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Theorem cdlemg11b 31124
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 995 . 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 960 . . . . . 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 1038 . . . . . 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 1010 . . . . . 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 30645 . . . . . 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 1184 . . . . 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 2404 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
15 simpl1l 1008 . . . . . . 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 29846 . . . . . . 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 1024 . . . . . . . . . 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 452 . . . . . . . . 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 29772 . . . . . . . . 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 30607 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  P  e.  ( Base `  K ) )  ->  ( G `  P )  e.  (
Base `  K )
)
232, 3, 21, 22syl3anc 1184 . . . . . . . 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 14434 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  ( G `  P )  e.  ( Base `  K
) )  ->  ( P  .\/  ( G `  P ) )  e.  ( Base `  K
) )
2517, 21, 23, 24syl3anc 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  .\/  ( G `  P ) )  e.  ( Base `  K
) )
26 simpl1r 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 ) ) )  ->  W  e.  H )
2714, 9lhpbase 30480 . . . . . . . 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 14435 . . . . . . 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 1184 . . . . . 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 1011 . . . . . . . 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 29772 . . . . . . . 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 14434 . . . . . . 7  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  ( P  .\/  Q )  e.  ( Base `  K
) )
3517, 21, 33, 34syl3anc 1184 . . . . . 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 14460 . . . . . . 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 1184 . . . . . 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 14444 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  P  e.  ( Base `  K )  /\  Q  e.  ( Base `  K
) )  ->  P  .<_  ( P  .\/  Q
) )
3917, 21, 33, 38syl3anc 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
) )
4014, 9, 10ltrncl 30607 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  G  e.  T  /\  Q  e.  ( Base `  K ) )  ->  ( G `  Q )  e.  (
Base `  K )
)
412, 3, 33, 40syl3anc 1184 . . . . . . . . 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 14444 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( G `  P )  e.  ( Base `  K
)  /\  ( G `  Q )  e.  (
Base `  K )
)  ->  ( G `  P )  .<_  ( ( G `  P ) 
.\/  ( G `  Q ) ) )
4317, 23, 41, 42syl3anc 1184 . . . . . . . 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 448 . . . . . . . 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 4198 . . . . . . 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 14446 . . . . . . . 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 1186 . . . . . . 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 888 . . . . . 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 14442 . . . . 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 4192 . . . 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 424 . . 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 2604 . 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   joincjn 14356   meetcmee 14357   Latclat 14429   Atomscatm 29746   HLchlt 29833   LHypclh 30466   LTrncltrn 30583   trLctrl 30640
This theorem is referenced by:  cdlemg12b  31126
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-map 6979  df-poset 14358  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-lat 14430  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641
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