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Theorem doca2N 31924
Description: Double orthocomplement of partial isomorphism A. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
doca2.h  |-  H  =  ( LHyp `  K
)
doca2.i  |-  I  =  ( ( DIsoA `  K
) `  W )
doca2.n  |-  ._|_  =  ( ( ocA `  K
) `  W )
Assertion
Ref Expression
doca2N  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (  ._|_  `  (  ._|_  `  (
I `  X )
) )  =  ( I `  X ) )

Proof of Theorem doca2N
StepHypRef Expression
1 hlol 30159 . . . . . . . . . . . . 13  |-  ( K  e.  HL  ->  K  e.  OL )
21ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  K  e.  OL )
3 eqid 2436 . . . . . . . . . . . . 13  |-  ( Base `  K )  =  (
Base `  K )
4 doca2.h . . . . . . . . . . . . 13  |-  H  =  ( LHyp `  K
)
5 doca2.i . . . . . . . . . . . . 13  |-  I  =  ( ( DIsoA `  K
) `  W )
63, 4, 5diadmclN 31835 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  X  e.  ( Base `  K
) )
73, 4lhpbase 30795 . . . . . . . . . . . . 13  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
87ad2antlr 708 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  W  e.  ( Base `  K
) )
9 eqid 2436 . . . . . . . . . . . . 13  |-  ( join `  K )  =  (
join `  K )
10 eqid 2436 . . . . . . . . . . . . 13  |-  ( meet `  K )  =  (
meet `  K )
11 eqid 2436 . . . . . . . . . . . . 13  |-  ( oc
`  K )  =  ( oc `  K
)
123, 9, 10, 11oldmm1 30015 . . . . . . . . . . . 12  |-  ( ( K  e.  OL  /\  X  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  ->  (
( oc `  K
) `  ( X
( meet `  K ) W ) )  =  ( ( ( oc
`  K ) `  X ) ( join `  K ) ( ( oc `  K ) `
 W ) ) )
132, 6, 8, 12syl3anc 1184 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
( oc `  K
) `  ( X
( meet `  K ) W ) )  =  ( ( ( oc
`  K ) `  X ) ( join `  K ) ( ( oc `  K ) `
 W ) ) )
1413oveq1d 6096 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
( ( oc `  K ) `  ( X ( meet `  K
) W ) ) ( meet `  K
) W )  =  ( ( ( ( oc `  K ) `
 X ) (
join `  K )
( ( oc `  K ) `  W
) ) ( meet `  K ) W ) )
1514eqcomd 2441 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
( ( ( oc
`  K ) `  X ) ( join `  K ) ( ( oc `  K ) `
 W ) ) ( meet `  K
) W )  =  ( ( ( oc
`  K ) `  ( X ( meet `  K
) W ) ) ( meet `  K
) W ) )
1615fveq2d 5732 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
( oc `  K
) `  ( (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  W ) ) (
meet `  K ) W ) )  =  ( ( oc `  K ) `  (
( ( oc `  K ) `  ( X ( meet `  K
) W ) ) ( meet `  K
) W ) ) )
17 hllat 30161 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  Lat )
1817ad2antrr 707 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  K  e.  Lat )
193, 10latmcl 14480 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  X  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  ->  ( X ( meet `  K
) W )  e.  ( Base `  K
) )
2018, 6, 8, 19syl3anc 1184 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  ( X ( meet `  K
) W )  e.  ( Base `  K
) )
213, 9, 10, 11oldmm2 30016 . . . . . . . . 9  |-  ( ( K  e.  OL  /\  ( X ( meet `  K
) W )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( ( oc `  K ) `  ( ( ( oc
`  K ) `  ( X ( meet `  K
) W ) ) ( meet `  K
) W ) )  =  ( ( X ( meet `  K
) W ) (
join `  K )
( ( oc `  K ) `  W
) ) )
222, 20, 8, 21syl3anc 1184 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
( oc `  K
) `  ( (
( oc `  K
) `  ( X
( meet `  K ) W ) ) (
meet `  K ) W ) )  =  ( ( X (
meet `  K ) W ) ( join `  K ) ( ( oc `  K ) `
 W ) ) )
2316, 22eqtrd 2468 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
( oc `  K
) `  ( (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  W ) ) (
