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

Theorem dochss 34845
Description: Subset law for orthocomplement. (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
dochss  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  (  ._|_  `  Y
)  C_  (  ._|_  `  X ) )

Proof of Theorem dochss
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 simp1l 1029 . . . . . 6  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  K  e.  HL )
2 hlclat 32836 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  CLat )
31, 2syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  K  e.  CLat )
4 ssrab2 3489 . . . . . 6  |-  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) }  C_  ( Base `  K
)
54a1i 11 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) }  C_  ( Base `  K
) )
6 simpll3 1046 . . . . . . . 8  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  /\  z  e.  ( Base `  K
) )  /\  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z )
)  ->  X  C_  Y
)
7 simpr 462 . . . . . . . 8  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  /\  z  e.  ( Base `  K
) )  /\  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z )
)  ->  Y  C_  (
( ( DIsoH `  K
) `  W ) `  z ) )
86, 7sstrd 3417 . . . . . . 7  |-  ( ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  /\  z  e.  ( Base `  K
) )  /\  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z )
)  ->  X  C_  (
( ( DIsoH `  K
) `  W ) `  z ) )
98ex 435 . . . . . 6  |-  ( ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  /\  z  e.  ( Base `  K
) )  ->  ( Y  C_  ( ( (
DIsoH `  K ) `  W ) `  z
)  ->  X  C_  (
( ( DIsoH `  K
) `  W ) `  z ) ) )
109ss2rabdv 3485 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) }  C_  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } )
11 eqid 2428 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
12 eqid 2428 . . . . . 6  |-  ( le
`  K )  =  ( le `  K
)
13 eqid 2428 . . . . . 6  |-  ( glb `  K )  =  ( glb `  K )
1411, 12, 13clatglbss 16316 . . . . 5  |-  ( ( K  e.  CLat  /\  {
z  e.  ( Base `  K )  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) }  C_  ( Base `  K
)  /\  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) }  C_  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } )  ->  (
( glb `  K
) `  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ( le
`  K ) ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) )
153, 5, 10, 14syl3anc 1264 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  ( ( glb `  K ) `  {
z  e.  ( Base `  K )  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ( le
`  K ) ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) )
16 hlop 32840 . . . . . 6  |-  ( K  e.  HL  ->  K  e.  OP )
171, 16syl 17 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  K  e.  OP )
1811, 13clatglbcl 16303 . . . . . 6  |-  ( ( K  e.  CLat  /\  {
z  e.  ( Base `  K )  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) }  C_  ( Base `  K
) )  ->  (
( glb `  K
) `  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } )  e.  (
Base `  K )
)
193, 4, 18sylancl 666 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  ( ( glb `  K ) `  {
z  e.  ( Base `  K )  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } )  e.  (
Base `  K )
)
20 ssrab2 3489 . . . . . 6  |-  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) }  C_  ( Base `  K
)
2111, 13clatglbcl 16303 . . . . . 6  |-  ( ( K  e.  CLat  /\  {
z  e.  ( Base `  K )  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) }  C_  ( Base `  K
) )  ->  (
( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } )  e.  (
Base `  K )
)
223, 20, 21sylancl 666 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  ( ( glb `  K ) `  {
z  e.  ( Base `  K )  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } )  e.  (
Base `  K )
)
23 eqid 2428 . . . . . 6  |-  ( oc
`  K )  =  ( oc `  K
