  e.  HL  /\  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 )  .\/  S )  =  ( ( ( P  .\/  Q
)  .\/  S )  ./\  ( T  .\/  S
) ) )
5446, 49, 533brtr4d 4202 . . . 4  |-  ( ( ( K  e.  HL  /\  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  .\/  Q )  .\/  S ) 
./\  T )  .\/  S ) )
55 hlol 29844 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  OL )
563, 55syl 16 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  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 )
571, 8, 9hlatjcl 29849 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  P  e.  A  /\  S  e.  A )  ->  ( P  .\/  S
)  e.  ( Base `  K ) )
583, 6, 12, 57syl3anc 1184 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  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 )
)
591, 8latjcl 14434 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  ( P  .\/  S )  e.  ( Base `  K
) )  ->  ( Q  .\/  ( P  .\/  S ) )  e.  (
Base `  K )
)
605, 30, 58, 59syl3anc 1184 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  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  .\/  ( P  .\/  S
) )  e.  (
Base `  K )
)
611, 8, 9hlatjcl 29849 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  ->  ( Q  .\/  T
)  e.  ( Base `  K ) )
623, 7, 13, 61syl3anc 1184 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  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 )
)
631, 16latmassOLD 29712 . . . . . . . . . 10  |-  ( ( K  e.  OL  /\  ( ( Q  .\/  ( P  .\/  S ) )  e.  ( Base `  K )  /\  ( Q  .\/  T )  e.  ( Base `  K
)  /\  T  e.  ( Base `  K )
) )  ->  (
( ( Q  .\/  ( P  .\/  S ) )  ./\  ( Q  .\/  T ) )  ./\  T )  =  ( ( Q  .\/  ( P 
.\/  S ) ) 
./\  ( ( Q 
.\/  T )  ./\  T ) ) )
6456, 60, 62, 24, 63syl13anc 1186 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  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  .\/  ( P  .\/  S ) ) 
./\  ( Q  .\/  T ) )  ./\  T
)  =  ( ( Q  .\/  ( P 
.\/  S ) ) 
./\  ( ( Q 
.\/  T )  ./\  T ) ) )
658, 9hlatjass 29852 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  S  e.  A
) )  ->  (
( P  .\/  Q
)  .\/  S )  =  ( P  .\/  ( Q  .\/  S ) ) )
663, 6, 7, 12, 65syl13anc 1186 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  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 )  =  ( P 
.\/  ( Q  .\/  S ) ) )
678, 9hlatj12 29853 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  S  e.  A
) )  ->  ( P  .\/  ( Q  .\/  S ) )  =  ( Q  .\/  ( P 
.\/  S ) ) )
683, 6, 7, 12, 67syl13anc 1186 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  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
) )  =  ( Q  .\/  ( P 
.\/  S ) ) )
6966, 68eqtr2d 2437 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  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  .\/  ( P  .\/  S
) )  =  ( ( P  .\/  Q
)  .\/  S )
)
702, 8, 9hlatlej2 29858 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  ->  T  .<_  ( Q  .\/  T ) )
713, 7, 13, 70syl3anc 1184 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  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 ) )
721, 2, 16latleeqm2 14464 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  T  e.  ( Base `  K )  /\  ( Q  .\/  T )  e.  ( Base `  K
) )  ->  ( T  .<_  ( Q  .\/  T )  <->  ( ( Q 
.\/  T )  ./\  T )  =  T ) )
735, 24, 62, 72syl3anc 1184 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  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 ) )
7471, 73mpbid 202 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  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 )
7569, 74oveq12d 6058 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  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  .\/  ( P  .\/  S ) )  ./\  (
( Q  .\/  T
)  ./\  T )
)  =  ( ( ( P  .\/  Q
)  .\/  S )  ./\  T ) )
7664, 75eqtr2d 2437 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  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  .\/  ( P  .\/  S ) )  ./\  ( Q  .\/  T ) )  ./\  T ) )
