Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dochocss Structured version   Unicode version

Theorem dochocss 32238
Description: Double negative law for orthocomplement of an arbitrary set of vectors. (Contributed by NM, 16-Apr-2014.)
Hypotheses
Ref Expression
dochss.h  |-  H  =  ( LHyp `  K
)
dochss.u  |-  U  =  ( ( DVecH `  K
) `  W )
dochss.v  |-  V  =  ( Base `  U
)
dochss.o  |-  ._|_  =  ( ( ocH `  K
) `  W )
Assertion
Ref Expression
dochocss  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  X  C_  (  ._|_  `  (  ._|_  `  X
) ) )

Proof of Theorem dochocss
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 ssintub 4070 . 2  |-  X  C_  |^|
{ z  e.  ran  ( ( DIsoH `  K
) `  W )  |  X  C_  z }
2 dochss.h . . . . 5  |-  H  =  ( LHyp `  K
)
3 eqid 2438 . . . . 5  |-  ( (
DIsoH `  K ) `  W )  =  ( ( DIsoH `  K ) `  W )
4 dochss.u . . . . 5  |-  U  =  ( ( DVecH `  K
) `  W )
5 dochss.v . . . . 5  |-  V  =  ( Base `  U
)
6 dochss.o . . . . 5  |-  ._|_  =  ( ( ocH `  K
) `  W )
72, 3, 4, 5, 6dochcl 32225 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  (  ._|_  `  X )  e.  ran  ( ( DIsoH `  K
) `  W )
)
8 eqid 2438 . . . . 5  |-  ( oc
`  K )  =  ( oc `  K
)
98, 2, 3, 6dochvalr 32229 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  (  ._|_  `  X
)  e.  ran  (
( DIsoH `  K ) `  W ) )  -> 
(  ._|_  `  (  ._|_  `  X ) )  =  ( ( ( DIsoH `  K ) `  W
) `  ( ( oc `  K ) `  ( `' ( ( DIsoH `  K ) `  W
) `  (  ._|_  `  X ) ) ) ) )
107, 9syldan 458 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  (  ._|_  `  (  ._|_  `  X ) )  =  ( ( ( DIsoH `  K ) `  W ) `  (
( oc `  K
) `  ( `' ( ( DIsoH `  K
) `  W ) `  (  ._|_  `  X
) ) ) ) )
118, 2, 3, 4, 5, 6dochval2 32224 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  (  ._|_  `  X )  =  ( ( ( DIsoH `  K
) `  W ) `  ( ( oc `  K ) `  ( `' ( ( DIsoH `  K ) `  W
) `  |^| { z  e.  ran  ( (
DIsoH `  K ) `  W )  |  X  C_  z } ) ) ) )
1211fveq2d 5735 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( `' ( ( DIsoH `  K
) `  W ) `  (  ._|_  `  X
) )  =  ( `' ( ( DIsoH `  K ) `  W
) `  ( (
( DIsoH `  K ) `  W ) `  (
( oc `  K
) `  ( `' ( ( DIsoH `  K
) `  W ) `  |^| { z  e. 
ran  ( ( DIsoH `  K ) `  W
)  |  X  C_  z } ) ) ) ) )
13 eqid 2438 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
14 eqid 2438 . . . . . . . . . . 11  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
1513, 2, 3, 4, 14dihf11 32139 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( DIsoH `  K
) `  W ) : ( Base `  K
) -1-1-> ( LSubSp `  U
) )
1615adantr 453 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( ( DIsoH `  K ) `  W ) : (
Base `  K ) -1-1-> ( LSubSp `  U )
)
17 f1f1orn 5688 . . . . . . . . 9  |-  ( ( ( DIsoH `  K ) `  W ) : (
Base `  K ) -1-1-> ( LSubSp `  U )  ->  ( ( DIsoH `  K
) `  W ) : ( Base `  K
)
-1-1-onto-> ran  ( ( DIsoH `  K
) `  W )
)
1816, 17syl 16 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( ( DIsoH `  K ) `  W ) : (
Base `  K ) -1-1-onto-> ran  ( ( DIsoH `  K
) `  W )
)
19 hlop 30234 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  OP )
2019ad2antrr 708 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  K  e.  OP )
21 simpl 445 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( K  e.  HL  /\  W  e.  H ) )
22 ssrab2 3430 . . . . . . . . . . . 12  |-  { z  e.  ran  ( (
DIsoH `  K ) `  W )  |  X  C_  z }  C_  ran  ( ( DIsoH `  K
) `  W )
2322a1i 11 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  { z  e.  ran  ( ( DIsoH `  K ) `  W
)  |  X  C_  z }  C_  ran  (
( DIsoH `  K ) `  W ) )
24 eqid 2438 . . . . . . . . . . . . . . . 16  |-  ( 1.
