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Theorem cdlemd2 30835
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 30000 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  Lat )
64, 5syl 16 . . . . . . . . 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 2435 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
10 cdlemd2.j . . . . . . . . . . 11  |-  .\/  =  ( join `  K )
11 cdlemd2.a . . . . . . . . . . 11  |-  A  =  ( Atoms `  K )
129, 10, 11hlatjcl 30003 . . . . . . . . . 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 30634 . . . . . . . . . 10  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
1714, 16syl 16 . . . . . . . . 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 2435 . . . . . . . . . 10  |-  ( meet `  K )  =  (
meet `  K )
199, 18latmcl 14468 . . . . . . . . 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 14494 . . . . . . . . 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 30770 . . . . . . . 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 30770 . . . . . . . 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 2470 . . . . . 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 6090 . . . . 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 29926 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
337, 32syl 16 . . . . . 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 30768 . . . . . 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 30768 . . . . . 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 2477 . . . 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 30003 . . . . . . . . . 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 14468 . . . . . . . . 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 14494 . . . . . . . . 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 30770 . . . . . . . 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 30770 . . . . . . . 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 2470 . . . . . 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 6090 . . . . 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 29926 . . . . . . 7  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
5440, 53syl 16 . . . . . 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 30768 . . . . . 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 30768 . . . . . 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 2477 . . . 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 6090 . . 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 14467 . . . . 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 14467 . . . . 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 30767 . . . 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 30767 . . . 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 2477 . 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 30834 . . . 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 5723 . 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 5723 . 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 2477 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 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   class class class wbr 4204   ` cfv 5445  (class class class)co 6072   Basecbs 13457   lecple 13524   joincjn 14389   meetcmee 14390   Latclat 14462   Atomscatm 29900   HLchlt 29987   LHypclh 30620   LTrncltrn 30737
This theorem is referenced by:  cdlemd4  30837  cdlemd5  30838
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-undef 6534  df-riota 6540  df-map 7011  df-poset 14391  df-plt 14403  df-lub 14419  df-glb 14420  df-join 14421  df-meet 14422  df-p0 14456  df-p1 14457  df-lat 14463  df-clat 14525  df-oposet 29813  df-ol 29815  df-oml 29816  df-covers 29903  df-ats 29904  df-atl 29935  df-cvlat 29959  df-hlat 29988  df-psubsp 30139  df-pmap 30140  df-padd 30432  df-lhyp 30624  df-laut 30625  df-ldil 30740  df-ltrn 30741
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