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Theorem cdlemd2 29656
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 985 . . . . . 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 987 . . . . . . . 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 1070 . . . . . . . 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 1068 . . . . . . . . . 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 28821 . . . . . . . . . 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 1074 . . . . . . . . . 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 989 . . . . . . . . . 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 2285 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
10 cdlemd2.j . . . . . . . . . . 11  |-  .\/  =  ( join `  K )
11 cdlemd2.a . . . . . . . . . . 11  |-  A  =  ( Atoms `  K )
129, 10, 11hlatjcl 28824 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  P  e.  A  /\  R  e.  A )  ->  ( P  .\/  R
)  e.  ( Base `  K ) )
134, 7, 8, 12syl3anc 1184 . . . . . . . . 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 1069 . . . . . . . . . 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 29455 . . . . . . . . . 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 2285 . . . . . . . . . 10  |-  ( meet `  K )  =  (
meet `  K )
199, 18latmcl 14152 . . . . . . . . 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 1184 . . . . . . . 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 14178 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( P  .\/  R )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  R ) (
meet `  K ) W )  .<_  W )
236, 13, 17, 22syl3anc 1184 . . . . . . . 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 29591 . . . . . . . 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 1188 . . . . . . 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 1071 . . . . . . . 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 29591 . . . . . . . 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 1188 . . . . . . 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 2320 . . . . . 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 5838 . . . . 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 28747 . . . . . . 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 29589 . . . . . 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 1188 . . . . 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 29589 . . . . . 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 1188 . . . . 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 2327 . . . 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 986 . . . . . 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 1076 . . . . . . . . . 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 28824 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  Q  e.  A  /\  R  e.  A )  ->  ( Q  .\/  R
)  e.  ( Base `  K ) )
424, 40, 8, 41syl3anc 1184 . . . . . . . . 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 14152 . . . . . . . . 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 1184 . . . . . . . 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 14178 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( Q  .\/  R )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( Q  .\/  R ) (
meet `  K ) W )  .<_  W )
466, 42, 17, 45syl3anc 1184 . . . . . . . 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 29591 . . . . . . . 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 1188 . . . . . . 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 29591 . . . . . . . 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 1188 . . . . . . 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 2320 . . . . . 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 5838 . . . . 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 28747 . . . . . . 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 29589 . . . . . 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 1188 . . . . 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 29589 . . . . . 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 1188 . . . . 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 2327 . . . 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 5838 . . 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 14151 . . . . 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 1184 . . . 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 14151 . . . . 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 1184 . . . 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 29588 . . . 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 1188 . . 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 29588 . . . 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 1188 . . 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 2327 . 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 990 . . . 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 991 . . . 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 1078 . . . . 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 1079 . . . . 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 1134 . . . 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 29655 . . . 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 1186 . . 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 5490 . 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 5490 . 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 2327 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 936    = wceq 1624    e. wcel 1685    =/= wne 2448   class class class wbr 4025   ` cfv 5222  (class class class)co 5820   Basecbs 13143   lecple 13210   joincjn 14073   meetcmee 14074   Latclat 14146   Atomscatm 28721   HLchlt 28808   LHypclh 29441   LTrncltrn 29558
This theorem is referenced by:  cdlemd4  29658  cdlemd5  29659
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-undef 6292  df-riota 6300  df-map 6770  df-poset 14075  df-plt 14087  df-lub 14103  df-glb 14104  df-join 14105  df-meet 14106  df-p0 14140  df-p1 14141  df-lat 14147  df-clat 14209  df-oposet 28634  df-ol 28636  df-oml 28637  df-covers 28724  df-ats 28725  df-atl 28756  df-cvlat 28780  df-hlat 28809  df-psubsp 28960  df-pmap 28961  df-padd 29253  df-lhyp 29445  df-laut 29446  df-ldil 29561  df-ltrn 29562
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