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Theorem cdlemg31c 30814
Description: Show that when  N is an atom, it is not under  W. TODO: Is there a shorter direct proof? Todo: should we eliminate  ( F `  P )  =/=  P here? (Contributed by NM, 29-May-2013.)
Hypotheses
Ref Expression
cdlemg12.l  |-  .<_  =  ( le `  K )
cdlemg12.j  |-  .\/  =  ( join `  K )
cdlemg12.m  |-  ./\  =  ( meet `  K )
cdlemg12.a  |-  A  =  ( Atoms `  K )
cdlemg12.h  |-  H  =  ( LHyp `  K
)
cdlemg12.t  |-  T  =  ( ( LTrn `  K
) `  W )
cdlemg12b.r  |-  R  =  ( ( trL `  K
) `  W )
cdlemg31.n  |-  N  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  F ) ) )
Assertion
Ref Expression
cdlemg31c  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  ->  -.  N  .<_  W )

Proof of Theorem cdlemg31c
StepHypRef Expression
1 simp11l 1068 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  ->  K  e.  HL )
2 simp11r 1069 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  ->  W  e.  H )
31, 2jca 519 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
4 simp13 989 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
5 simp31 993 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  -> 
v  =/=  ( R `
 F ) )
65necomd 2634 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  -> 
( R `  F
)  =/=  v )
7 simp12 988 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  -> 
( P  e.  A  /\  -.  P  .<_  W ) )
8 simp2r 984 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  ->  F  e.  T )
9 simp32 994 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  -> 
( F `  P
)  =/=  P )
10 cdlemg12.l . . . . 5  |-  .<_  =  ( le `  K )
11 cdlemg12.a . . . . 5  |-  A  =  ( Atoms `  K )
12 cdlemg12.h . . . . 5  |-  H  =  ( LHyp `  K
)
13 cdlemg12.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
14 cdlemg12b.r . . . . 5  |-  R  =  ( ( trL `  K
) `  W )
1510, 11, 12, 13, 14trlat 30284 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( F  e.  T  /\  ( F `  P )  =/=  P ) )  ->  ( R `  F )  e.  A
)
163, 7, 8, 9, 15syl112anc 1188 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  -> 
( R `  F
)  e.  A )
1710, 12, 13, 14trlle 30299 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  .<_  W )
183, 8, 17syl2anc 643 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  -> 
( R `  F
)  .<_  W )
19 simp2l 983 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  -> 
( v  e.  A  /\  v  .<_  W ) )
20 cdlemg12.j . . . 4  |-  .\/  =  ( join `  K )
2110, 20, 11, 12lhp2atnle 30148 . . 3  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  ( R `  F )  =/=  v
)  /\  ( ( R `  F )  e.  A  /\  ( R `  F )  .<_  W )  /\  (
v  e.  A  /\  v  .<_  W ) )  ->  -.  v  .<_  ( Q  .\/  ( R `
 F ) ) )
223, 4, 6, 16, 18, 19, 21syl321anc 1206 . 2  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  ->  -.  v  .<_  ( Q 
.\/  ( R `  F ) ) )
23 simp12l 1070 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  ->  P  e.  A )
24 simp13l 1072 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  ->  Q  e.  A )
25 simp2ll 1024 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  -> 
v  e.  A )
26 cdlemg12.m . . . . . . 7  |-  ./\  =  ( meet `  K )
27 cdlemg31.n . . . . . . 7  |-  N  =  ( ( P  .\/  v )  ./\  ( Q  .\/  ( R `  F ) ) )
2810, 20, 26, 11, 12, 13, 14, 27cdlemg31a 30812 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  (
v  e.  A  /\  F  e.  T )
)  ->  N  .<_  ( P  .\/  v ) )
291, 2, 23, 24, 25, 8, 28syl222anc 1200 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  ->  N  .<_  ( P  .\/  v ) )