meet `  K ) W ) )  =  ( ( X (
meet `  K ) W ) ( join `  K ) ( ( oc `  K ) `
 W ) ) )
2423oveq1d 6096 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
( ( oc `  K ) `  (
( ( ( oc
`  K ) `  X ) ( join `  K ) ( ( oc `  K ) `
 W ) ) ( meet `  K
) W ) ) ( join `  K
) ( ( oc
`  K ) `  W ) )  =  ( ( ( X ( meet `  K
) W ) (
join `  K )
( ( oc `  K ) `  W
) ) ( join `  K ) ( ( oc `  K ) `
 W ) ) )
25 hlop 30160 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  OP )
2625ad2antrr 707 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  K  e.  OP )
273, 11opoccl 29992 . . . . . . . . 9  |-  ( ( K  e.  OP  /\  W  e.  ( Base `  K ) )  -> 
( ( oc `  K ) `  W
)  e.  ( Base `  K ) )
2826, 8, 27syl2anc 643 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
( oc `  K
) `  W )  e.  ( Base `  K
) )
293, 9latjass 14524 . . . . . . . 8  |-  ( ( K  e.  Lat  /\  ( ( X (
meet `  K ) W )  e.  (
Base `  K )  /\  ( ( oc `  K ) `  W
)  e.  ( Base `  K )  /\  (
( oc `  K
) `  W )  e.  ( Base `  K
) ) )  -> 
( ( ( X ( meet `  K
) W ) (
join `  K )
( ( oc `  K ) `  W
) ) ( join `  K ) ( ( oc `  K ) `
 W ) )  =  ( ( X ( meet `  K
) W ) (
join `  K )
( ( ( oc
`  K ) `  W ) ( join `  K ) ( ( oc `  K ) `
 W ) ) ) )
3018, 20, 28, 28, 29syl13anc 1186 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
( ( X (
meet `  K ) W ) ( join `  K ) ( ( oc `  K ) `
 W ) ) ( join `  K
) ( ( oc
`  K ) `  W ) )  =  ( ( X (
meet `  K ) W ) ( join `  K ) ( ( ( oc `  K
) `  W )
( join `  K )
( ( oc `  K ) `  W
) ) ) )
313, 9latjidm 14503 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( ( oc `  K ) `  W
)  e.  ( Base `  K ) )  -> 
( ( ( oc
`  K ) `  W ) ( join `  K ) ( ( oc `  K ) `
 W ) )  =  ( ( oc
`  K ) `  W ) )
3218, 28, 31syl2anc 643 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
( ( oc `  K ) `  W
) ( join `  K
) ( ( oc
`  K ) `  W ) )  =  ( ( oc `  K ) `  W
) )
3332oveq2d 6097 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
( X ( meet `  K ) W ) ( join `  K
) ( ( ( oc `  K ) `
 W ) (
join `  K )
( ( oc `  K ) `  W
) ) )  =  ( ( X (
meet `  K ) W ) ( join `  K ) ( ( oc `  K ) `
 W ) ) )
3430, 33eqtrd 2468 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
( ( X (
meet `  K ) W ) ( join `  K ) ( ( oc `  K ) `
 W ) ) ( join `  K
) ( ( oc
`  K ) `  W ) )  =  ( ( X (
meet `  K ) W ) ( join `  K ) ( ( oc `  K ) `
 W ) ) )
3524, 34eqtrd 2468 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
( ( oc `  K ) `  (
( ( ( oc
`  K ) `  X ) ( join `  K ) ( ( oc `  K ) `
 W ) ) ( meet `  K
) W ) ) ( join `  K
) ( ( oc
`  K ) `  W ) )  =  ( ( X (
meet `  K ) W ) ( join `  K ) ( ( oc `  K ) `
 W ) ) )
3635oveq1d 6096 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
( ( ( oc
`  K ) `  ( ( ( ( oc `  K ) `
 X ) (
join `  K )
( ( oc `  K ) `  W
) ) ( meet `  K ) W ) ) ( join `  K
) ( ( oc
`  K ) `  W ) ) (
meet `  K ) W )  =  ( ( ( X (
meet `  K ) W ) ( join `  K ) ( ( oc `  K ) `
 W ) ) ( meet `  K
) W ) )
37 hloml 30155 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OML )
3837ad2antrr 707 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  K  e.  OML )
39 eqid 2436 . . . . . . 7  |-  ( le
`  K )  =  ( le `  K
)
403, 39, 10latmle2 14506 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  ->  ( X ( meet `  K
) W ) ( le `  K ) W )
4118, 6, 8, 40syl3anc 1184 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  ( X ( meet `  K
) W ) ( le `  K ) W )
423, 39, 9, 10, 11omlspjN 30059 . . . . 5  |-  ( ( K  e.  OML  /\  ( ( X (
meet `  K ) W )  e.  (
Base `  K )  /\  W  e.  ( Base `  K ) )  /\  ( X (
meet `  K ) W ) ( le
`  K ) W )  ->  ( (
( X ( meet `  K ) W ) ( join `  K
