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Theorem cdlemg31c 34236
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 1116 . . . 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 1117 . . . 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 534 . . 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 1037 . . 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 1041 . . . 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 2691 . . 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 1036 . . . 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 1032 . . . 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 1042 . . . 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 33705 . . . 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 1268 . . 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 33720 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T
)  ->  ( R `  F )  .<_  W )
183, 8, 17syl2anc 665 . . 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 1031 . . 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 33568 . . 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 1286 . 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 1118 . . . . . 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 1120 . . . . . 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 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 ) )  -> 
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 34234 . . . . . 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 1280 . . . . 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 466 . . . 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 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 )  ->  ( K  e.  HL  /\  W  e.  H ) )
32 simp112 1135 . . . . . . 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 1007 . . . . . . . 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 2691 . . . . . . 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 1118 . . . . . . 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 1142 . . . . . . 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 1006 . . . . . . 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 33568 . . . . . . 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 1285 . . . . . 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 1207 . . . . 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 2640 . . . 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 34235 . . . . 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 1280 . . . 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 466 . . 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 4446 . 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 663 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 370    /\ w3a 982    = wceq 1437    e. wcel 1872    =/= wne 2614   class class class wbr 4423   ` cfv 5601  (class class class)co 6306   lecple 15197   joincjn 16189   meetcmee 16190   Atomscatm 32799   HLchlt 32886   LHypclh 33519   LTrncltrn 33636   trLctrl 33694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-iin 4302  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6268  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-1st 6808  df-2nd 6809  df-map 7486  df-preset 16173  df-poset 16191  df-plt 16204  df-lub 16220  df-glb 16221  df-join 16222  df-meet 16223  df-p0 16285  df-p1 16286  df-lat 16292  df-clat 16354  df-oposet 32712  df-ol 32714  df-oml 32715  df-covers 32802  df-ats 32803  df-atl 32834  df-cvlat 32858  df-hlat 32887  df-psubsp 33038  df-pmap 33039  df-padd 33331  df-lhyp 33523  df-laut 33524  df-ldil 33639  df-ltrn 33640  df-trl 33695
This theorem is referenced by:  cdlemg31d  34237
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