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Theorem doca2N 31609
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 29844 . . . . . . . . . . . . 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 2404 . . . . . . . . . . . . 13  |-  ( Base `  K )  =  (
Base `  K )
4 doca2.h . . . . . . . . . . . . 13  |-  H  =  ( LHyp `  K
)
5 doca2.i . . . . . . . . . . . . 13  |-  I  =  ( ( DIsoA `  K
) `  W )
63, 4, 5diadmclN 31520 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  X  e.  ( Base `  K
) )
73, 4lhpbase 30480 . . . . . . . . . . . . 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 2404 . . . . . . . . . . . . 13  |-  ( join `  K )  =  (
join `  K )
10 eqid 2404 . . . . . . . . . . . . 13  |-  ( meet `  K )  =  (
meet `  K )
11 eqid 2404 . . . . . . . . . . . . 13  |-  ( oc
`  K )  =  ( oc `  K
)
123, 9, 10, 11oldmm1 29700 . . . . . . . . . . . 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 6055 . . . . . . . . . 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 2409 . . . . . . . . 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 5691 . . . . . . . 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 29846 . . . . . . . . . . 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 14435 . . . . . . . . . 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 29701 . . . . . . . . 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 2436 . . . . . . 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 6055 . . . . . 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 29845 . . . . . . . . . 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 29677 . . . . . . . . 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 14479 . . . . . . . 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 14458 . . . . . . . . 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 6056 . . . . . . 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 2436 . . . . . 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 2436 . . . . 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 6055 . . . 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 29840 . . . . . 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 2404 . . . . . . 7  |-  ( le
`  K )  =  ( le `  K
)
403, 39, 10latmle2 14461 . . . . . 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 29744 . . . . 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 31521 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  X
( le `  K
) W )
453, 39, 10latleeqm1 14463 . . . . . 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 2441 . . 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 5691 . 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 29677 . . . . . . 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 14434 . . . . . 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 14435 . . . . 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 14461 . . . . 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 31519 . . . . 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 2404 . . . 4  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
62 doca2.n . . . 4  |-  ._|_  =  ( ( ocA `  K
) `  W )
639, 10, 11, 4, 61, 5, 62diaocN 31608 . . 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 31608 . . 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 5691 . 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 2441 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 1649    e. wcel 1721   class class class wbr 4172   dom cdm 4837   ` cfv 5413  (class class class)co 6040   Basecbs 13424   lecple 13491   occoc 13492   joincjn 14356   meetcmee 14357   Latclat 14429   OPcops 29655   OLcol 29657   OMLcoml 29658   HLchlt 29833   LHypclh 30466   LTrncltrn 30583   DIsoAcdia 31511   ocAcocaN 31602
This theorem is referenced by:  doca3N  31610
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-undef 6502  df-riota 6508  df-map 6979  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-oposet 29659  df-cmtN 29660  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641  df-disoa 31512  df-docaN 31603
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