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Theorem cdlemd2 29555
Description: Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 29-May-2012.)
Hypotheses
Ref Expression
cdlemd2.l  |-  .<_  =  ( le `  K )
cdlemd2.j  |-  .\/  =  ( join `  K )
cdlemd2.a  |-  A  =  ( Atoms `  K )
cdlemd2.h  |-  H  =  ( LHyp `  K
)
cdlemd2.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
cdlemd2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  R )  =  ( G `  R ) )

Proof of Theorem cdlemd2
StepHypRef Expression
1 simp3l 988 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  P )  =  ( G `  P ) )
2 simp11 990 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
3 simp12l 1073 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  F  e.  T )
4 simp11l 1071 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  K  e.  HL )
5 hllat 28720 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  Lat )
64, 5syl 17 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  K  e.  Lat )
7 simp21l 1077 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  P  e.  A )
8 simp13 992 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  R  e.  A )
9 eqid 2258 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
10 cdlemd2.j . . . . . . . . . . 11  |-  .\/  =  ( join `  K )
11 cdlemd2.a . . . . . . . . . . 11  |-  A  =  ( Atoms `  K )
129, 10, 11hlatjcl 28723 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  ( P  .\/  R
)  e.  ( Base `  K ) )
134, 7, 8, 12syl3anc 1187 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( P  .\/  R )  e.  ( Base `  K
) )
14 simp11r 1072 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  W  e.  H )
15 cdlemd2.h . . . . . . . . . . 11  |-  H  =  ( LHyp `  K
)
169, 15lhpbase 29354 . . . . . . . . . 10  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1714, 16syl 17 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  W  e.  ( Base `  K
) )
18 eqid 2258 . . . . . . . . . 10  |-  ( meet `  K )  =  (
meet `  K )
199, 18latmcl 14119 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( P  .\/  R )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  R ) (
meet `  K ) W )  e.  (
Base `  K )
)
206, 13, 17, 19syl3anc 1187 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  (
( P  .\/  R
) ( meet `  K
) W )  e.  ( Base `  K
) )
21 cdlemd2.l . . . . . . . . . 10  |-  .<_  =  ( le `  K )
229, 21, 18latmle2 14145 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( P  .\/  R )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  R ) (
meet `  K ) W )  .<_  W )
236, 13, 17, 22syl3anc 1187 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  (
( P  .\/  R
) ( meet `  K
) W )  .<_  W )
24 cdlemd2.t . . . . . . . . 9  |-  T  =  ( ( LTrn `  K
) `  W )
259, 21, 15, 24ltrnval1 29490 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( ( P 
.\/  R ) (
meet `  K ) W )  e.  (
Base `  K )  /\  ( ( P  .\/  R ) ( meet `  K
) W )  .<_  W ) )  -> 
( F `  (
( P  .\/  R
) ( meet `  K
) W ) )  =  ( ( P 
.\/  R ) (
meet `  K ) W ) )
262, 3, 20, 23, 25syl112anc 1191 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  ( ( P  .\/  R ) (
meet `  K ) W ) )  =  ( ( P  .\/  R ) ( meet `  K
) W ) )
27 simp12r 1074 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  G  e.  T )
289, 21, 15, 24ltrnval1 29490 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( ( ( P 
.\/  R ) (
meet `  K ) W )  e.  (
Base `  K )  /\  ( ( P  .\/  R ) ( meet `  K
) W )  .<_  W ) )  -> 
( G `  (
( P  .\/  R
) ( meet `  K
) W ) )  =  ( ( P 
.\/  R ) (
meet `  K ) W ) )
292, 27, 20, 23, 28syl112anc 1191 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( G `  ( ( P  .\/  R ) (
meet `  K ) W ) )  =  ( ( P  .\/  R ) ( meet `  K
