Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cdlemg31c Structured version   Unicode version

Theorem cdlemg31c 36165
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 1108 . . . 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 1109 . . . 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 532 . . 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 1029 . . 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 1033 . . . 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 2714 . . 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 1028 . . . 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 1024 . . . 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 1034 . . . 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 35634 . . . 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 1233 . . 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 35649 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  .<_  W )
183, 8, 17syl2anc 661 . . 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 1023 . . 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 35497 . . 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 1251 . 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 1110 . . . . . 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 1112 . . . . . 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 1064 . . . . . 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 36163 . . . . . 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 1245 . . . . 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 465 . . . 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 1126 . . . . . . 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 1127 . . . . . . 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 999 . . . . . . . 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 2714 . . . . . . 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 1110 . . . . . . 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 1134 . . . . . . 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 998 . . . . . . 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 35497 . . . . . . 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 1250 . . . . . 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 1199 . . . . 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 2663 . . . 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 36164 . . . . 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 1245 . . . 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 465 . . 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 4459 . 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 659 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   class class class wbr 4437   ` cfv 5578  (class class class)co 6281   lecple 14581   joincjn 15447   meetcmee 15448   Atomscatm 34728   HLchlt 34815   LHypclh 35448   LTrncltrn 35565   trLctrl 35623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-map 7424  df-preset 15431  df-poset 15449  df-plt 15462  df-lub 15478  df-glb 15479  df-join 15480  df-meet 15481  df-p0 15543  df-p1 15544  df-lat 15550  df-clat 15612  df-oposet 34641  df-ol 34643  df-oml 34644  df-covers 34731  df-ats 34732  df-atl 34763  df-cvlat 34787  df-hlat 34816  df-psubsp 34967  df-pmap 34968  df-padd 35260  df-lhyp 35452  df-laut 35453  df-ldil 35568  df-ltrn 35569  df-trl 35624
This theorem is referenced by:  cdlemg31d  36166
  Copyright terms: Public domain W3C validator