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Theorem dochocss 36038
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 4293 . 2  |-  X  C_  |^|
{ z  e.  ran  ( ( DIsoH `  K
) `  W )  |  X  C_  z }
2 dochss.h . . . . 5  |-  H  =  ( LHyp `  K
)
3 eqid 2460 . . . . 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 36025 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  (  ._|_  `  X )  e.  ran  ( ( DIsoH `  K
) `  W )
)
8 eqid 2460 . . . . 5  |-  ( oc
`  K )  =  ( oc `  K
)
98, 2, 3, 6dochvalr 36029 . . . 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 470 . . 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 36024 . . . . . . . 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 5861 . . . . . . 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 2460 . . . . . . . . . . 11  |-  ( Base `  K )  =  (
Base `  K )
14 eqid 2460 . . . . . . . . . . 11  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
1513, 2, 3, 4, 14dihf11 35939 . . . . . . . . . 10  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( DIsoH `  K
) `  W ) : ( Base `  K
) -1-1-> ( LSubSp `  U
) )
1615adantr 465 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( ( DIsoH `  K ) `  W ) : (
Base `  K ) -1-1-> ( LSubSp `  U )
)
17 f1f1orn 5818 . . . . . . . . 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 34034 . . . . . . . . . 10  |-  ( K  e.  HL  ->  K  e.  OP )
2019ad2antrr 725 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  K  e.  OP )
21 simpl 457 . . . . . . . . . . 11  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( K  e.  HL  /\  W  e.  H ) )
22 ssrab2 3578 . . . . . . . . . . . 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 2460 . . . . . . . . . . . . . . . 16  |-  ( 1.
`  K )  =  ( 1. `  K
)
2524, 2, 3, 4, 5dih1 35958 . . . . . . . . . . . . . . 15  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( ( DIsoH `  K ) `  W
) `  ( 1. `  K ) )  =  V )
2625adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( (
( DIsoH `  K ) `  W ) `  ( 1. `  K ) )  =  V )
27 f1fn 5773 . . . . . . . . . . . . . . . 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 33857 . . . . . . . . . . . . . . . 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 6009 . . . . . . . . . . . . . . 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 661 . . . . . . . . . . . . . 14  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  ( (
( DIsoH `  K ) `  W ) `  ( 1. `  K ) )  e.  ran  ( (
DIsoH `  K ) `  W ) )
3326, 32eqeltrrd 2549 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  V  e.  ran  ( ( DIsoH `  K
) `  W )
)
34 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  X  C_  V
)
35 sseq2 3519 . . . . . . . . . . . . . 14  |-  ( z  =  V  ->  ( X  C_  z  <->  X  C_  V
) )
3635elrab 3254 . . . . . . . . . . . . 13  |-  ( V  e.  { z  e. 
ran  ( ( DIsoH `  K ) `  W
)  |  X  C_  z }  <->  ( V  e. 
ran  ( ( DIsoH `  K ) `  W
)  /\  X  C_  V
) )
3733, 34, 36sylanbrc 664 . . . . . . . . . . . 12  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  V  e.  { z  e.  ran  (
( DIsoH `  K ) `  W )  |  X  C_  z } )
38 ne0i 3784 . . . . . . . . . . . 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 36016 . . . . . . . . . . 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 1221 . . . . . . . . . 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 6167 . . . . . . . . . 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 661 . . . . . . . . 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 33866 . . . . . . . . 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 661 . . . . . . . 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 6161 . . . . . . . 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 661 . . . . . . 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 2501 . . . . . 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 5861 . . . . 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 33867 . . . . . 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 661 . . . . 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 2501 . . . 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 5861 . . 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 6162 . . . 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 661 . . 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 2506 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  |^| { z  e.  ran  ( (
DIsoH `  K ) `  W )  |  X  C_  z }  =  ( 
._|_  `  (  ._|_  `  X
) ) )
571, 56syl5sseq 3545 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  X  C_  (  ._|_  `  (  ._|_  `  X
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2655   {crab 2811    C_ wss 3469   (/)c0 3778   |^|cint 4275   `'ccnv 4991   ran crn 4993    Fn wfn 5574   -1-1->wf1 5576   -1-1-onto->wf1o 5578   ` cfv 5579   Basecbs 14479   occoc 14552   1.cp1 15514   LSubSpclss 17354   OPcops 33844   HLchlt 34022   LHypclh 34655   DVecHcdvh 35750   DIsoHcdih 35900   ocHcoch 36019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-riotaBAD 33631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-tpos 6945  df-undef 6992  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-sca 14560  df-vsca 14561  df-0g 14686  df-poset 15422  df-plt 15434  df-lub 15450  df-glb 15451  df-join 15452  df-meet 15453  df-p0 15515  df-p1 15516  df-lat 15522  df-clat 15584  df-mnd 15721  df-submnd 15771  df-grp 15851  df-minusg 15852  df-sbg 15853  df-subg 15986  df-cntz 16143  df-lsm 16445  df-cmn 16589  df-abl 16590  df-mgp 16925  df-ur 16937  df-rng 16981  df-oppr 17049  df-dvdsr 17067  df-unit 17068  df-invr 17098  df-dvr 17109  df-drng 17174  df-lmod 17290  df-lss 17355  df-lsp 17394  df-lvec 17525  df-lsatoms 33648  df-oposet 33848  df-ol 33850  df-oml 33851  df-covers 33938  df-ats 33939  df-atl 33970  df-cvlat 33994  df-hlat 34023  df-llines 34169  df-lplanes 34170  df-lvols 34171  df-lines 34172  df-psubsp 34174  df-pmap 34175  df-padd 34467  df-lhyp 34659  df-laut 34660  df-ldil 34775  df-ltrn 34776  df-trl 34830  df-tendo 35426  df-edring 35428  df-disoa 35701  df-dvech 35751  df-dib 35811  df-dic 35845  df-dih 35901  df-doch 36020
This theorem is referenced by:  dochsscl  36040  dochsat  36055  dochshpncl  36056  dochlkr  36057  dochdmj1  36062  dochnoncon  36063
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