) W ) )
3026, 29eqtr4d 2293 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  ( ( P  .\/  R ) (
meet `  K ) W ) )  =  ( G `  (
( P  .\/  R
) ( meet `  K
) W ) ) )
311, 30oveq12d 5810 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  (
( F `  P
)  .\/  ( F `  ( ( P  .\/  R ) ( meet `  K
) W ) ) )  =  ( ( G `  P ) 
.\/  ( G `  ( ( P  .\/  R ) ( meet `  K
) W ) ) ) )
329, 11atbase 28646 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
337, 32syl 17 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  P  e.  ( Base `  K
) )
349, 10, 15, 24ltrnj 29488 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  (
Base `  K )  /\  ( ( P  .\/  R ) ( meet `  K
) W )  e.  ( Base `  K
) ) )  -> 
( F `  ( P  .\/  ( ( P 
.\/  R ) (
meet `  K ) W ) ) )  =  ( ( F `
 P )  .\/  ( F `  ( ( P  .\/  R ) ( meet `  K
) W ) ) ) )
352, 3, 33, 20, 34syl112anc 1191 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  ( P  .\/  ( ( P  .\/  R ) ( meet `  K
) W ) ) )  =  ( ( F `  P ) 
.\/  ( F `  ( ( P  .\/  R ) ( meet `  K
) W ) ) ) )
369, 10, 15, 24ltrnj 29488 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( P  e.  (
Base `  K )  /\  ( ( P  .\/  R ) ( meet `  K
) W )  e.  ( Base `  K
) ) )  -> 
( G `  ( P  .\/  ( ( P 
.\/  R ) (
meet `  K ) W ) ) )  =  ( ( G `
 P )  .\/  ( G `  ( ( P  .\/  R ) ( meet `  K
) W ) ) ) )
372, 27, 33, 20, 36syl112anc 1191 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( G `  ( P  .\/  ( ( P  .\/  R ) ( meet `  K
) W ) ) )  =  ( ( G `  P ) 
.\/  ( G `  ( ( P  .\/  R ) ( meet `  K
) W ) ) ) )
3831, 35, 373eqtr4d 2300 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  ( P  .\/  ( ( P  .\/  R ) ( meet `  K
) W ) ) )  =  ( G `
 ( P  .\/  ( ( P  .\/  R ) ( meet `  K
) W ) ) ) )
39 simp3r 989 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  Q )  =  ( G `  Q ) )
40 simp22l 1079 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  Q  e.  A )
419, 10, 11hlatjcl 28723 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
424, 40, 8, 41syl3anc 1187 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( Q  .\/  R )  e.  ( Base `  K
) )
439, 18latmcl 14119 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( Q  .\/  R )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( Q  .\/  R ) (
meet `  K ) W )  e.  (
Base `  K )
)
446, 42, 17, 43syl3anc 1187 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  (
( Q  .\/  R
) ( meet `  K
) W )  e.  ( Base `  K
) )
459, 21, 18latmle2 14145 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( Q  .\/  R )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( Q  .\/  R ) (
meet `  K ) W )  .<_  W )
466, 42, 17, 45syl3anc 1187 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  (
( Q  .\/  R
) ( meet `  K
) W )  .<_  W )
479, 21, 15, 24ltrnval1 29490 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( ( Q 
.\/  R ) (
meet `  K ) W )  e.  (
Base `  K )  /\  ( ( Q  .\/  R ) ( meet `  K
) W )  .<_  W ) )  -> 
( F `  (
( Q  .\/  R
) ( meet `  K
) W ) )  =  ( ( Q 
.\/  R ) (
meet `  K ) W ) )
482, 3, 44, 46, 47syl112anc 1191 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  ( ( Q  .\/  R ) (
meet `  K ) W ) )  =  ( ( Q  .\/  R ) ( meet `  K
) W ) )
499, 21, 15, 24ltrnval1 29490 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( ( ( Q 
.\/  R ) (
meet `  K ) W )  e.  (
Base `  K )  /\  ( ( Q  .\/  R ) ( meet `  K
) W )  .<_  W ) )  -> 
( G `  (
( Q  .\/  R
) ( meet `  K
) W ) )  =  ( ( Q 
.\/  R ) (
meet `  K ) W ) )
502, 27, 44, 46, 49syl112anc 1191 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( G `  ( ( Q  .\/  R ) (
meet `  K ) W ) )  =  ( ( Q  .\/  R ) ( meet `  K
) W ) )