772, 8, 9hlatlej1 29857 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  T  e.  A )  ->  Q  .<_  ( Q  .\/  T ) )
783, 7, 13, 77syl3anc 1184 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  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 ) )
791, 2, 8, 16, 9atmod1i1 30339 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  ( Q  e.  A  /\  ( P  .\/  S
)  e.  ( Base `  K )  /\  ( Q  .\/  T )  e.  ( Base `  K
) )  /\  Q  .<_  ( Q  .\/  T
) )  ->  ( Q  .\/  ( ( P 
.\/  S )  ./\  ( Q  .\/  T ) ) )  =  ( ( Q  .\/  ( P  .\/  S ) ) 
./\  ( Q  .\/  T ) ) )
803, 7, 58, 62, 78, 79syl131anc 1197 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  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  .\/  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) ) )  =  ( ( Q 
.\/  ( P  .\/  S ) )  ./\  ( Q  .\/  T ) ) )
812, 8, 9hlatlej2 29858 . . . . . . . . . . . 12  |-  ( ( K  e.  HL  /\  U  e.  A  /\  Q  e.  A )  ->  Q  .<_  ( U  .\/  Q ) )
823, 31, 7, 81syl3anc 1184 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  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  .<_  ( U  .\/  Q ) )
83 simp13 989 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  Q  =  R  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )
84 simp12 988 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  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 )
8584oveq1d 6055 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  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  .\/  U )  =  ( R  .\/  U ) )
868, 9hlatjcom 29850 . . . . . . . . . . . . . 14  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  U  e.  A )  ->  ( Q  .\/  U
)  =  ( U 
.\/  Q ) )
873, 7, 31, 86syl3anc 1184 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  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  .\/  U )  =  ( U  .\/  Q ) )
8885, 87eqtr3d 2438 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  Q  =  R  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( R  .\/  U )  =  ( U  .\/  Q ) )
8983, 88breqtrd 4196 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  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 ) )  .<_  ( U  .\/  Q ) )
901, 16latmcl 14435 . . . . . . . . . . . . 13  |-  ( ( K  e.  Lat  /\  ( P  .\/  S )  e.  ( Base `  K
)  /\  ( Q  .\/  T )  e.  (
Base `  K )
)  ->  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  e.  ( Base `  K ) )
915, 58, 62, 90syl3anc 1184 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  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 ) )
921, 8, 9hlatjcl 29849 . . . . . . . . . . . . 13  |-  ( ( K  e.  HL  /\  U  e.  A  /\  Q  e.  A )  ->  ( U  .\/  Q
)  e.  ( Base `  K ) )
933, 31, 7, 92syl3anc 1184 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  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  .\/  Q )  e.  (
Base `  K )
)
941, 2, 8latjle12 14446 . . . . . . . . . . . 12  |-  ( ( K  e.  Lat  /\  ( Q  e.  ( Base `  K )  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  e.  ( Base `  K
)  /\  ( U  .\/  Q )  e.  (
Base `  K )
) )  ->  (
( Q  .<_  ( U 
.\/  Q )  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( U  .\/  Q ) )  <->  ( Q  .\/  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) ) ) 
.<_  ( U  .\/  Q
) ) )
955, 30, 91, 93, 94syl13anc 1186 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  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  .<_  ( U  .\/  Q )  /\  ( ( P  .\/  S ) 
./\  ( Q  .\/  T ) )  .<_  ( U 
.\/  Q ) )  <-> 
( Q  .\/  (
( P  .\/  S
)  ./\  ( Q  .\/  T ) ) ) 
.<_  ( U  .\/  Q
) ) )
9682, 89, 95mpbi2and 888 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  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  .\/  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) ) ) 
.<_  ( U  .\/  Q
) )
