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Theorem cdleme22d 35540
Description: Part of proof of Lemma E in [Crawley] p. 113, 3rd paragraph, 9th line on p. 115. (Contributed by NM, 4-Dec-2012.)
Hypotheses
Ref Expression
cdleme22.l  |-  .<_  =  ( le `  K )
cdleme22.j  |-  .\/  =  ( join `  K )
cdleme22.m  |-  ./\  =  ( meet `  K )
cdleme22.a  |-  A  =  ( Atoms `  K )
cdleme22.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
cdleme22d  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  V  =  ( ( S  .\/  T )  ./\  W ) )

Proof of Theorem cdleme22d
StepHypRef Expression
1 simp3r 1025 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  S  .<_  ( T  .\/  V ) )
2 simp1l 1020 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  K  e.  HL )
3 simp22l 1115 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  T  e.  A )
4 simp23l 1117 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  V  e.  A )
5 cdleme22.l . . . . . . . 8  |-  .<_  =  ( le `  K )
6 cdleme22.j . . . . . . . 8  |-  .\/  =  ( join `  K )
7 cdleme22.a . . . . . . . 8  |-  A  =  ( Atoms `  K )
85, 6, 7hlatlej1 34572 . . . . . . 7  |-  ( ( K  e.  HL  /\  T  e.  A  /\  V  e.  A )  ->  T  .<_  ( T  .\/  V ) )
92, 3, 4, 8syl3anc 1228 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  T  .<_  ( T  .\/  V ) )
10 hllat 34561 . . . . . . . 8  |-  ( K  e.  HL  ->  K  e.  Lat )
112, 10syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  K  e.  Lat )
12 simp21l 1113 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  S  e.  A )
13 eqid 2467 . . . . . . . . 9  |-  ( Base `  K )  =  (
Base `  K )
1413, 7atbase 34487 . . . . . . . 8  |-  ( S  e.  A  ->  S  e.  ( Base `  K
) )
1512, 14syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  S  e.  ( Base `  K ) )
1613, 7atbase 34487 . . . . . . . 8  |-  ( T  e.  A  ->  T  e.  ( Base `  K
) )
173, 16syl 16 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  T  e.  ( Base `  K ) )
1813, 6, 7hlatjcl 34564 . . . . . . . 8  |-  ( ( K  e.  HL  /\  T  e.  A  /\  V  e.  A )  ->  ( T  .\/  V
)  e.  ( Base `  K ) )
192, 3, 4, 18syl3anc 1228 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( T  .\/  V
)  e.  ( Base `  K ) )
2013, 5, 6latjle12 15566 . . . . . . 7  |-  ( ( K  e.  Lat  /\  ( S  e.  ( Base `  K )  /\  T  e.  ( Base `  K )  /\  ( T  .\/  V )  e.  ( Base `  K
) ) )  -> 
( ( S  .<_  ( T  .\/  V )  /\  T  .<_  ( T 
.\/  V ) )  <-> 
( S  .\/  T
)  .<_  ( T  .\/  V ) ) )
2111, 15, 17, 19, 20syl13anc 1230 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( ( S  .<_  ( T  .\/  V )  /\  T  .<_  ( T 
.\/  V ) )  <-> 
( S  .\/  T
)  .<_  ( T  .\/  V ) ) )
221, 9, 21mpbi2and 919 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( S  .\/  T
)  .<_  ( T  .\/  V ) )
2313, 6, 7hlatjcl 34564 . . . . . . 7  |-  ( ( K  e.  HL  /\  S  e.  A  /\  T  e.  A )  ->  ( S  .\/  T
)  e.  ( Base `  K ) )
242, 12, 3, 23syl3anc 1228 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( S  .\/  T
)  e.  ( Base `  K ) )
25 simp1r 1021 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  W  e.  H )
26 cdleme22.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
2713, 26lhpbase 35195 . . . . . . 7  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
2825, 27syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  W  e.  ( Base `  K ) )
29 cdleme22.m . . . . . . 7  |-  ./\  =  ( meet `  K )
3013, 5, 29latmlem1 15585 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( ( S  .\/  T )  e.  ( Base `  K )  /\  ( T  .\/  V )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
) )  ->  (
( S  .\/  T
)  .<_  ( T  .\/  V )  ->  ( ( S  .\/  T )  ./\  W )  .<_  ( ( T  .\/  V )  ./\  W ) ) )
3111, 24, 19, 28, 30syl13anc 1230 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( ( S  .\/  T )  .<_  ( T  .\/  V )  ->  (
( S  .\/  T
)  ./\  W )  .<_  ( ( T  .\/  V )  ./\  W )
) )
3222, 31mpd 15 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( ( S  .\/  T )  ./\  W )  .<_  ( ( T  .\/  V )  ./\  W )
)
33 simp1 996 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
34 simp22 1030 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( T  e.  A  /\  -.  T  .<_  W ) )
35 eqid 2467 . . . . . . . 8  |-  ( 0.
