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Theorem doca2N 31316
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 29551 . . . . . . . . . . . . 13  |-  ( K  e.  HL  ->  K  e.  OL )
21ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  K  e.  OL )
3 eqid 2283 . . . . . . . . . . . . 13  |-  ( Base `  K )  =  (
Base `  K )
4 doca2.h . . . . . . . . . . . . 13  |-  H  =  ( LHyp `  K
)
5 doca2.i . . . . . . . . . . . . 13  |-  I  =  ( ( DIsoA `  K
) `  W )
63, 4, 5diadmclN 31227 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  X  e.  ( Base `  K
) )
73, 4lhpbase 30187 . . . . . . . . . . . . 13  |-  ( W  e.  H  ->  W  e.  ( Base `  K
) )
87ad2antlr 707 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  W  e.  ( Base `  K
) )
9 eqid 2283 . . . . . . . . . . . . 13  |-  ( join `  K )  =  (
join `  K )
10 eqid 2283 . . . . . . . . . . . . 13  |-  ( meet `  K )  =  (
meet `  K )
11 eqid 2283 . . . . . . . . . . . . 13  |-  ( oc
`  K )  =  ( oc `  K
)
123, 9, 10, 11oldmm1 29407 . . . . . . . . . . . 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 1182 . . . . . . . . . . 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 5873 . . . . . . . . . 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 2288 . . . . . . . . 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 5529 . . . . . . . 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 29553 . . . . . . . . . . 11  |-  ( K  e.  HL  ->  K  e.  Lat )
1817ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  K  e.  Lat )
193, 10latmcl 14157 . . . . . . . . . 10  |-  ( ( K  e.  Lat  /\  X  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  ->  ( X ( meet `  K
) W )  e.  ( Base `  K
) )
2018, 6, 8, 19syl3anc 1182 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  ( X ( meet `  K
) W )  e.  ( Base `  K
) )
213, 9, 10, 11oldmm2 29408 . . . . . . . . 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 1182 . . . . . . . 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 2315 . . . . . . 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 5873 . . . . . 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 29552 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  OP )
2625ad2antrr 706 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  K  e.  OP )
273, 11opoccl 29384 . . . . . . . . 9  |-  ( ( K  e.  OP  /\  W  e.  ( Base `  K ) )  -> 
( ( oc `  K ) `  W
)  e.  ( Base `  K ) )
2826, 8, 27syl2anc 642 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
( oc `  K
) `  W )  e.  ( Base `  K
) )
293, 9latjass 14201 . . . . . . . 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 1184 . . . . . . 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 14180 . . . . . . . . 9  |-  ( ( K  e.  Lat  /\  ( ( oc `  K ) `  W
)  e.  ( Base `  K ) )  -> 
( ( ( oc
`  K ) `  W ) ( join `  K ) ( ( oc `  K ) `
 W ) )  =  ( ( oc
`  K ) `  W ) )
3218, 28, 31syl2anc 642 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
( ( oc `  K ) `  W
) ( join `  K
) ( ( oc
`  K ) `  W ) )  =  ( ( oc `  K ) `  W
) )
3332oveq2d 5874 . . . . . . 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 2315 . . . . . 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 2315 . . . . 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 5873 . . . 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 29547 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OML )
3837ad2antrr 706 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  K  e.  OML )
39 eqid 2283 . . . . . . 7  |-  ( le
`  K )  =  ( le `  K
)
403, 39, 10latmle2 14183 . . . . . 6  |-  ( ( K  e.  Lat  /\  X  e.  ( Base `  K )  /\  W  e.  ( Base `  K
) )  ->  ( X ( meet `  K
) W ) ( le `  K ) W )
4118, 6, 8, 40syl3anc 1182 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  ( X ( meet `  K
) W ) ( le `  K ) W )
423, 39, 9, 10, 11omlspjN 29451 . . . . 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 1187 . . . 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 31228 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  X
( le `  K
) W )
453, 39, 10latleeqm1 14185 . . . . . 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 1182 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  ( X ( le `  K ) W  <->  ( X
( meet `  K ) W )  =  X ) )
4744, 46mpbid 201 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  ( X ( meet `  K
) W )  =  X )
4836, 43, 473eqtrrd 2320 . . 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 5529 . 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 29384 . . . . . . 7  |-  ( ( K  e.  OP  /\  X  e.  ( Base `  K ) )  -> 
( ( oc `  K ) `  X
)  e.  ( Base `  K ) )
5126, 6, 50syl2anc 642 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
( oc `  K
) `  X )  e.  ( Base `  K
) )
523, 9latjcl 14156 . . . . . 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 1182 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  e.  dom  I )  ->  (
( ( oc `  K ) `  X
) ( join `  K
) ( ( oc
`  K ) `  W ) )  e.  ( Base `  K
) )
543, 10latmcl 14157 . . . . 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 1182 . . . 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 14183 . . . . 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 1182 . . . 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 31226 . . . . 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 451 . . . 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 888 . . 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 2283 . . . 4  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
62 doca2.n . . . 4  |-  ._|_  =  ( ( ocA `  K
) `  W )
639, 10, 11, 4, 61, 5, 62diaocN 31315 . . 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 456 . 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 31315 . . 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 5529 . 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 2320 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   class class class wbr 4023   dom cdm 4689   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   occoc 13216   joincjn 14078   meetcmee 14079   Latclat 14151   OPcops 29362   OLcol 29364   OMLcoml 29365   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   DIsoAcdia 31218   ocAcocaN 31309
This theorem is referenced by:  doca3N  31317
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-undef 6298  df-riota 6304  df-map 6774  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-oposet 29366  df-cmtN 29367  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348  df-disoa 31219  df-docaN 31310
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