)
2411, 12, 23oplecon3b 32678 . . . . 5  |-  ( ( K  e.  OP  /\  ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } )  e.  (
Base `  K )  /\  ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } )  e.  (
Base `  K )
)  ->  ( (
( glb `  K
) `  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ( le
`  K ) ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } )  <->  ( ( oc `  K ) `  ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ) ( le `  K ) ( ( oc `  K ) `  (
( glb `  K
) `  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ) ) )
2517, 19, 22, 24syl3anc 1264 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  ( ( ( glb `  K ) `
 { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ( le
`  K ) ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } )  <->  ( ( oc `  K ) `  ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ) ( le `  K ) ( ( oc `  K ) `  (
( glb `  K
) `  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ) ) )
2615, 25mpbid 213 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  ( ( oc
`  K ) `  ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ) ( le `  K ) ( ( oc `  K ) `  (
( glb `  K
) `  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ) )
27 simp1 1005 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  ( K  e.  HL  /\  W  e.  H ) )
2811, 23opoccl 32672 . . . . 5  |-  ( ( K  e.  OP  /\  ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } )  e.  (
Base `  K )
)  ->  ( ( oc `  K ) `  ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) )  e.  ( Base `  K
) )
2917, 22, 28syl2anc 665 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  ( ( oc
`  K ) `  ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) )  e.  ( Base `  K
) )
3011, 23opoccl 32672 . . . . 5  |-  ( ( K  e.  OP  /\  ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } )  e.  (
Base `  K )
)  ->  ( ( oc `  K ) `  ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) )  e.  ( Base `  K
) )
3117, 19, 30syl2anc 665 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  ( ( oc
`  K ) `  ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) )  e.  ( Base `  K
) )
32 dochss.h . . . . 5  |-  H  =  ( LHyp `  K
)
33 eqid 2428 . . . . 5  |-  ( (
DIsoH `  K ) `  W )  =  ( ( DIsoH `  K ) `  W )
3411, 12, 32, 33dihord 34744 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( ( oc
`  K ) `  ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) )  e.  ( Base `  K
)  /\  ( ( oc `  K ) `  ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) )  e.  ( Base `  K
) )  ->  (
( ( ( DIsoH `  K ) `  W
) `  ( ( oc `  K ) `  ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ) ) 
C_  ( ( (
DIsoH `  K ) `  W ) `  (
( oc `  K
) `  ( ( glb `  K ) `  { z  e.  (
Base `  K )  |  X  C_  ( ( ( DIsoH `  K ) `  W ) `  z
) } ) ) )  <->  ( ( oc
`  K ) `  ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ) ( le `  K ) ( ( oc `  K ) `  (
( glb `  K
) `  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ) ) )
3527, 29, 31, 34syl3anc 1264 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  ( ( ( ( DIsoH `  K ) `  W ) `  (
( oc `  K
) `  ( ( glb `  K ) `  { z  e.  (
Base `  K )  |  Y  C_  ( ( ( DIsoH `  K ) `  W ) `  z
) } ) ) )  C_  ( (
( DIsoH `  K ) `  W ) `  (
( oc `  K
) `  ( ( glb `  K ) `  { z  e.  (
Base `  K )  |  X  C_  ( ( ( DIsoH `  K ) `  W ) `  z
) } ) ) )  <->  ( ( oc
`  K ) `  ( ( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ) ( le `  K ) ( ( oc `  K ) `  (
( glb `  K
) `  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ) ) )
3626, 35mpbird 235 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  ( ( (
DIsoH `  K ) `  W ) `  (
( oc `  K
) `  ( ( glb `  K ) `  { z  e.  (
Base `  K )  |  Y  C_  ( ( ( DIsoH `  K ) `  W ) `  z
) } ) ) )  C_  ( (
( DIsoH `  K ) `  W ) `  (
( oc `  K
) `  ( ( glb `  K ) `  { z  e.  (
Base `  K )  |  X  C_  ( ( ( DIsoH `  K ) `  W ) `  z