9780, 96eqbrtrrd 4194 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  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  .\/  ( P  .\/  S ) )  ./\  ( Q  .\/  T ) ) 
.<_  ( U  .\/  Q
) )
982, 8, 9hlatlej1 29857 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  T  e.  A  /\  U  e.  A )  ->  T  .<_  ( T  .\/  U ) )
993, 13, 31, 98syl3anc 1184 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  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 ) )
1001, 16latmcl 14435 . . . . . . . . . . 11  |-  ( ( K  e.  Lat  /\  ( Q  .\/  ( P 
.\/  S ) )  e.  ( Base `  K
)  /\  ( Q  .\/  T )  e.  (
Base `  K )
)  ->  ( ( Q  .\/  ( P  .\/  S ) )  ./\  ( Q  .\/  T ) )  e.  ( Base `  K
) )
1015, 60, 62, 100syl3anc 1184 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  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  .\/  ( P  .\/  S ) )  ./\  ( Q  .\/  T ) )  e.  ( Base `  K
) )
1021, 2, 16latmlem12 14467 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  ( ( ( Q 
.\/  ( P  .\/  S ) )  ./\  ( Q  .\/  T ) )  e.  ( Base `  K
)  /\  ( U  .\/  Q )  e.  (
Base `  K )
)  /\  ( T  e.  ( Base `  K
)  /\  ( T  .\/  U )  e.  (
Base `  K )
) )  ->  (
( ( ( Q 
.\/  ( P  .\/  S ) )  ./\  ( Q  .\/  T ) ) 
.<_  ( U  .\/  Q
)  /\  T  .<_  ( T  .\/  U ) )  ->  ( (
( Q  .\/  ( P  .\/  S ) ) 
./\  ( Q  .\/  T ) )  ./\  T
)  .<_  ( ( U 
.\/  Q )  ./\  ( T  .\/  U ) ) ) )
1035, 101, 93, 24, 33, 102syl122anc 1193 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  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  .\/  ( P  .\/  S ) )  ./\  ( Q  .\/  T ) )  .<_  ( U  .\/  Q )  /\  T  .<_  ( T 
.\/  U ) )  ->  ( ( ( Q  .\/  ( P 
.\/  S ) ) 
./\  ( Q  .\/  T ) )  ./\  T
)  .<_  ( ( U 
.\/  Q )  ./\  ( T  .\/  U ) ) ) )
10497, 99, 103mp2and 661 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  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  .\/  ( P  .\/  S ) ) 
./\  ( Q  .\/  T ) )  ./\  T
)  .<_  ( ( U 
.\/  Q )  ./\  ( T  .\/  U ) ) )
10576, 104eqbrtrd 4192 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  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 )  .<_  ( ( U  .\/  Q ) 
./\  ( T  .\/  U ) ) )
1062, 8, 9hlatlej2 29858 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  T  e.  A  /\  U  e.  A )  ->  U  .<_  ( T  .\/  U ) )
1073, 13, 31, 106syl3anc 1184 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  Q  =  R  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  U  .<_  ( T  .\/  U ) )
1081, 2, 8, 16, 9atmod1i1 30339 . . . . . . . . 9  |-  ( ( K  e.  HL  /\  ( U  e.  A  /\  Q  e.  ( Base `  K )  /\  ( T  .\/  U )  e.  ( Base `  K
) )  /\  U  .<_  ( T  .\/  U
) )  ->  ( U  .\/  ( Q  ./\  ( T  .\/  U ) ) )  =  ( ( U  .\/  Q
)  ./\  ( T  .\/  U ) ) )
1093, 31, 30, 33, 107, 108syl131anc 1197 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  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  .\/  ( Q  ./\  ( T  .\/  U ) ) )  =  ( ( U  .\/  Q ) 
./\  ( T  .\/  U ) ) )
1101, 9atbase 29772 . . . . . . . . . 10  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
11131, 110syl 16 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  Q  =  R  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  U  e.  ( Base `  K )
)
1121, 8latjcom 14443 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  U  e.  ( Base `  K )  /\  ( Q  ./\  ( T  .\/  U ) )  e.  (
Base `  K )
)  ->  ( U  .\/  ( Q  ./\  ( T  .\/  U ) ) )  =  ( ( Q  ./\  ( T  .\/  U ) )  .\/  U ) )
1135, 111, 35, 112syl3anc 1184 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  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  .\/  ( Q  ./\  ( T  .\/  U ) ) )  =  ( ( Q  ./\  ( T  .\/  U ) )  .\/  U ) )
114109, 113eqtr3d 2438 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  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  .\/  Q )  ./\  ( T  .\/  U ) )  =  ( ( Q  ./\  ( T  .\/  U ) )  .\/  U ) )
115105, 114breqtrd 4196 . . . . . 6  |-  ( ( ( K  e.  HL  /\  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  ./\  ( T  .\/  U ) )  .\/  U ) )