5148, 50eqtr4d 2293 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  ( ( Q  .\/  R ) (
meet `  K ) W ) )  =  ( G `  (
( Q  .\/  R
) ( meet `  K
) W ) ) )
5239, 51oveq12d 5810 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  (
( F `  Q
)  .\/  ( F `  ( ( Q  .\/  R ) ( meet `  K
) W ) ) )  =  ( ( G `  Q ) 
.\/  ( G `  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) )
539, 11atbase 28646 . . . . . . 7  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
5440, 53syl 17 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  Q  e.  ( Base `  K
) )
559, 10, 15, 24ltrnj 29488 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( Q  e.  (
Base `  K )  /\  ( ( Q  .\/  R ) ( meet `  K
) W )  e.  ( Base `  K
) ) )  -> 
( F `  ( Q  .\/  ( ( Q 
.\/  R ) (
meet `  K ) W ) ) )  =  ( ( F `
 Q )  .\/  ( F `  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) )
562, 3, 54, 44, 55syl112anc 1191 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) ) )  =  ( ( F `  Q ) 
.\/  ( F `  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) )
579, 10, 15, 24ltrnj 29488 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( Q  e.  (
Base `  K )  /\  ( ( Q  .\/  R ) ( meet `  K
) W )  e.  ( Base `  K
) ) )  -> 
( G `  ( Q  .\/  ( ( Q 
.\/  R ) (
meet `  K ) W ) ) )  =  ( ( G `
 Q )  .\/  ( G `  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) )
582, 27, 54, 44, 57syl112anc 1191 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( G `  ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) ) )  =  ( ( G `  Q ) 
.\/  ( G `  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) )
5952, 56, 583eqtr4d 2300 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) ) )  =  ( G `
 ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) )
6038, 59oveq12d 5810 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  (
( F `  ( P  .\/  ( ( P 
.\/  R ) (
meet `  K ) W ) ) ) ( meet `  K
) ( F `  ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) )  =  ( ( G `  ( P  .\/  ( ( P 
.\/  R ) (
meet `  K ) W ) ) ) ( meet `  K
) ( G `  ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) ) )
619, 10latjcl 14118 . . . . 5  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  (
( P  .\/  R
) ( meet `  K
) W )  e.  ( Base `  K
) )  ->  ( P  .\/  ( ( P 
.\/  R ) (
meet `  K ) W ) )  e.  ( Base `  K
) )
626, 33, 20, 61syl3anc 1187 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( P  .\/  ( ( P 
.\/  R ) (
meet `  K ) W ) )  e.  ( Base `  K
) )
639, 10latjcl 14118 . . . . 5  |-  ( ( K  e.  Lat  /\  Q  e.  ( Base `  K )  /\  (
( Q  .\/  R
) ( meet `  K
) W )  e.  ( Base `  K
) )  ->  ( Q  .\/  ( ( Q 
.\/  R ) (
meet `  K ) W ) )  e.  ( Base `  K
) )
646, 54, 44, 63syl3anc 1187 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( Q  .\/  ( ( Q 
.\/  R ) (
meet `  K ) W ) )  e.  ( Base `  K
) )
659, 18, 15, 24ltrnm 29487 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( ( P  .\/  ( ( P  .\/  R ) ( meet `  K
) W ) )  e.  ( Base `  K
)  /\  ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) )  e.  ( Base `  K
) ) )  -> 
( F `  (
( P  .\/  (
( P  .\/  R
) ( meet `  K
) W ) ) ( meet `  K
) ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) )  =  ( ( F `  ( P  .\/  ( ( P 
.\/  R ) (
meet `  K ) W ) ) ) ( meet `  K
) ( F `  ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) ) )
662, 3, 62, 64, 65syl112anc 1191 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  ( ( P  .\/  ( ( P 
.\/  R ) (
meet `  K ) W ) ) (
meet `  K )
( Q  .\/  (
( Q  .\/  R
) ( meet `  K
) W ) ) ) )  =  ( ( F `  ( P  .\/  ( ( P 
.\/  R ) (
meet `  K ) W ) ) ) ( meet `  K
) ( F `  ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) ) )
