Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hdmap1l6d Structured version   Unicode version

Theorem hdmap1l6d 36611
Description: Lemmma for hdmap1l6 36619. (Contributed by NM, 1-May-2015.)
Hypotheses
Ref Expression
hdmap1l6.h  |-  H  =  ( LHyp `  K
)
hdmap1l6.u  |-  U  =  ( ( DVecH `  K
) `  W )
hdmap1l6.v  |-  V  =  ( Base `  U
)
hdmap1l6.p  |-  .+  =  ( +g  `  U )
hdmap1l6.s  |-  .-  =  ( -g `  U )
hdmap1l6c.o  |-  .0.  =  ( 0g `  U )
hdmap1l6.n  |-  N  =  ( LSpan `  U )
hdmap1l6.c  |-  C  =  ( (LCDual `  K
) `  W )
hdmap1l6.d  |-  D  =  ( Base `  C
)
hdmap1l6.a  |-  .+b  =  ( +g  `  C )
hdmap1l6.r  |-  R  =  ( -g `  C
)
hdmap1l6.q  |-  Q  =  ( 0g `  C
)
hdmap1l6.l  |-  L  =  ( LSpan `  C )
hdmap1l6.m  |-  M  =  ( (mapd `  K
) `  W )
hdmap1l6.i  |-  I  =  ( (HDMap1 `  K
) `  W )
hdmap1l6.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
hdmap1l6.f  |-  ( ph  ->  F  e.  D )
hdmap1l6cl.x  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
hdmap1l6.mn  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( L `  { F } ) )
hdmap1l6d.xn  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
hdmap1l6d.yz  |-  ( ph  ->  ( N `  { Y } )  =  ( N `  { Z } ) )
hdmap1l6d.y  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
hdmap1l6d.z  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
hdmap1l6d.w  |-  ( ph  ->  w  e.  ( V 
\  {  .0.  }
) )
hdmap1l6d.wn  |-  ( ph  ->  -.  w  e.  ( N `  { X ,  Y } ) )
Assertion
Ref Expression
hdmap1l6d  |-  ( ph  ->  ( I `  <. X ,  F ,  ( w  .+  ( Y 
.+  Z ) )
>. )  =  (
( I `  <. X ,  F ,  w >. )  .+b  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )
) )

Proof of Theorem hdmap1l6d
StepHypRef Expression
1 hdmap1l6.h . . . . . 6  |-  H  =  ( LHyp `  K
)
2 hdmap1l6.c . . . . . 6  |-  C  =  ( (LCDual `  K
) `  W )
3 hdmap1l6.k . . . . . 6  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
41, 2, 3lcdlmod 36389 . . . . 5  |-  ( ph  ->  C  e.  LMod )
5 hdmap1l6.u . . . . . 6  |-  U  =  ( ( DVecH `  K
) `  W )
6 hdmap1l6.v . . . . . 6  |-  V  =  ( Base `  U
)
7 hdmap1l6c.o . . . . . 6  |-  .0.  =  ( 0g `  U )
8 hdmap1l6.n . . . . . 6  |-  N  =  ( LSpan `  U )
9 hdmap1l6.d . . . . . 6  |-  D  =  ( Base `  C
)
10 hdmap1l6.l . . . . . 6  |-  L  =  ( LSpan `  C )
11 hdmap1l6.m . . . . . 6  |-  M  =  ( (mapd `  K
) `  W )
12 hdmap1l6.i . . . . . 6  |-  I  =  ( (HDMap1 `  K
) `  W )
13 hdmap1l6.f . . . . . 6  |-  ( ph  ->  F  e.  D )
14 hdmap1l6.mn . . . . . 6  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( L `  { F } ) )
151, 5, 3dvhlvec 35906 . . . . . . . . 9  |-  ( ph  ->  U  e.  LVec )
16 hdmap1l6d.w . . . . . . . . . 10  |-  ( ph  ->  w  e.  ( V 
\  {  .0.  }
) )
1716eldifad 3488 . . . . . . . . 9  |-  ( ph  ->  w  e.  V )
18 hdmap1l6cl.x . . . . . . . . . 10  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
1918eldifad 3488 . . . . . . . . 9  |-  ( ph  ->  X  e.  V )
20 hdmap1l6d.y . . . . . . . . . 10  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
2120eldifad 3488 . . . . . . . . 9  |-  ( ph  ->  Y  e.  V )
22 hdmap1l6d.wn . . . . . . . . 9  |-  ( ph  ->  -.  w  e.  ( N `  { X ,  Y } ) )
236, 8, 15, 17, 19, 21, 22lspindpi 17561 . . . . . . . 8  |-  ( ph  ->  ( ( N `  { w } )  =/=  ( N `  { X } )  /\  ( N `  { w } )  =/=  ( N `  { Y } ) ) )
