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Theorem cdlemb 29233
Description: Given two atoms not less than or equal to an element covered by 1, there is a third. Lemma B in [Crawley] p. 112. (Contributed by NM, 8-May-2012.)
Hypotheses
Ref Expression
cdlemb.b  |-  B  =  ( Base `  K
)
cdlemb.l  |-  .<_  =  ( le `  K )
cdlemb.j  |-  .\/  =  ( join `  K )
cdlemb.u  |-  .1.  =  ( 1. `  K )
cdlemb.c  |-  C  =  (  <o  `  K )
cdlemb.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
cdlemb  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  E. r  e.  A  ( -.  r  .<_  X  /\  -.  r  .<_  ( P  .\/  Q ) ) )
Distinct variable groups:    A, r    B, r    C, r    .\/ , r    K, r    .<_ , r    P, r    Q, r    .1. , r    X, r

Proof of Theorem cdlemb
StepHypRef Expression
1 simp11 990 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  K  e.  HL )
2 simp12 991 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  P  e.  A )
3 simp13 992 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  Q  e.  A )
4 simp2l 986 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  X  e.  B )
5 simp2r 987 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  P  =/=  Q )
6 simp31 996 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  X C  .1.  )
7 simp32 997 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  -.  P  .<_  X )
8 cdlemb.b . . . . 5  |-  B  =  ( Base `  K
)
9 cdlemb.l . . . . 5  |-  .<_  =  ( le `  K )
10 cdlemb.j . . . . 5  |-  .\/  =  ( join `  K )
11 eqid 2258 . . . . 5  |-  ( meet `  K )  =  (
meet `  K )
12 cdlemb.u . . . . 5  |-  .1.  =  ( 1. `  K )
13 cdlemb.c . . . . 5  |-  C  =  (  <o  `  K )
14 cdlemb.a . . . . 5  |-  A  =  ( Atoms `  K )
158, 9, 10, 11, 12, 13, 141cvrat 28915 . . . 4  |-  ( ( K  e.  HL  /\  ( P  e.  A  /\  Q  e.  A  /\  X  e.  B
)  /\  ( P  =/=  Q  /\  X C  .1.  /\  -.  P  .<_  X ) )  -> 
( ( P  .\/  Q ) ( meet `  K
) X )  e.  A )
161, 2, 3, 4, 5, 6, 7, 15syl133anc 1210 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  -> 
( ( P  .\/  Q ) ( meet `  K
) X )  e.  A )
17 hllat 28803 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  Lat )
181, 17syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  K  e.  Lat )
198, 14atbase 28729 . . . . . . 7  |-  ( P  e.  A  ->  P  e.  B )
202, 19syl 17 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  P  e.  B )
218, 14atbase 28729 . . . . . . 7  |-  ( Q  e.  A  ->  Q  e.  B )
223, 21syl 17 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  Q  e.  B )
238, 10latjcl 14119 . . . . . 6  |-  ( ( K  e.  Lat  /\  P  e.  B  /\  Q  e.  B )  ->  ( P  .\/  Q
)  e.  B )
2418, 20, 22, 23syl3anc 1187 . . . . 5  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  -> 
( P  .\/  Q
)  e.  B )
258, 9, 11latmle2 14146 . . . . 5  |-  ( ( K  e.  Lat  /\  ( P  .\/  Q )  e.  B  /\  X  e.  B )  ->  (
( P  .\/  Q
) ( meet `  K
) X )  .<_  X )
2618, 24, 4, 25syl3anc 1187 . . . 4  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  -> 
( ( P  .\/  Q ) ( meet `  K
) X )  .<_  X )
27 eqid 2258 . . . . 5  |-  ( lt
`  K )  =  ( lt `  K
)
288, 9, 27, 12, 13, 141cvratlt 28913 . . . 4  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  Q ) ( meet `  K
) X )  e.  A  /\  X  e.  B )  /\  ( X C  .1.  /\  (
( P  .\/  Q
) ( meet `  K
) X )  .<_  X ) )  -> 
( ( P  .\/  Q ) ( meet `  K
) X ) ( lt `  K ) X )
291, 16, 4, 6, 26, 28syl32anc 1195 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  -> 
( ( P  .\/  Q ) ( meet `  K
) X ) ( lt `  K ) X )
308, 27, 142atlt 28878 . . 3  |-  ( ( ( K  e.  HL  /\  ( ( P  .\/  Q ) ( meet `  K
) X )  e.  A  /\  X  e.  B )  /\  (
( P  .\/  Q
) ( meet `  K
) X ) ( lt `  K ) X )  ->  E. u  e.  A  ( u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) )
311, 16, 4, 29, 30syl31anc 1190 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  E. u  e.  A  ( u  =/=  (
( P  .\/  Q
) ( meet `  K
) X )  /\  u ( lt `  K ) X ) )
32 simpl11 1035 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q )  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  (
u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) ) )  ->  K  e.  HL )
33 simpl12 1036 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q )  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  (
u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) ) )  ->  P  e.  A )
34 simprl 735 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q )  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  (
u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) ) )  ->  u  e.  A )
35 simpl32 1042 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q )  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  (
u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) ) )  ->  -.  P  .<_  X )
36 simprrr 744 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q )  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  (
u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) ) )  ->  u
( lt `  K
) X )
37 simpl2l 1013 . . . . . . . . . . 11  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q )  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  (
u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) ) )  ->  X  e.  B )
389, 27pltle 14058 . . . . . . . . . . 11  |-  ( ( K  e.  HL  /\  u  e.  A  /\  X  e.  B )  ->  ( u ( lt
`  K ) X  ->  u  .<_  X ) )
