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Theorem cdlemg11b 31501
Description: TODO: FIX COMMENT (Contributed by NM, 5-May-2013.)
Hypotheses
Ref Expression
cdlemg8.l  |-  .<_  =  ( le `  K )
cdlemg8.j  |-  .\/  =  ( join `  K )
cdlemg8.m  |-  ./\  =  ( meet `  K )
cdlemg8.a  |-  A  =  ( Atoms `  K )
cdlemg8.h  |-  H  =  ( LHyp `  K
)
cdlemg8.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg10.r  |-  R  =  ( ( trL `  K
) `  W )
Assertion
Ref Expression
cdlemg11b  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  ( G  e.  T  /\  P  =/=  Q  /\  -.  ( R `  G )  .<_  ( P 
.\/  Q ) ) )  ->  ( P  .\/  Q )  =/=  (
( G `  P
)  .\/  ( G `  Q ) ) )

Proof of Theorem cdlemg11b
StepHypRef Expression
1 simp33 996 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  ( G  e.  T  /\  P  =/=  Q  /\  -.  ( R `  G )  .<_  ( P 
.\/  Q ) ) )  ->  -.  ( R `  G )  .<_  ( P  .\/  Q
) )
2 simpl1 961 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( P  .\/  Q )  =  ( ( G `
 P )  .\/  ( G `  Q ) ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
3 simpl31 1039 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( P  .\/  Q )  =  ( ( G `
 P )  .\/  ( G `  Q ) ) )  ->  G  e.  T )
4 simpl2l 1011 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( P  .\/  Q )  =  ( ( G `
 P )  .\/  ( G `  Q ) ) )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
5 cdlemg8.l . . . . . . 7  |-  .<_  =  ( le `  K )
6 cdlemg8.j . . . . . . 7  |-  .\/  =  ( join `  K )
7 cdlemg8.m . . . . . . 7  |-  ./\  =  ( meet `  K )
8 cdlemg8.a . . . . . . 7  |-  A  =  ( Atoms `  K )
9 cdlemg8.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
10 cdlemg8.t . . . . . . 7  |-  T  =  ( ( LTrn `  K
) `  W )
11 cdlemg10.r . . . . . . 7  |-  R  =  ( ( trL `  K
) `  W )
125, 6, 7, 8, 9, 10, 11trlval2 31022 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( R `  G )  =  ( ( P  .\/  ( G `  P )
)  ./\  W )
)
132, 3, 4, 12syl3anc 1185 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( P  .\/  Q )  =  ( ( G `
 P )  .\/  ( G `  Q ) ) )  ->  ( R `  G )  =  ( ( P 
.\/  ( G `  P ) )  ./\  W ) )
14 eqid 2438 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
15 simpl1l 1009 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( P  .\/  Q )  =  ( ( G `
 P )  .\/  ( G `  Q ) ) )  ->  K  e.  HL )
16 hllat 30223 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
1715, 16syl 16 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( P  .\/  Q )  =  ( ( G `
 P )  .\/  ( G `  Q ) ) )  ->  K  e.  Lat )
18 simp2ll 1025 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  ( G  e.  T  /\  P  =/=  Q  /\  -.  ( R `  G )  .<_  ( P 
.\/  Q ) ) )  ->  P  e.  A )
1918adantr 453 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( P  .\/  Q )  =  ( ( G `
 P )  .\/  ( G `  Q ) ) )  ->  P  e.  A )
2014, 8atbase 30149 . . . . . . . . 9  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
2119, 20syl 16 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( P  .\/  Q )  =  ( ( G `
 P )  .\/  ( G `  Q ) ) )  ->  P  e.  ( Base `  K
) )
2214, 9, 10ltrncl 30984 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  P  e.  ( Base `  K ) )  ->  ( G `  P )  e.  (