1161, 8latjcl 14434 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( Q  ./\  ( T 
.\/  U ) )  e.  ( Base `  K
)  /\  U  e.  ( Base `  K )
)  ->  ( ( Q  ./\  ( T  .\/  U ) )  .\/  U
)  e.  ( Base `  K ) )
1175, 35, 111, 116syl3anc 1184 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  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  .\/  U ) )  .\/  U
)  e.  ( Base `  K ) )
1181, 2, 8latjlej1 14449 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( ( ( ( P  .\/  Q ) 
.\/  S )  ./\  T )  e.  ( Base `  K )  /\  (
( Q  ./\  ( T  .\/  U ) ) 
.\/  U )  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
) )  ->  (
( ( ( P 
.\/  Q )  .\/  S )  ./\  T )  .<_  ( ( Q  ./\  ( T  .\/  U ) )  .\/  U )  ->  ( ( ( ( P  .\/  Q
)  .\/  S )  ./\  T )  .\/  S
)  .<_  ( ( ( Q  ./\  ( T  .\/  U ) )  .\/  U )  .\/  S ) ) )
1195, 26, 117, 20, 118syl13anc 1186 . . . . . 6  |-  ( ( ( K  e.  HL  /\  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  ./\  ( T  .\/  U ) )  .\/  U )  ->  ( ( ( ( P  .\/  Q
)  .\/  S )  ./\  T )  .\/  S
)  .<_  ( ( ( Q  ./\  ( T  .\/  U ) )  .\/  U )  .\/  S ) ) )
120115, 119mpd 15 . . . . 5  |-  ( ( ( K  e.  HL  /\  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 )  .\/  S )  .<_  ( (
( Q  ./\  ( T  .\/  U ) ) 
.\/  U )  .\/  S ) )
1211, 8latjass 14479 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( ( Q  ./\  ( T  .\/  U ) )  e.  ( Base `  K )  /\  U  e.  ( Base `  K
)  /\  S  e.  ( Base `  K )
) )  ->  (
( ( Q  ./\  ( T  .\/  U ) )  .\/  U ) 
.\/  S )  =  ( ( Q  ./\  ( T  .\/  U ) )  .\/  ( U 
.\/  S ) ) )
1225, 35, 111, 20, 121syl13anc 1186 . . . . 5  |-  ( ( ( K  e.  HL  /\  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  .\/  U ) ) 
.\/  U )  .\/  S )  =  ( ( Q  ./\  ( T  .\/  U ) )  .\/  ( U  .\/  S ) ) )
123120, 122breqtrd 4196 . . . 4  |-  ( ( ( K  e.  HL  /\  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 )  .\/  S )  .<_  ( ( Q  ./\  ( T  .\/  U ) )  .\/  ( U  .\/  S ) ) )
1241, 2, 5, 18, 28, 39, 54, 123lattrd 14442 . . 3  |-  ( ( ( K  e.  HL  /\  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  ./\  ( T  .\/  U ) )  .\/  ( U  .\/  S ) ) )
1251, 2, 16latmle1 14460 . . . 4  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  ( Base `  K
)  /\  ( S  .\/  T )  e.  (
Base `  K )
)  ->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( P  .\/  Q ) )
1265, 11, 15, 125syl3anc 1184 . . 3  |-  ( ( ( K  e.  HL  /\  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  .\/  Q ) )
1271, 2, 16latlem12 14462 . . . 4  |-  ( ( K  e.  Lat  /\  ( ( ( P 
.\/  Q )  ./\  ( S  .\/  T ) )  e.  ( Base `  K )  /\  (
( Q  ./\  ( T  .\/  U ) ) 
.\/  ( U  .\/  S ) )  e.  (
Base `  K )  /\  ( P  .\/  Q
)  e.  ( Base `  K ) ) )  ->  ( ( ( ( P  .\/  Q
)  ./\  ( S  .\/  T ) )  .<_  ( ( Q  ./\  ( T  .\/  U ) )  .\/  ( U 
.\/  S ) )  /\  ( ( P 
.\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( P  .\/  Q ) )  <->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( (
( Q  ./\  ( T  .\/  U ) ) 
.\/  ( U  .\/  S ) )  ./\  ( P  .\/  Q ) ) ) )
1285, 18, 39, 11, 127syl13anc 1186 . . 3  |-  ( ( ( K  e.  HL  /\  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  ./\  ( T  .\/  U ) )  .\/  ( U 
.\/  S ) )  /\  ( ( P 
.\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( P  .\/  Q ) )  <->  ( ( P  .\/  Q )  ./\  ( S  .\/  T ) )  .<_  ( (
( Q  ./\  ( T  .\/  U ) ) 
.\/  ( U  .\/  S ) )  ./\  ( P  .\/  Q ) ) ) )
129124, 126, 128mpbi2and 888 . 2  |-  ( ( ( K  e.  HL  /\  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  ./\  ( T  .\/  U ) ) 
.\/  ( U  .\/  S ) )  ./\  ( P  .\/  Q ) ) )
1301, 9atbase 29772 . . . . . 6  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
1316, 130syl 16 . . . . 5  |-  ( ( ( K  e.  HL  /\  Q  =  R  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  P  e.  ( Base `  K )
)