3029adantr 452 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  /\  N  .<_  W )  ->  N  .<_  ( P  .\/  v ) )
31 simp111 1086 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  /\  N  .<_  W  /\  N  =/=  v )  ->  ( K  e.  HL  /\  W  e.  H ) )
32 simp112 1087 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  /\  N  .<_  W  /\  N  =/=  v )  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
33 simp3 959 . . . . . . . 8  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  /\  N  .<_  W  /\  N  =/=  v )  ->  N  =/=  v )
3433necomd 2634 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  /\  N  .<_  W  /\  N  =/=  v )  ->  v  =/=  N )
35 simp12l 1070 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  /\  N  .<_  W  /\  N  =/=  v )  ->  (
v  e.  A  /\  v  .<_  W ) )
36 simp133 1094 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  /\  N  .<_  W  /\  N  =/=  v )  ->  N  e.  A )
37 simp2 958 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  /\  N  .<_  W  /\  N  =/=  v )  ->  N  .<_  W )
3810, 20, 11, 12lhp2atnle 30148 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  v  =/=  N
)  /\  ( v  e.  A  /\  v  .<_  W )  /\  ( N  e.  A  /\  N  .<_  W ) )  ->  -.  N  .<_  ( P  .\/  v ) )
3931, 32, 34, 35, 36, 37, 38syl312anc 1205 . . . . . 6  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  /\  N  .<_  W  /\  N  =/=  v )  ->  -.  N  .<_  ( P  .\/  v ) )
40393expia 1155 . . . . 5  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  /\  N  .<_  W )  -> 
( N  =/=  v  ->  -.  N  .<_  ( P 
.\/  v ) ) )
4140necon4ad 2612 . . . 4  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  /\  N  .<_  W )  -> 
( N  .<_  ( P 
.\/  v )  ->  N  =  v )
)
4230, 41mpd 15 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  /\  N  .<_  W )  ->  N  =  v )
4310, 20, 26, 11, 12, 13, 14, 27cdlemg31b 30813 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  Q  e.  A )  /\  (
v  e.  A  /\  F  e.  T )
)  ->  N  .<_  ( Q  .\/  ( R `
 F ) ) )
441, 2, 23, 24, 25, 8, 43syl222anc 1200 . . . 4  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  ->  N  .<_  ( Q  .\/  ( R `  F ) ) )
4544adantr 452 . . 3  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  /\  N  .<_  W )  ->  N  .<_  ( Q  .\/  ( R `  F ) ) )
4642, 45eqbrtrrd 4176 . 2  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  /\  N  .<_  W )  -> 
v  .<_  ( Q  .\/  ( R `  F ) ) )
4722, 46mtand 641 1  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( P  e.  A  /\  -.  P  .<_  W )  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  /\  ( ( v  e.  A  /\  v  .<_  W )  /\  F  e.  T )  /\  (
v  =/=  ( R `
 F )  /\  ( F `  P )  =/=  P  /\  N  e.  A ) )  ->  -.  N  .<_  W )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2551   class class class wbr 4154   ` cfv 5395  (class class class)co 6021   lecple 13464   joincjn 14329   meetcmee 14330   Atomscatm 29379   HLchlt 29466   LHypclh 30099   LTrncltrn 30216   trLctrl 30273
This theorem is referenced by:  cdlemg31d  30815
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-undef 6480  df-riota 6486  df-map 6957  df-poset 14331  df-plt 14343  df-lub 14359  df-glb 14360  df-join 14361  df-meet 14362  df-p0 14396  df-p1 14397  df-lat 14403  df-clat 14465  df-oposet 29292  df-ol 29294  df-oml 29295  df-covers 29382  df-ats 29383  df-atl 29414  df-cvlat 29438  df-hlat 29467  df-psubsp 29618  df-pmap 29619  df-padd 29911  df-lhyp 30103  df-laut 30104  df-ldil 30219  df-ltrn 30220  df-trl 30274
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