`  K )  =  ( 1. `  K
)
2524, 2, 3, 4, 5dih1 32158 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( DIsoH `  K ) `  W
) `  ( 1. `  K ) )  =  V )
2625adantr 453 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( (
( DIsoH `  K ) `  W ) `  ( 1. `  K ) )  =  V )
27 f1fn 5643 . . . . . . . . . . . . . . . 16  |-  ( ( ( DIsoH `  K ) `  W ) : (
Base `  K ) -1-1-> ( LSubSp `  U )  ->  ( ( DIsoH `  K
) `  W )  Fn  ( Base `  K
) )
2816, 27syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( ( DIsoH `  K ) `  W )  Fn  ( Base `  K ) )
2913, 24op1cl 30057 . . . . . . . . . . . . . . . 16  |-  ( K  e.  OP  ->  ( 1. `  K )  e.  ( Base `  K
) )
3020, 29syl 16 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( 1. `  K )  e.  (
Base `  K )
)
31 fnfvelrn 5870 . . . . . . . . . . . . . . 15  |-  ( ( ( ( DIsoH `  K
) `  W )  Fn  ( Base `  K
)  /\  ( 1. `  K )  e.  (
Base `  K )
)  ->  ( (
( DIsoH `  K ) `  W ) `  ( 1. `  K ) )  e.  ran  ( (
DIsoH `  K ) `  W ) )
3228, 30, 31syl2anc 644 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( (
( DIsoH `  K ) `  W ) `  ( 1. `  K ) )  e.  ran  ( (
DIsoH `  K ) `  W ) )
3326, 32eqeltrrd 2513 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  V  e.  ran  ( ( DIsoH `  K
) `  W )
)
34 simpr 449 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  X  C_  V
)
35 sseq2 3372 . . . . . . . . . . . . . 14  |-  ( z  =  V  ->  ( X  C_  z  <->  X  C_  V
) )
3635elrab 3094 . . . . . . . . . . . . 13  |-  ( V  e.  { z  e. 
ran  ( ( DIsoH `  K ) `  W
)  |  X  C_  z }  <->  ( V  e. 
ran  ( ( DIsoH `  K ) `  W
)  /\  X  C_  V
) )
3733, 34, 36sylanbrc 647 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  V  e.  { z  e.  ran  (
( DIsoH `  K ) `  W )  |  X  C_  z } )
38 ne0i 3636 . . . . . . . . . . . 12  |-  ( V  e.  { z  e. 
ran  ( ( DIsoH `  K ) `  W
)  |  X  C_  z }  ->  { z  e.  ran  ( (
DIsoH `  K ) `  W )  |  X  C_  z }  =/=  (/) )
3937, 38syl 16 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  { z  e.  ran  ( ( DIsoH `  K ) `  W
)  |  X  C_  z }  =/=  (/) )
402, 3dihintcl 32216 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( { z  e.  ran  ( (
DIsoH `  K ) `  W )  |  X  C_  z }  C_  ran  ( ( DIsoH `  K
) `  W )  /\  { z  e.  ran  ( ( DIsoH `  K
) `  W )  |  X  C_  z }  =/=  (/) ) )  ->  |^| { z  e.  ran  ( ( DIsoH `  K
) `  W )  |  X  C_  z }  e.  ran  ( (
DIsoH `  K ) `  W ) )
4121, 23, 39, 40syl12anc 1183 . . . . . . . . . 10  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  |^| { z  e.  ran  ( (
DIsoH `  K ) `  W )  |  X  C_  z }  e.  ran  ( ( DIsoH `  K
) `  W )
)
42 f1ocnvdm 6021 . . . . . . . . . 10  |-  ( ( ( ( DIsoH `  K
) `  W ) : ( Base `  K
)
-1-1-onto-> ran  ( ( DIsoH `  K
) `  W )  /\  |^| { z  e. 
ran  ( ( DIsoH `  K ) `  W
)  |  X  C_  z }  e.  ran  ( ( DIsoH `  K
) `  W )
)  ->  ( `' ( ( DIsoH `  K
) `  W ) `  |^| { z  e. 
ran  ( ( DIsoH `  K ) `  W
)  |  X  C_  z } )  e.  (
Base `  K )
)
4318, 41, 42syl2anc 644 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( `' ( ( DIsoH `  K
) `  W ) `  |^| { z  e. 