1321, 2, 8, 16latmlej12 14475 . . . . 5  |-  ( ( K  e.  Lat  /\  ( Q  e.  ( Base `  K )  /\  ( T  .\/  U )  e.  ( Base `  K
)  /\  P  e.  ( Base `  K )
) )  ->  ( Q  ./\  ( T  .\/  U ) )  .<_  ( P 
.\/  Q ) )
1335, 30, 33, 131, 132syl13anc 1186 . . . 4  |-  ( ( ( K  e.  HL  /\  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  .\/  U
) )  .<_  ( P 
.\/  Q ) )
1341, 2, 8, 16, 9llnmod1i2 30342 . . . 4  |-  ( ( ( K  e.  HL  /\  ( Q  ./\  ( T  .\/  U ) )  e.  ( Base `  K
)  /\  ( P  .\/  Q )  e.  (
Base `  K )
)  /\  ( U  e.  A  /\  S  e.  A )  /\  ( Q  ./\  ( T  .\/  U ) )  .<_  ( P 
.\/  Q ) )  ->  ( ( Q 
./\  ( T  .\/  U ) )  .\/  (
( U  .\/  S
)  ./\  ( P  .\/  Q ) ) )  =  ( ( ( Q  ./\  ( T  .\/  U ) )  .\/  ( U  .\/  S ) )  ./\  ( P  .\/  Q ) ) )
1353, 35, 11, 31, 12, 133, 134syl321anc 1206 . . 3  |-  ( ( ( K  e.  HL  /\  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  .\/  U ) )  .\/  (
( U  .\/  S
)  ./\  ( P  .\/  Q ) ) )  =  ( ( ( Q  ./\  ( T  .\/  U ) )  .\/  ( U  .\/  S ) )  ./\  ( P  .\/  Q ) ) )
1368, 9hlatjidm 29851 . . . . . . 7  |-  ( ( K  e.  HL  /\  Q  e.  A )  ->  ( Q  .\/  Q
)  =  Q )
1373, 7, 136syl2anc 643 . . . . . 6  |-  ( ( ( K  e.  HL  /\  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 )
13884oveq2d 6056 . . . . . 6  |-  ( ( ( K  e.  HL  /\  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  .\/  R ) )
139137, 138eqtr3d 2438 . . . . 5  |-  ( ( ( K  e.  HL  /\  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 ) )
140139oveq1d 6055 . . . 4  |-  ( ( ( K  e.  HL  /\  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  .\/  U
) )  =  ( ( Q  .\/  R
)  ./\  ( T  .\/  U ) ) )
1411, 16latmcom 14459 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( U  .\/  S )  e.  ( Base `  K
)  /\  ( P  .\/  Q )  e.  (
Base `  K )
)  ->  ( ( U  .\/  S )  ./\  ( P  .\/  Q ) )  =  ( ( P  .\/  Q ) 
./\  ( U  .\/  S ) ) )
1425, 37, 11, 141syl3anc 1184 . . . . 5  |-  ( ( ( K  e.  HL  /\  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 )  ./\  ( P  .\/  Q ) )  =  ( ( P  .\/  Q ) 
./\  ( U  .\/  S ) ) )
1438, 9hlatjcom 29850 . . . . . . . 8  |-  ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  ->  ( P  .\/  Q
)  =  ( Q 
.\/  P ) )
1443, 6, 7, 143syl3anc 1184 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  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 )  =  ( Q  .\/  P ) )
14584oveq1d 6055 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  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  .\/  P )  =  ( R  .\/  P ) )
146144, 145eqtrd 2436 . . . . . 6  |-  ( ( ( K  e.  HL  /\  Q  =  R  /\  ( ( P  .\/  S )  ./\  ( Q  .\/  T ) )  .<_  ( R  .\/  U ) )  /\  ( P  e.  A  /\  Q  e.  A  /\  R  e.  A )  /\  ( S  e.  A  /\  T  e.  A  /\  U  e.  A )
)  ->  ( P  .\/  Q )  =  ( R  .\/  P ) )
147146oveq1d 6055 . . . . 5  |-  ( ( ( K  e.  HL  /\  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 )  ./\  ( U  .\/  S ) )  =  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) )
148142, 147eqtrd 2436 . . . 4  |-  ( ( ( K  e.  HL  /\  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 )  ./\  ( P  .\/  Q ) )  =  ( ( R  .\/  P ) 
./\  ( U  .\/  S ) ) )
149140, 148oveq12d 6058 . . 3  |-  ( ( ( K  e.  HL  /\  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  .\/  U ) )  .\/  (
( U  .\/  S
)  ./\  ( P  .\/  Q ) ) )  =  ( ( ( Q  .\/  R ) 
./\  ( T  .\/  U ) )  .\/  (
( R  .\/  P
)  ./\  ( U  .\/  S ) ) ) )
150135, 149eqtr3d 2438 . 2  |-  ( ( ( K  e.  HL  /\  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  .\/  U ) ) 
.\/  ( U  .\/  S ) )  ./\  ( P  .\/  Q ) )  =  ( ( ( Q  .\/  R ) 
./\  ( T  .\/  U ) )  .\/  (
( R  .\/  P
)  ./\  ( U  .\/  S ) ) ) )
151129, 150breqtrd 4196 1  |-  ( ( ( K  e.  HL  /\  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:  dalawlem13  30365
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
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