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Theorem cdlemg31c 30888
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 1066 . . . 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 1067 . . . 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 518 . . 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 987 . . 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 991 . . . 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 2529 . . 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 986 . . . 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 982 . . . 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 992 . . . 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 30358 . . . 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 1186 . . 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 30373 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  .<_  W )
183, 8, 17syl2anc 642 . . 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 981 . . 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 30222 . . 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 1204 . 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 1068 . . . . . 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 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 ) )  ->  Q  e.  A )
25 simp2ll 1022 . . . . . 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 30886 . . . . . 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 1198 . . . . 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 451 . . . 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 1084 . . . . . . 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 1085 . . . . . . 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 957 . . . . . . . 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 2529 . . . . . . 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 1068 . . . . . . 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 1092 . . . . . . 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 956 . . . . . . 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 30222 . . . . . . 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 1203 . . . . . 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 1153 . . . . 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 2507 . . . 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 14 . . 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 30887 . . . . 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 1198 . . . 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 451 . . 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 4045 . 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 640 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   lecple 13215   joincjn 14078   meetcmee 14079   Atomscatm 29453   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   trLctrl 30347
This theorem is referenced by:  cdlemg31d  30889
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-iin 3908  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-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348
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