ran  ( ( DIsoH `  K ) `  W
)  |  X  C_  z } )  e.  (
Base `  K )
)
4413, 8opoccl 30066 . . . . . . . . 9  |-  ( ( K  e.  OP  /\  ( `' ( ( DIsoH `  K ) `  W
) `  |^| { z  e.  ran  ( (
DIsoH `  K ) `  W )  |  X  C_  z } )  e.  ( Base `  K
) )  ->  (
( oc `  K
) `  ( `' ( ( DIsoH `  K
) `  W ) `  |^| { z  e. 
ran  ( ( DIsoH `  K ) `  W
)  |  X  C_  z } ) )  e.  ( Base `  K
) )
4520, 43, 44syl2anc 644 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( ( oc `  K ) `  ( `' ( ( DIsoH `  K ) `  W
) `  |^| { z  e.  ran  ( (
DIsoH `  K ) `  W )  |  X  C_  z } ) )  e.  ( Base `  K
) )
46 f1ocnvfv1 6017 . . . . . . . 8  |-  ( ( ( ( DIsoH `  K
) `  W ) : ( Base `  K
)
-1-1-onto-> ran  ( ( DIsoH `  K
) `  W )  /\  ( ( oc `  K ) `  ( `' ( ( DIsoH `  K ) `  W
) `  |^| { z  e.  ran  ( (
DIsoH `  K ) `  W )  |  X  C_  z } ) )  e.  ( Base `  K
) )  ->  ( `' ( ( DIsoH `  K ) `  W
) `  ( (
( DIsoH `  K ) `  W ) `  (
( oc `  K
) `  ( `' ( ( DIsoH `  K
) `  W ) `  |^| { z  e. 
ran  ( ( DIsoH `  K ) `  W
)  |  X  C_  z } ) ) ) )  =  ( ( oc `  K ) `
 ( `' ( ( DIsoH `  K ) `  W ) `  |^| { z  e.  ran  (
( DIsoH `  K ) `  W )  |  X  C_  z } ) ) )
4718, 45, 46syl2anc 644 . . . . . . 7  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( `' ( ( DIsoH `  K
) `  W ) `  ( ( ( DIsoH `  K ) `  W
) `  ( ( oc `  K ) `  ( `' ( ( DIsoH `  K ) `  W
) `  |^| { z  e.  ran  ( (
DIsoH `  K ) `  W )  |  X  C_  z } ) ) ) )  =  ( ( oc `  K
) `  ( `' ( ( DIsoH `  K
) `  W ) `  |^| { z  e. 
ran  ( ( DIsoH `  K ) `  W
)  |  X  C_  z } ) ) )
4812, 47eqtrd 2470 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( `' ( ( DIsoH `  K
) `  W ) `  (  ._|_  `  X
) )  =  ( ( oc `  K
) `  ( `' ( ( DIsoH `  K
) `  W ) `  |^| { z  e. 
ran  ( ( DIsoH `  K ) `  W
)  |  X  C_  z } ) ) )
4948fveq2d 5735 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( ( oc `  K ) `  ( `' ( ( DIsoH `  K ) `  W
) `  (  ._|_  `  X ) ) )  =  ( ( oc
`  K ) `  ( ( oc `  K ) `  ( `' ( ( DIsoH `  K ) `  W
) `  |^| { z  e.  ran  ( (
DIsoH `  K ) `  W )  |  X  C_  z } ) ) ) )
5013, 8opococ 30067 . . . . . 6  |-  ( ( K  e.  OP  /\  ( `' ( ( DIsoH `  K ) `  W
) `  |^| { z  e.  ran  ( (
DIsoH `  K ) `  W )  |  X  C_  z } )  e.  ( Base `  K
) )  ->  (
( oc `  K
) `  ( ( oc `  K ) `  ( `' ( ( DIsoH `  K ) `  W
) `  |^| { z  e.  ran  ( (
DIsoH `  K ) `  W )  |  X  C_  z } ) ) )  =  ( `' ( ( DIsoH `  K
) `  W ) `  |^| { z  e. 
ran  ( ( DIsoH `  K ) `  W
)  |  X  C_  z } ) )
5120, 43, 50syl2anc 644 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( ( oc `  K ) `  ( ( oc `  K ) `  ( `' ( ( DIsoH `  K ) `  W
) `  |^| { z  e.  ran  ( (
DIsoH `  K ) `  W )  |  X  C_  z } ) ) )  =  ( `' ( ( DIsoH `  K
) `  W ) `  |^| { z  e. 