679, 18, 15, 24ltrnm 29487 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( ( P  .\/  ( ( P  .\/  R ) ( meet `  K
) W ) )  e.  ( Base `  K
)  /\  ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) )  e.  ( Base `  K
) ) )  -> 
( G `  (
( P  .\/  (
( P  .\/  R
) ( meet `  K
) W ) ) ( meet `  K
) ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) )  =  ( ( G `  ( P  .\/  ( ( P 
.\/  R ) (
meet `  K ) W ) ) ) ( meet `  K
) ( G `  ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) ) )
682, 27, 62, 64, 67syl112anc 1191 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( G `  ( ( P  .\/  ( ( P 
.\/  R ) (
meet `  K ) W ) ) (
meet `  K )
( Q  .\/  (
( Q  .\/  R
) ( meet `  K
) W ) ) ) )  =  ( ( G `  ( P  .\/  ( ( P 
.\/  R ) (
meet `  K ) W ) ) ) ( meet `  K
) ( G `  ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) ) )
6960, 66, 683eqtr4d 2300 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  ( ( P  .\/  ( ( P 
.\/  R ) (
meet `  K ) W ) ) (
meet `  K )
( Q  .\/  (
( Q  .\/  R
) ( meet `  K
) W ) ) ) )  =  ( G `  ( ( P  .\/  ( ( P  .\/  R ) ( meet `  K
) W ) ) ( meet `  K
) ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) ) )
70 simp21 993 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
71 simp22 994 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
72 simp23l 1081 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  P  =/=  Q )
73 simp23r 1082 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  -.  R  .<_  ( P  .\/  Q ) )
748, 72, 733jca 1137 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( R  e.  A  /\  P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )
7521, 10, 18, 11, 15cdlemd1 29554 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R  e.  A  /\  P  =/= 
Q  /\  -.  R  .<_  ( P  .\/  Q
) ) ) )  ->  R  =  ( ( P  .\/  (
( P  .\/  R
) ( meet `  K
) W ) ) ( meet `  K
) ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) )
762, 70, 71, 74, 75syl13anc 1189 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  R  =  ( ( P 
.\/  ( ( P 
.\/  R ) (
meet `  K ) W ) ) (
meet `  K )
( Q  .\/  (
( Q  .\/  R
) ( meet `  K
) W ) ) ) )
7776fveq2d 5462 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  R )  =  ( F `  ( ( P  .\/  ( ( P  .\/  R ) ( meet `  K
) W ) ) ( meet `  K
) ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) ) )
7876fveq2d 5462 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( G `  R )  =  ( G `  ( ( P  .\/  ( ( P  .\/  R ) ( meet `  K
) W ) ) ( meet `  K
) ( Q  .\/  ( ( Q  .\/  R ) ( meet `  K
) W ) ) ) ) )
7969, 77, 783eqtr4d 2300 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( F  e.  T  /\  G  e.  T )  /\  R  e.  A
)  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( P  =/=  Q  /\  -.  R  .<_  ( P  .\/  Q ) ) )  /\  ( ( F `  P )  =  ( G `  P )  /\  ( F `  Q )  =  ( G `  Q ) ) )  ->  ( F `  R )  =  ( G `  R ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2421   class class class wbr 3997   ` cfv 4673  (class class class)co 5792   Basecbs 13110   lecple 13177   joincjn 14040   meetcmee 14041   Latclat 14113   Atomscatm 28620   HLchlt 28707   LHypclh 29340   LTrncltrn 29457
This theorem is referenced by:  cdlemd4  29557  cdlemd5  29558
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-map 6742  df-poset 14042  df-plt 14054  df-lub 14070  df-glb 14071  df-join 14072  df-meet 14073  df-p0 14107  df-p1 14108  df-lat 14114  df-clat 14176  df-oposet 28533  df-ol 28535  df-oml 28536  df-covers 28623  df-ats 28624  df-atl 28655  df-cvlat 28679  df-hlat 28708  df-psubsp 28859  df-pmap 28860  df-padd 29152  df-lhyp 29344  df-laut 29345  df-ldil 29460  df-ltrn 29461
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