`  K )  =  ( 0. `  K
)
365, 29, 35, 7, 26lhpmat 35227 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( T  e.  A  /\  -.  T  .<_  W ) )  -> 
( T  ./\  W
)  =  ( 0.
`  K ) )
3733, 34, 36syl2anc 661 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( T  ./\  W
)  =  ( 0.
`  K ) )
3837oveq1d 6310 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( ( T  ./\  W )  .\/  V )  =  ( ( 0.
`  K )  .\/  V ) )
39 simp23r 1118 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  V  .<_  W )
4013, 5, 6, 29, 7atmod4i1 35063 . . . . . 6  |-  ( ( K  e.  HL  /\  ( V  e.  A  /\  T  e.  ( Base `  K )  /\  W  e.  ( Base `  K ) )  /\  V  .<_  W )  -> 
( ( T  ./\  W )  .\/  V )  =  ( ( T 
.\/  V )  ./\  W ) )
412, 4, 17, 28, 39, 40syl131anc 1241 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( ( T  ./\  W )  .\/  V )  =  ( ( T 
.\/  V )  ./\  W ) )
42 hlol 34559 . . . . . . 7  |-  ( K  e.  HL  ->  K  e.  OL )
432, 42syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  K  e.  OL )
4413, 7atbase 34487 . . . . . . 7  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
454, 44syl 16 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  V  e.  ( Base `  K ) )
4613, 6, 35olj02 34424 . . . . . 6  |-  ( ( K  e.  OL  /\  V  e.  ( Base `  K ) )  -> 
( ( 0. `  K )  .\/  V
)  =  V )
4743, 45, 46syl2anc 661 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( ( 0. `  K )  .\/  V
)  =  V )
4838, 41, 473eqtr3d 2516 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( ( T  .\/  V )  ./\  W )  =  V )
4932, 48breqtrd 4477 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( ( S  .\/  T )  ./\  W )  .<_  V )
50 hlatl 34558 . . . . 5  |-  ( K  e.  HL  ->  K  e.  AtLat )
512, 50syl 16 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  K  e.  AtLat )
52 simp21r 1114 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  -.  S  .<_  W )
53 simp3l 1024 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  S  =/=  T )
545, 6, 29, 7, 26lhpat 35240 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  S  =/=  T ) )  ->  ( ( S 
.\/  T )  ./\  W )  e.  A )
552, 25, 12, 52, 3, 53, 54syl222anc 1244 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( ( S  .\/  T )  ./\  W )  e.  A )
565, 7atcmp 34509 . . . 4  |-  ( ( K  e.  AtLat  /\  (
( S  .\/  T
)  ./\  W )  e.  A  /\  V  e.  A )  ->  (
( ( S  .\/  T )  ./\  W )  .<_  V  <->  ( ( S 
.\/  T )  ./\  W )  =  V ) )
5751, 55, 4, 56syl3anc 1228 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( ( ( S 
.\/  T )  ./\  W )  .<_  V  <->  ( ( S  .\/  T )  ./\  W )  =  V ) )
5849, 57mpbid 210 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  -> 
( ( S  .\/  T )  ./\  W )  =  V )
5958eqcomd 2475 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( S  e.  A  /\  -.  S  .<_  W )  /\  ( T  e.  A  /\  -.  T  .<_  W )  /\  ( V  e.  A  /\  V  .<_  W ) )  /\  ( S  =/=  T  /\  S  .<_  ( T  .\/  V
) ) )  ->  V  =  ( ( S  .\/  T )  ./\  W ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   Basecbs 14507   lecple 14579   joincjn 15448   meetcmee 15449   0.cp0 15541   Latclat 15549   OLcol 34372   Atomscatm 34461   AtLatcal 34462   HLchlt 34548   LHypclh 35181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-poset 15450  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 34374  df-ol 34376  df-oml 34377  df-covers 34464  df-ats 34465  df-atl 34496  df-cvlat 34520  df-hlat 34549  df-psubsp 34700  df-pmap 34701  df-padd 34993  df-lhyp 35185
This theorem is referenced by:  cdleme22g  35545
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