2423simpld 459 . . . . . . 7  |-  ( ph  ->  ( N `  {
w } )  =/=  ( N `  { X } ) )
2524necomd 2738 . . . . . 6  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { w } ) )
261, 5, 6, 7, 8, 2, 9, 10, 11, 12, 3, 13, 14, 25, 18, 17hdmap1cl 36602 . . . . 5  |-  ( ph  ->  ( I `  <. X ,  F ,  w >. )  e.  D )
27 hdmap1l6.a . . . . . 6  |-  .+b  =  ( +g  `  C )
28 hdmap1l6.q . . . . . 6  |-  Q  =  ( 0g `  C
)
299, 27, 28lmod0vrid 17326 . . . . 5  |-  ( ( C  e.  LMod  /\  (
I `  <. X ,  F ,  w >. )  e.  D )  -> 
( ( I `  <. X ,  F ,  w >. )  .+b  Q
)  =  ( I `
 <. X ,  F ,  w >. ) )
304, 26, 29syl2anc 661 . . . 4  |-  ( ph  ->  ( ( I `  <. X ,  F ,  w >. )  .+b  Q
)  =  ( I `
 <. X ,  F ,  w >. ) )
3130adantr 465 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( (
I `  <. X ,  F ,  w >. ) 
.+b  Q )  =  ( I `  <. X ,  F ,  w >. ) )
32 oteq3 4224 . . . . . 6  |-  ( ( Y  .+  Z )  =  .0.  ->  <. X ,  F ,  ( Y  .+  Z ) >.  =  <. X ,  F ,  .0.  >.
)
3332fveq2d 5868 . . . . 5  |-  ( ( Y  .+  Z )  =  .0.  ->  (
I `  <. X ,  F ,  ( Y  .+  Z ) >. )  =  ( I `  <. X ,  F ,  .0.  >. ) )
341, 5, 6, 7, 2, 9, 28, 12, 3, 13, 19hdmap1val0 36597 . . . . 5  |-  ( ph  ->  ( I `  <. X ,  F ,  .0.  >.
)  =  Q )
3533, 34sylan9eqr 2530 . . . 4  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )  =  Q )
3635oveq2d 6298 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( (
I `  <. X ,  F ,  w >. ) 
.+b  ( I `  <. X ,  F , 
( Y  .+  Z
) >. ) )  =  ( ( I `  <. X ,  F ,  w >. )  .+b  Q
) )
37 oveq2 6290 . . . . . 6  |-  ( ( Y  .+  Z )  =  .0.  ->  (
w  .+  ( Y  .+  Z ) )  =  ( w  .+  .0.  ) )
381, 5, 3dvhlmod 35907 . . . . . . 7  |-  ( ph  ->  U  e.  LMod )
39 hdmap1l6.p . . . . . . . 8  |-  .+  =  ( +g  `  U )
406, 39, 7lmod0vrid 17326 . . . . . . 7  |-  ( ( U  e.  LMod  /\  w  e.  V )  ->  (
w  .+  .0.  )  =  w )
4138, 17, 40syl2anc 661 . . . . . 6  |-  ( ph  ->  ( w  .+  .0.  )  =  w )
4237, 41sylan9eqr 2530 . . . . 5  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( w  .+  ( Y  .+  Z
) )  =  w )
4342oteq3d 4227 . . . 4  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  <. X ,  F ,  ( w  .+  ( Y  .+  Z
) ) >.  =  <. X ,  F ,  w >. )
4443fveq2d 5868 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( I `  <. X ,  F ,  ( w  .+  ( Y  .+  Z ) ) >. )  =  ( I `  <. X ,  F ,  w >. ) )
4531, 36, 443eqtr4rd 2519 . 2  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( I `  <. X ,  F ,  ( w  .+  ( Y  .+  Z ) ) >. )  =  ( ( I `  <. X ,  F ,  w >. )  .+b  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )
) )
46 hdmap1l6.s . . 3  |-  .-  =  ( -g `  U )
47 hdmap1l6.r . . 3  |-  R  =  ( -g `  C
)
483adantr 465 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( K  e.  HL  /\  W  e.  H ) )
4913adantr 465 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  F  e.  D
)
5018adantr 465 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  X  e.  ( V  \  {  .0.  } ) )
5114adantr 465 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( M `  ( N `  { X } ) )  =  ( L `  { F } ) )
5216adantr 465 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  w  e.  ( V  \  {  .0.  } ) )
