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

Theorem dochocss 31556
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 3880 . 2  |-  X  C_  |^|
{ z  e.  ran  ( ( DIsoH `  K
) `  W )  |  X  C_  z }
2 dochss.h . . . . 5  |-  H  =  ( LHyp `  K
)
3 eqid 2283 . . . . 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 31543 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  (  ._|_  `  X )  e.  ran  ( ( DIsoH `  K
) `  W )
)
8 eqid 2283 . . . . 5  |-  ( oc
`  K )  =  ( oc `  K
)
98, 2, 3, 6dochvalr 31547 . . . 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 456 . . 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 31542 . . . . . . . 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 5529 . . . . . . 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 2283 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
14 eqid 2283 . . . . . . . . . . 11  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
1513, 2, 3, 4, 14dihf11 31457 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( DIsoH `  K
) `  W ) : ( Base `  K
) -1-1-> ( LSubSp `  U
) )
1615adantr 451 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( ( DIsoH `  K ) `  W ) : (
Base `  K ) -1-1-> ( LSubSp `  U )
)
17 f1f1orn 5483 . . . . . . . . 9  |-  ( ( ( DIsoH `  K ) `  W ) : (
Base `  K ) -1-1-> ( LSubSp `  U )  ->  ( ( DIsoH `  K
) `  W ) : ( Base `  K
)
-1-1-onto-> ran  ( ( DIsoH `  K
) `  W )
)
1816, 17syl 15 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( ( DIsoH `  K ) `  W ) : (
Base `  K ) -1-1-onto-> ran  ( ( DIsoH `  K
) `  W )
)
19 hlop 29552 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  OP )
2019ad2antrr 706 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  K  e.  OP )
21 simpl 443 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( K  e.  HL  /\  W  e.  H ) )
22 ssrab2 3258 . . . . . . . . . . . 12  |-  { z  e.  ran  ( (
DIsoH `  K ) `  W )  |  X  C_  z }  C_  ran  ( ( DIsoH `  K
) `  W )
2322a1i 10 . . . . . . . . . . 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 2283 . . . . . . . . . . . . . . . 16  |-  ( 1.
`  K )  =  ( 1. `  K
)
2524, 2, 3, 4, 5dih1 31476 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( DIsoH `  K ) `  W
) `  ( 1. `  K ) )  =  V )
2625adantr 451 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( (
( DIsoH `  K ) `  W ) `  ( 1. `  K ) )  =  V )
27 f1fn 5438 . . . . . . . . . . . . . . . 16  |-  ( ( ( DIsoH `  K ) `  W ) : (
Base `  K ) -1-1-> ( LSubSp `  U )  ->  ( ( DIsoH `  K
) `  W )  Fn  ( Base `  K
) )
2816, 27syl 15 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( ( DIsoH `  K ) `  W )  Fn  ( Base `  K ) )
2913, 24op1cl 29375 . . . . . . . . . . . . . . . 16  |-  ( K  e.  OP  ->  ( 1. `  K )  e.  ( Base `  K
) )
3020, 29syl 15 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( 1. `  K )  e.  (
Base `  K )
)
31 fnfvelrn 5662 . . . . . . . . . . . . . . 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 642 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( (
( DIsoH `  K ) `  W ) `  ( 1. `  K ) )  e.  ran  ( (
DIsoH `  K ) `  W ) )
3326, 32eqeltrrd 2358 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  V  e.  ran  ( ( DIsoH `  K
) `  W )
)
34 simpr 447 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  X  C_  V
)
35 sseq2 3200 . . . . . . . . . . . . . 14  |-  ( z  =  V  ->  ( X  C_  z  <->  X  C_  V
) )
3635elrab 2923 . . . . . . . . . . . . 13  |-  ( V  e.  { z  e. 
ran  ( ( DIsoH `  K ) `  W
)  |  X  C_  z }  <->  ( V  e. 
ran  ( ( DIsoH `  K ) `  W
)  /\  X  C_  V
) )
3733, 34, 36sylanbrc 645 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  V  e.  { z  e.  ran  (
( DIsoH `  K ) `  W )  |  X  C_  z } )
38 ne0i 3461 . . . . . . . . . . . 12  |-  ( V  e.  { z  e. 
ran  ( ( DIsoH `  K ) `  W
)  |  X  C_  z }  ->  { z  e.  ran  ( (
DIsoH `  K ) `  W )  |  X  C_  z }  =/=  (/) )
3937, 38syl 15 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  { z  e.  ran  ( ( DIsoH `  K ) `  W
)  |  X  C_  z }  =/=  (/) )
402, 3dihintcl 31534 . . . . . . . . . . 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 1180 . . . . . . . . . 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 5796 . . . . . . . . . 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 642 . . . . . . . . 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 29384 . . . . . . . . 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 642 . . . . . . . 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 5792 . . . . . . . 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 642 . . . . . . 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 2315 . . . . . 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 5529 . . . . 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 29385 . . . . . 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 642 . . . . 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 2315 . . . 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 5529 . . 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 5793 . . . 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 642 . . 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 2320 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  |^| { z  e.  ran  ( (
DIsoH `  K ) `  W )  |  X  C_  z }  =  ( 
._|_  `  (  ._|_  `  X
) ) )
571, 56syl5sseq 3226 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  X  C_  (  ._|_  `  (  ._|_  `  X
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   {crab 2547    C_ wss 3152   (/)c0 3455   |^|cint 3862   `'ccnv 4688   ran crn 4690    Fn wfn 5250   -1-1->wf1 5252   -1-1-onto->wf1o 5254   ` cfv 5255   Basecbs 13148   occoc 13216   1.cp1 14144   LSubSpclss 15689   OPcops 29362   HLchlt 29540   LHypclh 30173   DVecHcdvh 31268   DIsoHcdih 31418   ocHcoch 31537
This theorem is referenced by:  dochsscl  31558  dochsat  31573  dochshpncl  31574  dochlkr  31575  dochdmj1  31580  dochnoncon  31581
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  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  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-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-tpos 6234  df-undef 6298  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-0g 13404  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-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cntz 14793  df-lsm 14947  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856  df-lsatoms 29166  df-oposet 29366  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-tendo 30944  df-edring 30946  df-disoa 31219  df-dvech 31269  df-dib 31329  df-dic 31363  df-dih 31419  df-doch 31538
  Copyright terms: Public domain W3C validator