) ( ( oc
`  K ) `  W ) ) (
meet `  K ) W )  =  ( X ( meet `  K
) W ) )
4338, 20, 8, 41, 42syl121anc 1189 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
( ( X (
meet `  K ) W ) ( join `  K ) ( ( oc `  K ) `
 W ) ) ( meet `  K
) W )  =  ( X ( meet `  K ) W ) )
4439, 4, 5diadmleN 31836 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  X
( le `  K
) W )
453, 39, 10latleeqm1 14508 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  ->  ( X ( le `  K ) W  <->  ( X
( meet `  K ) W )  =  X ) )
4618, 6, 8, 45syl3anc 1184 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  ( X ( le `  K ) W  <->  ( X
( meet `  K ) W )  =  X ) )
4744, 46mpbid 202 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  ( X ( meet `  K
) W )  =  X )
4836, 43, 473eqtrrd 2473 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  X  =  ( ( ( ( oc `  K
) `  ( (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  W ) ) (
meet `  K ) W ) ) (
join `  K )
( ( oc `  K ) `  W
) ) ( meet `  K ) W ) )
4948fveq2d 5732 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
I `  X )  =  ( I `  ( ( ( ( oc `  K ) `
 ( ( ( ( oc `  K
) `  X )
( join `  K )
( ( oc `  K ) `  W
) ) ( meet `  K ) W ) ) ( join `  K
) ( ( oc
`  K ) `  W ) ) (
meet `  K ) W ) ) )
503, 11opoccl 29992 . . . . . . 7  |-  ( ( K  e.  OP  /\  X  e.  ( Base `  K ) )  -> 
( ( oc `  K ) `  X
)  e.  ( Base `  K ) )
5126, 6, 50syl2anc 643 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
( oc `  K
) `  X )  e.  ( Base `  K
) )
523, 9latjcl 14479 . . . . . 6  |-  ( ( K  e.  Lat  /\  ( ( oc `  K ) `  X
)  e.  ( Base `  K )  /\  (
( oc `  K
) `  W )  e.  ( Base `  K
) )  ->  (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  W ) )  e.  ( Base `  K
) )
5318, 51, 28, 52syl3anc 1184 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  W ) )  e.  ( Base `  K
) )
543, 10latmcl 14480 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( ( oc
`  K ) `  X ) ( join `  K ) ( ( oc `  K ) `
 W ) )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  W ) ) (
meet `  K ) W )  e.  (
Base `  K )
)
5518, 53, 8, 54syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
( ( ( oc
`  K ) `  X ) ( join `  K ) ( ( oc `  K ) `
 W ) ) ( meet `  K
) W )  e.  ( Base `  K
) )
563, 39, 10latmle2 14506 . . . . 5  |-  ( ( K  e.  Lat  /\  ( ( ( oc
`  K ) `  X ) ( join `  K ) ( ( oc `  K ) `
 W ) )  e.  ( Base `  K
)  /\  W  e.  ( Base `  K )
)  ->  ( (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  W ) ) (
meet `  K ) W ) ( le
`  K ) W )
5718, 53, 8, 56syl3anc 1184 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
( ( ( oc
`  K ) `  X ) ( join `  K ) ( ( oc `  K ) `
 W ) ) ( meet `  K
) W ) ( le `  K ) W )
583, 39, 4, 5diaeldm 31834 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( ( ( oc `  K
) `  X )
( join `  K )
( ( oc `  K ) `  W
) ) ( meet `  K ) W )  e.  dom  I  <->  ( (
( ( ( oc
`  K ) `  X ) ( join `  K ) ( ( oc `  K ) `
 W ) ) ( meet `  K
) W )  e.  ( Base `  K
)  /\  ( (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  W ) ) (
meet `  K ) W ) ( le
`  K ) W ) ) )
5958adantr 452 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
( ( ( ( oc `  K ) `
 X ) (
join `  K )
( ( oc `  K ) `  W
) ) ( meet `  K ) W )  e.  dom  I  <->  ( (
( ( ( oc
`  K ) `  X ) ( join `  K ) ( ( oc `  K ) `
 W ) ) ( meet `  K
) W )  e.  ( Base `  K
)  /\  ( (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  W ) ) (
meet `  K ) W ) ( le
`  K ) W ) ) )
6055, 57, 59mpbir2and 889 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
( ( ( oc
`  K ) `  X ) ( join `  K ) ( ( oc `  K ) `
 W ) ) ( meet `  K
) W )  e. 