3932, 34, 37, 38syl3anc 1187 . . . . . . . . . 10  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q )  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  (
u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) ) )  ->  (
u ( lt `  K ) X  ->  u  .<_  X ) )
4036, 39mpd 16 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q )  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  (
u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) ) )  ->  u  .<_  X )
41 breq1 4000 . . . . . . . . 9  |-  ( P  =  u  ->  ( P  .<_  X  <->  u  .<_  X ) )
4240, 41syl5ibrcom 215 . . . . . . . 8  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q )  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  (
u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) ) )  ->  ( P  =  u  ->  P 
.<_  X ) )
4342necon3bd 2458 . . . . . . 7  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q )  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  (
u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) ) )  ->  ( -.  P  .<_  X  ->  P  =/=  u ) )
4435, 43mpd 16 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q )  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  (
u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) ) )  ->  P  =/=  u )
459, 10, 14hlsupr 28825 . . . . . 6  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  u  e.  A )  /\  P  =/=  u
)  ->  E. r  e.  A  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) )
4632, 33, 34, 44, 45syl31anc 1190 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q )  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  (
u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) ) )  ->  E. r  e.  A  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u ) ) )
47 eqid 2258 . . . . . . . . . 10  |-  ( ( P  .\/  Q ) ( meet `  K
) X )  =  ( ( P  .\/  Q ) ( meet `  K
) X )
488, 9, 10, 12, 13, 14, 27, 11, 47cdlemblem 29232 . . . . . . . . 9  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q )  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  (
u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) )  /\  ( r  e.  A  /\  (
r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u
) ) ) )  ->  ( -.  r  .<_  X  /\  -.  r  .<_  ( P  .\/  Q
) ) )
49483exp 1155 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  -> 
( ( u  e.  A  /\  ( u  =/=  ( ( P 
.\/  Q ) (
meet `  K ) X )  /\  u
( lt `  K
) X ) )  ->  ( ( r  e.  A  /\  (
r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u
) ) )  -> 
( -.  r  .<_  X  /\  -.  r  .<_  ( P  .\/  Q ) ) ) ) )
5049exp4a 592 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  -> 
( ( u  e.  A  /\  ( u  =/=  ( ( P 
.\/  Q ) (
meet `  K ) X )  /\  u
( lt `  K
) X ) )  ->  ( r  e.  A  ->  ( (
r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P  .\/  u
) )  ->  ( -.  r  .<_  X  /\  -.  r  .<_  ( P 
.\/  Q ) ) ) ) ) )
5150imp 420 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q )  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  (
u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) ) )  ->  (
r  e.  A  -> 
( ( r  =/= 
P  /\  r  =/=  u  /\  r  .<_  ( P 
.\/  u ) )  ->  ( -.  r  .<_  X  /\  -.  r  .<_  ( P  .\/  Q
) ) ) ) )
5251reximdvai 2628 . . . . 5  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q )  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  (
u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) ) )  ->  ( E. r  e.  A  ( r  =/=  P  /\  r  =/=  u  /\  r  .<_  ( P 
.\/  u ) )  ->  E. r  e.  A  ( -.  r  .<_  X  /\  -.  r  .<_  ( P  .\/  Q ) ) ) )
5346, 52mpd 16 . . . 4  |-  ( ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q )  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  /\  ( u  e.  A  /\  (
u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X ) ) )  ->  E. r  e.  A  ( -.  r  .<_  X  /\  -.  r  .<_  ( P  .\/  Q ) ) )
5453exp32 591 . . 3  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  -> 
( u  e.  A  ->  ( ( u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X )  ->  E. r  e.  A  ( -.  r  .<_  X  /\  -.  r  .<_  ( P  .\/  Q ) ) ) ) )
5554rexlimdv 2641 . 2  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  -> 
( E. u  e.  A  ( u  =/=  ( ( P  .\/  Q ) ( meet `  K
) X )  /\  u ( lt `  K ) X )  ->  E. r  e.  A  ( -.  r  .<_  X  /\  -.  r  .<_  ( P  .\/  Q ) ) ) )
5631, 55mpd 16 1  |-  ( ( ( K  e.  HL  /\  P  e.  A  /\  Q  e.  A )  /\  ( X  e.  B  /\  P  =/=  Q
)  /\  ( X C  .1.  /\  -.  P  .<_  X  /\  -.  Q  .<_  X ) )  ->  E. r  e.  A  ( -.  r  .<_  X  /\  -.  r  .<_  ( P  .\/  Q ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2421   E.wrex 2519   class class class wbr 3997   ` cfv 4673  (class class class)co 5792   Basecbs 13111   lecple 13178   ltcplt 14038   joincjn 14041   meetcmee 14042   1.cp1 14107   Latclat 14114    <o ccvr 28702   Atomscatm 28703   HLchlt 28790
This theorem is referenced by:  cdlemb2  29480
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-iota 6225  df-undef 6264  df-riota 6272  df-poset 14043  df-plt 14055  df-lub 14071  df-glb 14072  df-join 14073  df-meet 14074  df-p0 14108  df-p1 14109  df-lat 14115  df-clat 14177  df-oposet 28616  df-ol 28618  df-oml 28619  df-covers 28706  df-ats 28707  df-atl 28738  df-cvlat 28762  df-hlat 28791
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