53 hdmap1l6d.z . . . . . . 7  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
5453eldifad 3488 . . . . . 6  |-  ( ph  ->  Z  e.  V )
556, 39lmodvacl 17309 . . . . . 6  |-  ( ( U  e.  LMod  /\  Y  e.  V  /\  Z  e.  V )  ->  ( Y  .+  Z )  e.  V )
5638, 21, 54, 55syl3anc 1228 . . . . 5  |-  ( ph  ->  ( Y  .+  Z
)  e.  V )
5756anim1i 568 . . . 4  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( ( Y 
.+  Z )  e.  V  /\  ( Y 
.+  Z )  =/= 
.0.  ) )
58 eldifsn 4152 . . . 4  |-  ( ( Y  .+  Z )  e.  ( V  \  {  .0.  } )  <->  ( ( Y  .+  Z )  e.  V  /\  ( Y 
.+  Z )  =/= 
.0.  ) )
5957, 58sylibr 212 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( Y  .+  Z )  e.  ( V  \  {  .0.  } ) )
60 hdmap1l6d.yz . . . . . . 7  |-  ( ph  ->  ( N `  { Y } )  =  ( N `  { Z } ) )
61 hdmap1l6d.xn . . . . . . . . 9  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
626, 8, 15, 19, 21, 54, 61lspindpi 17561 . . . . . . . 8  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  /\  ( N `  { X } )  =/=  ( N `  { Z } ) ) )
6362simpld 459 . . . . . . 7  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
646, 39, 7, 8, 15, 18, 20, 53, 16, 60, 63, 22mapdindp1 36517 . . . . . 6  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { ( Y  .+  Z ) } ) )
656, 39, 7, 8, 15, 18, 20, 53, 16, 60, 63, 22mapdindp2 36518 . . . . . 6  |-  ( ph  ->  -.  w  e.  ( N `  { X ,  ( Y  .+  Z ) } ) )
666, 7, 8, 15, 18, 56, 17, 64, 65lspindp1 17562 . . . . 5  |-  ( ph  ->  ( ( N `  { w } )  =/=  ( N `  { ( Y  .+  Z ) } )  /\  -.  X  e.  ( N `  {
w ,  ( Y 
.+  Z ) } ) ) )
6766simprd 463 . . . 4  |-  ( ph  ->  -.  X  e.  ( N `  { w ,  ( Y  .+  Z ) } ) )
6867adantr 465 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  -.  X  e.  ( N `  { w ,  ( Y  .+  Z ) } ) )
6923simprd 463 . . . . . . . . 9  |-  ( ph  ->  ( N `  {
w } )  =/=  ( N `  { Y } ) )
706, 7, 8, 15, 16, 21, 69lspsnne1 17546 . . . . . . . 8  |-  ( ph  ->  -.  w  e.  ( N `  { Y } ) )
71 eqid 2467 . . . . . . . . . 10  |-  ( LSSum `  U )  =  (
LSSum `  U )
726, 8, 71, 38, 21, 54lsmpr 17518 . . . . . . . . 9  |-  ( ph  ->  ( N `  { Y ,  Z }
)  =  ( ( N `  { Y } ) ( LSSum `  U ) ( N `
 { Z }
) ) )
7360oveq2d 6298 . . . . . . . . 9  |-  ( ph  ->  ( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Y } ) )  =  ( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Z } ) ) )
74 eqid 2467 . . . . . . . . . . . . 13  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
756, 74, 8lspsncl 17406 . . . . . . . . . . . 12  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
7638, 21, 75syl2anc 661 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
7774lsssubg 17386 . . . . . . . . . . 11  |-  ( ( U  e.  LMod  /\  ( N `  { Y } )  e.  (
LSubSp `  U ) )  ->  ( N `  { Y } )  e.  (SubGrp `  U )
)
7838, 76, 77syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( N `  { Y } )  e.  (SubGrp `  U ) )
7971lsmidm 16478 . . . . . . . . . 10  |-  ( ( N `  { Y } )  e.  (SubGrp `  U )  ->  (
( N `  { Y } ) ( LSSum `  U ) ( N `
 { Y }
) )  =  ( N `  { Y } ) )
8078, 79syl 16 . . . . . . . . 9  |-  ( ph  ->  ( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Y } ) )  =  ( N `  { Y } ) )
8172, 73, 803eqtr2d 2514 . . . . . . . 8  |-  ( ph  ->  ( N `  { Y ,  Z }
)  =  ( N `