ran  ( ( DIsoH `  K ) `  W
)  |  X  C_  z } ) )
5249, 51eqtrd 2470 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( ( oc `  K ) `  ( `' ( ( DIsoH `  K ) `  W
) `  (  ._|_  `  X ) ) )  =  ( `' ( ( DIsoH `  K ) `  W ) `  |^| { z  e.  ran  (
( DIsoH `  K ) `  W )  |  X  C_  z } ) )
5352fveq2d 5735 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( (
( DIsoH `  K ) `  W ) `  (
( oc `  K
) `  ( `' ( ( DIsoH `  K
) `  W ) `  (  ._|_  `  X
) ) ) )  =  ( ( (
DIsoH `  K ) `  W ) `  ( `' ( ( DIsoH `  K ) `  W
) `  |^| { z  e.  ran  ( (
DIsoH `  K ) `  W )  |  X  C_  z } ) ) )
54 f1ocnvfv2 6018 . . . 4  |-  ( ( ( ( DIsoH `  K
) `  W ) : ( Base `  K
)
-1-1-onto-> ran  ( ( DIsoH `  K
) `  W )  /\  |^| { z  e. 
ran  ( ( DIsoH `  K ) `  W
)  |  X  C_  z }  e.  ran  ( ( DIsoH `  K
) `  W )
)  ->  ( (
( DIsoH `  K ) `  W ) `  ( `' ( ( DIsoH `  K ) `  W
) `  |^| { z  e.  ran  ( (
DIsoH `  K ) `  W )  |  X  C_  z } ) )  =  |^| { z  e.  ran  ( (
DIsoH `  K ) `  W )  |  X  C_  z } )
5518, 41, 54syl2anc 644 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( (
( DIsoH `  K ) `  W ) `  ( `' ( ( DIsoH `  K ) `  W
) `  |^| { z  e.  ran  ( (
DIsoH `  K ) `  W )  |  X  C_  z } ) )  =  |^| { z  e.  ran  ( (
DIsoH `  K ) `  W )  |  X  C_  z } )
5610, 53, 553eqtrrd 2475 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  |^| { z  e.  ran  ( (
DIsoH `  K ) `  W )  |  X  C_  z }  =  ( 
._|_  `  (  ._|_  `  X
) ) )
571, 56syl5sseq 3398 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  X  C_  (  ._|_  `  (  ._|_  `  X
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   {crab 2711    C_ wss 3322   (/)c0 3630   |^|cint 4052   `'ccnv 4880   ran crn 4882    Fn wfn 5452   -1-1->wf1 5454   -1-1-onto->wf1o 5456   ` cfv 5457   Basecbs 13474   occoc 13542   1.cp1 14472   LSubSpclss 16013   OPcops 30044   HLchlt 30222   LHypclh 30855   DVecHcdvh 31950   DIsoHcdih 32100   ocHcoch 32219
This theorem is referenced by:  dochsscl  32240  dochsat  32255  dochshpncl  32256  dochlkr  32257  dochdmj1  32262  dochnoncon  32263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-fal 1330  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-tpos 6482  df-undef 6546  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-3 10064  df-4 10065  df-5 10066  df-6 10067  df-n0 10227  df-z 10288  df-uz 10494  df-fz 11049  df-struct 13476  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-sca 13550  df-vsca 13551  df-0g 13732  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-p1 14474  df-lat 14480  df-clat 14542  df-mnd 14695  df-submnd 14744  df-grp 14817  df-minusg 14818  df-sbg 14819  df-subg 14946  df-cntz 15121  df-lsm 15275  df-cmn 15419  df-abl 15420  df-mgp 15654  df-rng 15668  df-ur 15670  df-oppr 15733  df-dvdsr 15751  df-unit 15752  df-invr 15782  df-dvr 15793  df-drng 15842  df-lmod 15957  df-lss 16014  df-lsp 16053  df-lvec 16180  df-lsatoms 29848  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194  df-hlat 30223  df-llines 30369  df-lplanes 30370  df-lvols 30371  df-lines 30372  df-psubsp 30374  df-pmap 30375  df-padd 30667  df-lhyp 30859  df-laut 30860  df-ldil 30975  df-ltrn 30976  df-trl 31030  df-tendo 31626  df-edring 31628  df-disoa 31901  df-dvech 31951  df-dib 32011  df-dic 32045  df-dih 32101  df-doch 32220
  Copyright terms: Public domain W3C validator