Base `  K )
)
232, 3, 21, 22syl3anc 1185 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( P  .\/  Q )  =  ( ( G `
 P )  .\/  ( G `  Q ) ) )  ->  ( G `  P )  e.  ( Base `  K
) )
2414, 6latjcl 14481 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  ( G `  P )  e.  ( Base `  K
) )  ->  ( P  .\/  ( G `  P ) )  e.  ( Base `  K
) )
2517, 21, 23, 24syl3anc 1185 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( P  .\/  Q )  =  ( ( G `
 P )  .\/  ( G `  Q ) ) )  ->  ( P  .\/  ( G `  P ) )  e.  ( Base `  K
) )
26 simpl1r 1010 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( P  .\/  Q )  =  ( ( G `
 P )  .\/  ( G `  Q ) ) )  ->  W  e.  H )
2714, 9lhpbase 30857 . . . . . . . 8  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2826, 27syl 16 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( P  .\/  Q )  =  ( ( G `
 P )  .\/  ( G `  Q ) ) )  ->  W  e.  ( Base `  K
) )
2914, 7latmcl 14482 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  ( G `
 P ) )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  ( G `  P ) )  ./\  W )  e.  ( Base `  K ) )
3017, 25, 28, 29syl3anc 1185 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( P  .\/  Q )  =  ( ( G `
 P )  .\/  ( G `  Q ) ) )  ->  (
( P  .\/  ( G `  P )
)  ./\  W )  e.  ( Base `  K
) )
31 simpl2r 1012 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( P  .\/  Q )  =  ( ( G `
 P )  .\/  ( G `  Q ) ) )  ->  Q  e.  A )
3214, 8atbase 30149 . . . . . . . 8  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
3331, 32syl 16 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( P  .\/  Q )  =  ( ( G `
 P )  .\/  ( G `  Q ) ) )  ->  Q  e.  ( Base `  K
) )
3414, 6latjcl 14481 . . . . . . 7  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
3517, 21, 33, 34syl3anc 1185 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( P  .\/  Q )  =  ( ( G `
 P )  .\/  ( G `  Q ) ) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
3614, 5, 7latmle1 14507 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( P  .\/  ( G `
 P ) )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( P  .\/  ( G `  P ) )  ./\  W )  .<_  ( P  .\/  ( G `  P
) ) )
3717, 25, 28, 36syl3anc 1185 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( P  .\/  Q )  =  ( ( G `
 P )  .\/  ( G `  Q ) ) )  ->  (
( P  .\/  ( G `  P )
)  ./\  W )  .<_  ( P  .\/  ( G `  P )
) )
3814, 5, 6latlej1 14491 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  P  .<_  ( P  .\/  Q
) )
3917, 21, 33, 38syl3anc 1185 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( P  .\/  Q )  =  ( ( G `
 P )  .\/  ( G `  Q ) ) )  ->  P  .<_  ( P  .\/  Q
) )
4014, 9, 10ltrncl 30984 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  Q  e.  ( Base `  K ) )  ->  ( G `  Q )  e.  (
Base `  K )
)
412, 3, 33, 40syl3anc 1185 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( P  .\/  Q )  =  ( ( G `
 P )  .\/  ( G `  Q ) ) )  ->  ( G `  Q )  e.  ( Base `  K
) )
4214, 5, 6latlej1 14491 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( G `  P )  e.  ( Base `  K
)  /\  ( G `  Q )  e.  (
Base `  K )
)  ->  ( G `  P )  .<_  ( ( G `  P ) 
.\/  ( G `  Q ) ) )