 { Y }
) )
8270, 81neleqtrrd 2580 . . . . . . 7  |-  ( ph  ->  -.  w  e.  ( N `  { Y ,  Z } ) )
836, 39, 8, 38, 21, 54, 17, 82lspindp4 17566 . . . . . 6  |-  ( ph  ->  -.  w  e.  ( N `  { Y ,  ( Y  .+  Z ) } ) )
846, 8, 15, 17, 21, 56, 83lspindpi 17561 . . . . 5  |-  ( ph  ->  ( ( N `  { w } )  =/=  ( N `  { Y } )  /\  ( N `  { w } )  =/=  ( N `  { ( Y  .+  Z ) } ) ) )
8584simprd 463 . . . 4  |-  ( ph  ->  ( N `  {
w } )  =/=  ( N `  {
( Y  .+  Z
) } ) )
8685adantr 465 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( N `  { w } )  =/=  ( N `  { ( Y  .+  Z ) } ) )
87 eqidd 2468 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( I `  <. X ,  F ,  w >. )  =  ( I `  <. X ,  F ,  w >. ) )
88 eqidd 2468 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( I `  <. X ,  F , 
( Y  .+  Z
) >. )  =  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )
)
891, 5, 6, 39, 46, 7, 8, 2, 9, 27, 47, 28, 10, 11, 12, 48, 49, 50, 51, 52, 59, 68, 86, 87, 88hdmap1l6a 36607 . 2  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( I `  <. X ,  F , 
( w  .+  ( Y  .+  Z ) )
>. )  =  (
( I `  <. X ,  F ,  w >. )  .+b  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )
) )
9045, 89pm2.61dane 2785 1  |-  ( ph  ->  ( I `  <. X ,  F ,  ( w  .+  ( Y 
.+  Z ) )
>. )  =  (
( I `  <. X ,  F ,  w >. )  .+b  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662    \ cdif 3473   {csn 4027   {cpr 4029   <.cotp 4035   ` cfv 5586  (class class class)co 6282   Basecbs 14486   +g cplusg 14551   0gc0g 14691   -gcsg 15726  SubGrpcsubg 15990   LSSumclsm 16450   LModclmod 17295   LSubSpclss 17361   LSpanclspn 17400   HLchlt 34147   LHypclh 34780   DVecHcdvh 35875  LCDualclcd 36383  mapdcmpd 36421  HDMap1chdma1 36589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-riotaBAD 33756
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-ot 4036  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-tpos 6952  df-undef 6999  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-sca 14567  df-vsca 14568  df-0g 14693  df-mre 14837  df-mrc 14838  df-acs 14840  df-poset 15429  df-plt 15441  df-lub 15457  df-glb 15458  df-join 15459  df-meet 15460  df-p0 15522  df-p1 15523  df-lat 15529  df-clat 15591  df-mnd 15728  df-submnd 15778  df-grp 15858  df-minusg 15859  df-sbg 15860  df-subg 15993  df-cntz 16150  df-oppg 16176  df-lsm 16452  df-cmn 16596  df-abl 16597  df-mgp 16932  df-ur 16944  df-rng 16988  df-oppr 17056  df-dvdsr 17074  df-unit 17075  df-invr 17105  df-dvr 17116  df-drng 17181  df-lmod 17297  df-lss 17362  df-lsp 17401  df-lvec 17532  df-lsatoms 33773  df-lshyp 33774  df-lcv 33816  df-lfl 33855  df-lkr 33883  df-ldual 33921  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-llines 34294  df-lplanes 34295  df-lvols 34296  df-lines 34297  df-psubsp 34299  df-pmap 34300  df-padd 34592  df-lhyp 34784  df-laut 34785  df-ldil 34900  df-ltrn 34901  df-trl 34955  df-tgrp 35539  df-tendo 35551  df-edring 35553  df-dveca 35799  df-disoa 35826  df-dvech 35876  df-dib 35936  df-dic 35970  df-dih 36026  df-doch 36145  df-djh 36192  df-lcdual 36384  df-mapd 36422  df-hdmap1 36591
This theorem is referenced by:  hdmap1l6g  36614
  Copyright terms: Public domain W3C validator