dom  I )
61 eqid 2436 . . . 4  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
62 doca2.n . . . 4  |-  ._|_  =  ( ( ocA `  K
) `  W )
639, 10, 11, 4, 61, 5, 62diaocN 31923 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( ( ( oc `  K
) `  X )
( join `  K )
( ( oc `  K ) `  W
) ) ( meet `  K ) W )  e.  dom  I )  ->  ( I `  ( ( ( ( oc `  K ) `
 ( ( ( ( oc `  K
) `  X )
( join `  K )
( ( oc `  K ) `  W
) ) ( meet `  K ) W ) ) ( join `  K
) ( ( oc
`  K ) `  W ) ) (
meet `  K ) W ) )  =  (  ._|_  `  ( I `
 ( ( ( ( oc `  K
) `  X )
( join `  K )
( ( oc `  K ) `  W
) ) ( meet `  K ) W ) ) ) )
6460, 63syldan 457 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
I `  ( (
( ( oc `  K ) `  (
( ( ( oc
`  K ) `  X ) ( join `  K ) ( ( oc `  K ) `
 W ) ) ( meet `  K
) W ) ) ( join `  K
) ( ( oc
`  K ) `  W ) ) (
meet `  K ) W ) )  =  (  ._|_  `  ( I `
 ( ( ( ( oc `  K
) `  X )
( join `  K )
( ( oc `  K ) `  W
) ) ( meet `  K ) W ) ) ) )
659, 10, 11, 4, 61, 5, 62diaocN 31923 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
I `  ( (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  W ) ) (
meet `  K ) W ) )  =  (  ._|_  `  ( I `
 X ) ) )
6665fveq2d 5732 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (  ._|_  `  ( I `  ( ( ( ( oc `  K ) `
 X ) (
join `  K )
( ( oc `  K ) `  W
) ) ( meet `  K ) W ) ) )  =  ( 
._|_  `  (  ._|_  `  (
I `  X )
) ) )
6749, 64, 663eqtrrd 2473 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (  ._|_  `  (  ._|_  `  (
I `  X )
) )  =  ( I `  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4212   dom cdm 4878   ` cfv 5454  (class class class)co 6081   Basecbs 13469   lecple 13536   occoc 13537   joincjn 14401   meetcmee 14402   Latclat 14474   OPcops 29970   OLcol 29972   OMLcoml 29973   HLchlt 30148   LHypclh 30781   LTrncltrn 30898   DIsoAcdia 31826   ocAcocaN 31917
This theorem is referenced by:  doca3N  31925
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-undef 6543  df-riota 6549  df-map 7020  df-poset 14403  df-plt 14415  df-lub 14431  df-glb 14432  df-join 14433  df-meet 14434  df-p0 14468  df-p1 14469  df-lat 14475  df-clat 14537  df-oposet 29974  df-cmtN 29975  df-ol 29976  df-oml 29977  df-covers 30064  df-ats 30065  df-atl 30096  df-cvlat 30120  df-hlat 30149  df-llines 30295  df-lplanes 30296  df-lvols 30297  df-lines 30298  df-psubsp 30300  df-pmap 30301  df-padd 30593  df-lhyp 30785  df-laut 30786  df-ldil 30901  df-ltrn 30902  df-trl 30956  df-disoa 31827  df-docaN 31918
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