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Theorem dochocss 34386
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 4245 . 2  |-  X  C_  |^|
{ z  e.  ran  ( ( DIsoH `  K
) `  W )  |  X  C_  z }
2 dochss.h . . . . 5  |-  H  =  ( LHyp `  K
)
3 eqid 2402 . . . . 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 34373 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  (  ._|_  `  X )  e.  ran  ( ( DIsoH `  K
) `  W )
)
8 eqid 2402 . . . . 5  |-  ( oc
`  K )  =  ( oc `  K
)
98, 2, 3, 6dochvalr 34377 . . . 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 468 . . 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 34372 . . . . . . . 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 5853 . . . . . . 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 2402 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
14 eqid 2402 . . . . . . . . . . 11  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
1513, 2, 3, 4, 14dihf11 34287 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( DIsoH `  K
) `  W ) : ( Base `  K
) -1-1-> ( LSubSp `  U
) )
1615adantr 463 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( ( DIsoH `  K ) `  W ) : (
Base `  K ) -1-1-> ( LSubSp `  U )
)
17 f1f1orn 5810 . . . . . . . . 9  |-  ( ( ( DIsoH `  K ) `  W ) : (
Base `  K ) -1-1-> ( LSubSp `  U )  ->  ( ( DIsoH `  K
) `  W ) : ( Base `  K
)
-1-1-onto-> ran  ( ( DIsoH `  K
) `  W )
)
1816, 17syl 17 . . . . . . . 8  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( ( DIsoH `  K ) `  W ) : (
Base `  K ) -1-1-onto-> ran  ( ( DIsoH `  K
) `  W )
)
19 hlop 32380 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  OP )
2019ad2antrr 724 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  K  e.  OP )
21 simpl 455 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( K  e.  HL  /\  W  e.  H ) )
22 ssrab2 3524 . . . . . . . . . . . 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 2402 . . . . . . . . . . . . . . . 16  |-  ( 1.
`  K )  =  ( 1. `  K
)
2524, 2, 3, 4, 5dih1 34306 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( DIsoH `  K ) `  W
) `  ( 1. `  K ) )  =  V )
2625adantr 463 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( (
( DIsoH `  K ) `  W ) `  ( 1. `  K ) )  =  V )
27 f1fn 5765 . . . . . . . . . . . . . . . 16  |-  ( ( ( DIsoH `  K ) `  W ) : (
Base `  K ) -1-1-> ( LSubSp `  U )  ->  ( ( DIsoH `  K
) `  W )  Fn  ( Base `  K
) )
2816, 27syl 17 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( ( DIsoH `  K ) `  W )  Fn  ( Base `  K ) )
2913, 24op1cl 32203 . . . . . . . . . . . . . . . 16  |-  ( K  e.  OP  ->  ( 1. `  K )  e.  ( Base `  K
) )
3020, 29syl 17 . . . . . . . . . . . . . . 15  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( 1. `  K )  e.  (
Base `  K )
)
31 fnfvelrn 6006 . . . . . . . . . . . . . . 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 659 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( (
( DIsoH `  K ) `  W ) `  ( 1. `  K ) )  e.  ran  ( (
DIsoH `  K ) `  W ) )
3326, 32eqeltrrd 2491 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  V  e.  ran  ( ( DIsoH `  K
) `  W )
)
34 simpr 459 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  X  C_  V
)
35 sseq2 3464 . . . . . . . . . . . . . 14  |-  ( z  =  V  ->  ( X  C_  z  <->  X  C_  V
) )
3635elrab 3207 . . . . . . . . . . . . 13  |-  ( V  e.  { z  e. 
ran  ( ( DIsoH `  K ) `  W
)  |  X  C_  z }  <->  ( V  e. 
ran  ( ( DIsoH `  K ) `  W
)  /\  X  C_  V
) )
3733, 34, 36sylanbrc 662 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  V  e.  { z  e.  ran  (
( DIsoH `  K ) `  W )  |  X  C_  z } )
38 ne0i 3744 . . . . . . . . . . . 12  |-  ( V  e.  { z  e. 
ran  ( ( DIsoH `  K ) `  W
)  |  X  C_  z }  ->  { z  e.  ran  ( (
DIsoH `  K ) `  W )  |  X  C_  z }  =/=  (/) )
3937, 38syl 17 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  { z  e.  ran  ( ( DIsoH `  K ) `  W
)  |  X  C_  z }  =/=  (/) )
402, 3dihintcl 34364 . . . . . . . . . . 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 1228 . . . . . . . . . 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 6171 . . . . . . . . . 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 659 . . . . . . . . 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 32212 . . . . . . . . 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 659 . . . . . . . 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 6163 . . . . . . . 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 659 . . . . . . 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 2443 . . . . . 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 5853 . . . . 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 32213 . . . . . 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 659 . . . . 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 2443 . . . 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 5853 . . 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 6164 . . . 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 659 . . 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 2448 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  |^| { z  e.  ran  ( (
DIsoH `  K ) `  W )  |  X  C_  z }  =  ( 
._|_  `  (  ._|_  `  X
) ) )
571, 56syl5sseq 3490 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  X  C_  (  ._|_  `  (  ._|_  `  X
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   {crab 2758    C_ wss 3414   (/)c0 3738   |^|cint 4227   `'ccnv 4822   ran crn 4824    Fn wfn 5564   -1-1->wf1 5566   -1-1-onto->wf1o 5568   ` cfv 5569   Basecbs 14841   occoc 14917   1.cp1 15992   LSubSpclss 17898   OPcops 32190   HLchlt 32368   LHypclh 33001   DVecHcdvh 34098   DIsoHcdih 34248   ocHcoch 34367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-riotaBAD 31977
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-1st 6784  df-2nd 6785  df-tpos 6958  df-undef 7005  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-n0 10837  df-z 10906  df-uz 11128  df-fz 11727  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-sca 14925  df-vsca 14926  df-0g 15056  df-preset 15881  df-poset 15899  df-plt 15912  df-lub 15928  df-glb 15929  df-join 15930  df-meet 15931  df-p0 15993  df-p1 15994  df-lat 16000  df-clat 16062  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-submnd 16291  df-grp 16381  df-minusg 16382  df-sbg 16383  df-subg 16522  df-cntz 16679  df-lsm 16980  df-cmn 17124  df-abl 17125  df-mgp 17462  df-ur 17474  df-ring 17520  df-oppr 17592  df-dvdsr 17610  df-unit 17611  df-invr 17641  df-dvr 17652  df-drng 17718  df-lmod 17834  df-lss 17899  df-lsp 17938  df-lvec 18069  df-lsatoms 31994  df-oposet 32194  df-ol 32196  df-oml 32197  df-covers 32284  df-ats 32285  df-atl 32316  df-cvlat 32340  df-hlat 32369  df-llines 32515  df-lplanes 32516  df-lvols 32517  df-lines 32518  df-psubsp 32520  df-pmap 32521  df-padd 32813  df-lhyp 33005  df-laut 33006  df-ldil 33121  df-ltrn 33122  df-trl 33177  df-tendo 33774  df-edring 33776  df-disoa 34049  df-dvech 34099  df-dib 34159  df-dic 34193  df-dih 34249  df-doch 34368
This theorem is referenced by:  dochsscl  34388  dochsat  34403  dochshpncl  34404  dochlkr  34405  dochdmj1  34410  dochnoncon  34411
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