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Theorem cdlemg11b 30831
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 993 . 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 958 . . . . . 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 1036 . . . . . 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 1008 . . . . . 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 30352 . . . . . 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 1182 . . . . 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 2283 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
15 simpl1l 1006 . . . . . . 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 29553 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  Lat )
1715, 16syl 15 . . . . . 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 1022 . . . . . . . . . 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 451 . . . . . . . . 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 29479 . . . . . . . . 9  |-  ( P  e.  A  ->  P  e.  ( Base `  K
) )
2119, 20syl 15 . . . . . . . 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 30314 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  P  e.  ( Base `  K ) )  ->  ( G `  P )  e.  (
Base `  K )
)
232, 3, 21, 22syl3anc 1182 . . . . . . . 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 14156 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  ( G `  P )  e.  ( Base `  K
) )  ->  ( P  .\/  ( G `  P ) )  e.  ( Base `  K
) )
2517, 21, 23, 24syl3anc 1182 . . . . . . 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 1007 . . . . . . . 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 30187 . . . . . . . 8  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2826, 27syl 15 . . . . . . 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 14157 . . . . . . 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 1182 . . . . . 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 1009 . . . . . . . 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 29479 . . . . . . . 8  |-  ( Q  e.  A  ->  Q  e.  ( Base `  K
) )
3331, 32syl 15 . . . . . . 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 14156 . . . . . . 7  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
3517, 21, 33, 34syl3anc 1182 . . . . . 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 14182 . . . . . . 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 1182 . . . . . 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 14166 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  P  .<_  ( P  .\/  Q
) )
3917, 21, 33, 38syl3anc 1182 . . . . . . 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 30314 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  Q  e.  ( Base `  K ) )  ->  ( G `  Q )  e.  (
Base `  K )
)
412, 3, 33, 40syl3anc 1182 . . . . . . . . 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 14166 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( G `  P )  e.  ( Base `  K
)  /\  ( G `  Q )  e.  (
Base `  K )
)  ->  ( G `  P )  .<_  ( ( G `  P ) 
.\/  ( G `  Q ) ) )
4317, 23, 41, 42syl3anc 1182 . . . . . . . 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 447 . . . . . . . 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 4049 . . . . . . 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 14168 . . . . . . . 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 1184 . . . . . . 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 887 . . . . . 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 14164 . . . . 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 4043 . . . 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 423 . . 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 2483 . 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 14 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   joincjn 14078   meetcmee 14079   Latclat 14151   Atomscatm 29453   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   trLctrl 30347
This theorem is referenced by:  cdlemg12b  30833
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-lat 14152  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348
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