) } ) ) ) )
37 dochss.u . . . 4  |-  U  =  ( ( DVecH `  K
) `  W )
38 dochss.v . . . 4  |-  V  =  ( Base `  U
)
39 dochss.o . . . 4  |-  ._|_  =  ( ( ocH `  K
) `  W )
4011, 13, 23, 32, 33, 37, 38, 39dochval 34831 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V
)  ->  (  ._|_  `  Y )  =  ( ( ( DIsoH `  K
) `  W ) `  ( ( oc `  K ) `  (
( glb `  K
) `  { z  e.  ( Base `  K
)  |  Y  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ) ) )
41403adant3 1025 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  (  ._|_  `  Y
)  =  ( ( ( DIsoH `  K ) `  W ) `  (
( oc `  K
) `  ( ( glb `  K ) `  { z  e.  (
Base `  K )  |  Y  C_  ( ( ( DIsoH `  K ) `  W ) `  z
) } ) ) ) )
42 simp3 1007 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  X  C_  Y
)
43 simp2 1006 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  Y  C_  V
)
4442, 43sstrd 3417 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  X  C_  V
)
4511, 13, 23, 32, 33, 37, 38, 39dochval 34831 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  X  C_  V
)  ->  (  ._|_  `  X )  =  ( ( ( DIsoH `  K
) `  W ) `  ( ( oc `  K ) `  (
( glb `  K
) `  { z  e.  ( Base `  K
)  |  X  C_  ( ( ( DIsoH `  K ) `  W
) `  z ) } ) ) ) )
4627, 44, 45syl2anc 665 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  (  ._|_  `  X
)  =  ( ( ( DIsoH `  K ) `  W ) `  (
( oc `  K
) `  ( ( glb `  K ) `  { z  e.  (
Base `  K )  |  X  C_  ( ( ( DIsoH `  K ) `  W ) `  z
) } ) ) ) )
4736, 41, 463sstr4d 3450 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  Y  C_  V  /\  X  C_  Y )  ->  (  ._|_  `  Y
)  C_  (  ._|_  `  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   {crab 2718    C_ wss 3379   class class class wbr 4366   ` cfv 5544   Basecbs 15064   lecple 15140   occoc 15141   glbcglb 16131   CLatccla 16296   OPcops 32650   HLchlt 32828   LHypclh 33461   DVecHcdvh 34558   DIsoHcdih 34708   ocHcoch 34827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-riotaBAD 32437
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-iin 4245  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-1st 6751  df-2nd 6752  df-tpos 6928  df-undef 6975  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-1o 7137  df-oadd 7141  df-er 7318  df-map 7429  df-en 7525  df-dom 7526  df-sdom 7527  df-fin 7528  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-nn 10561  df-2 10619  df-3 10620  df-4 10621  df-5 10622  df-6 10623  df-n0 10821  df-z 10889  df-uz 11111  df-fz 11736  df-struct 15066  df-ndx 15067  df-slot 15068  df-base 15069  df-sets 15070  df-ress 15071  df-plusg 15146  df-mulr 15147  df-sca 15149  df-vsca 15150  df-0g 15283  df-preset 16116  df-poset 16134  df-plt 16147  df-lub 16163  df-glb 16164  df-join 16165  df-meet 16166  df-p0 16228  df-p1 16229  df-lat 16235  df-clat 16297  df-mgm 16431  df-sgrp 16470  df-mnd 16480  df-submnd 16526  df-grp 16616  df-minusg 16617  df-sbg 16618  df-subg 16757  df-cntz 16914  df-lsm 17231  df-cmn 17375  df-abl 17376  df-mgp 17667  df-ur 17679  df-ring 17725  df-oppr 17794  df-dvdsr 17812  df-unit 17813  df-invr 17843  df-dvr 17854  df-drng 17920  df-lmod 18036  df-lss 18099  df-lsp 18138  df-lvec 18269  df-oposet 32654  df-ol 32656  df-oml 32657  df-covers 32744  df-ats 32745  df-atl 32776  df-cvlat 32800  df-hlat 32829  df-llines 32975  df-lplanes 32976  df-lvols 32977  df-lines 32978  df-psubsp 32980  df-pmap 32981  df-padd 33273  df-lhyp 33465  df-laut 33466  df-ldil 33581  df-ltrn 33582  df-trl 33637  df-tendo 34234  df-edring 34236  df-disoa 34509  df-dvech 34559  df-dib 34619  df-dic 34653  df-dih 34709  df-doch 34828
This theorem is referenced by:  dochsscl  34848  dochord  34850  dihoml4  34857  dochocsp  34859  dochdmj1  34870  dochpolN  34970  lclkrlem2p  35002  lclkrslem1  35017  lclkrslem2  35018  lcfrvalsnN  35021  mapdsn  35121
  Copyright terms: Public domain W3C validator