4317, 23, 41, 42syl3anc 1185 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( P  .\/  Q )  =  ( ( G `
 P )  .\/  ( G `  Q ) ) )  ->  ( G `  P )  .<_  ( ( G `  P )  .\/  ( G `  Q )
) )
44 simpr 449 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( P  .\/  Q )  =  ( ( G `
 P )  .\/  ( G `  Q ) ) )  ->  ( P  .\/  Q )  =  ( ( G `  P )  .\/  ( G `  Q )
) )
4543, 44breqtrrd 4240 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( P  .\/  Q )  =  ( ( G `
 P )  .\/  ( G `  Q ) ) )  ->  ( G `  P )  .<_  ( P  .\/  Q
) )
4614, 5, 6latjle12 14493 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( P  e.  ( Base `  K )  /\  ( G `  P )  e.  ( Base `  K
)  /\  ( P  .\/  Q )  e.  (
Base `  K )
) )  ->  (
( P  .<_  ( P 
.\/  Q )  /\  ( G `  P ) 
.<_  ( P  .\/  Q
) )  <->  ( P  .\/  ( G `  P
) )  .<_  ( P 
.\/  Q ) ) )
4717, 21, 23, 35, 46syl13anc 1187 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( P  .\/  Q )  =  ( ( G `
 P )  .\/  ( G `  Q ) ) )  ->  (
( P  .<_  ( P 
.\/  Q )  /\  ( G `  P ) 
.<_  ( P  .\/  Q
) )  <->  ( P  .\/  ( G `  P
) )  .<_  ( P 
.\/  Q ) ) )
4839, 45, 47mpbi2and 889 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( P  .\/  Q )  =  ( ( G `
 P )  .\/  ( G `  Q ) ) )  ->  ( P  .\/  ( G `  P ) )  .<_  ( P  .\/  Q ) )
4914, 5, 17, 30, 25, 35, 37, 48lattrd 14489 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( P  .\/  Q )  =  ( ( G `
 P )  .\/  ( G `  Q ) ) )  ->  (
( P  .\/  ( G `  P )
)  ./\  W )  .<_  ( P  .\/  Q
) )
5013, 49eqbrtrd 4234 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  (
( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A
)  /\  ( G  e.  T  /\  P  =/= 
Q  /\  -.  ( R `  G )  .<_  ( P  .\/  Q
) ) )  /\  ( P  .\/  Q )  =  ( ( G `
 P )  .\/  ( G `  Q ) ) )  ->  ( R `  G )  .<_  ( P  .\/  Q
) )
5150ex 425 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  ( G  e.  T  /\  P  =/=  Q  /\  -.  ( R `  G )  .<_  ( P 
.\/  Q ) ) )  ->  ( ( P  .\/  Q )  =  ( ( G `  P )  .\/  ( G `  Q )
)  ->  ( R `  G )  .<_  ( P 
.\/  Q ) ) )
5251necon3bd 2640 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  ( G  e.  T  /\  P  =/=  Q  /\  -.  ( R `  G )  .<_  ( P 
.\/  Q ) ) )  ->  ( -.  ( R `  G ) 
.<_  ( P  .\/  Q
)  ->  ( P  .\/  Q )  =/=  (
( G `  P
)  .\/  ( G `  Q ) ) ) )
531, 52mpd 15 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( P  e.  A  /\  -.  P  .<_  W )  /\  Q  e.  A )  /\  ( G  e.  T  /\  P  =/=  Q  /\  -.  ( R `  G )  .<_  ( P 
.\/  Q ) ) )  ->  ( P  .\/  Q )  =/=  (
( G `  P
)  .\/  ( G `  Q ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   Basecbs 13471   lecple 13538   joincjn 14403   meetcmee 14404   Latclat 14476   Atomscatm 30123   HLchlt 30210   LHypclh 30843   LTrncltrn 30960   trLctrl 31017
This theorem is referenced by:  cdlemg12b  31503
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-undef 6545  df-riota 6551  df-map 7022  df-poset 14405  df-lub 14433  df-glb 14434  df-join 14435  df-meet 14436  df-lat 14477  df-ats 30127  df-atl 30158  df-cvlat 30182  df-hlat 30211  df-lhyp 30847  df-laut 30848  df-ldil 30963  df-ltrn 30964  df-trl 31018
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