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Theorem hdmap1l6d 35091
Description: Lemmma for hdmap1l6 35099. (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 34869 . . . . 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 34386 . . . . . . . . 9  |-  ( ph  ->  U  e.  LVec )
16 hdmap1l6d.w . . . . . . . . . 10  |-  ( ph  ->  w  e.  ( V 
\  {  .0.  }
) )
1716eldifad 3454 . . . . . . . . 9  |-  ( ph  ->  w  e.  V )
18 hdmap1l6cl.x . . . . . . . . . 10  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
1918eldifad 3454 . . . . . . . . 9  |-  ( ph  ->  X  e.  V )
20 hdmap1l6d.y . . . . . . . . . 10  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
2120eldifad 3454 . . . . . . . . 9  |-  ( ph  ->  Y  e.  V )
22 hdmap1l6d.wn . . . . . . . . 9  |-  ( ph  ->  -.  w  e.  ( N `  { X ,  Y } ) )
236, 8, 15, 17, 19, 21, 22lspindpi 18290 . . . . . . . 8  |-  ( ph  ->  ( ( N `  { w } )  =/=  ( N `  { X } )  /\  ( N `  { w } )  =/=  ( N `  { Y } ) ) )
2423simpld 460 . . . . . . 7  |-  ( ph  ->  ( N `  {
w } )  =/=  ( N `  { X } ) )
2524necomd 2702 . . . . . 6  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { w } ) )
261, 5, 6, 7, 8, 2, 9, 10, 11, 12, 3, 13, 14, 25, 18, 17hdmap1cl 35082 . . . . 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 18057 . . . . 5  |-  ( ( C  e.  LMod  /\  (
I `  <. X ,  F ,  w >. )  e.  D )  -> 
( ( I `  <. X ,  F ,  w >. )  .+b  Q
)  =  ( I `
 <. X ,  F ,  w >. ) )
304, 26, 29syl2anc 665 . . . 4  |-  ( ph  ->  ( ( I `  <. X ,  F ,  w >. )  .+b  Q
)  =  ( I `
 <. X ,  F ,  w >. ) )
3130adantr 466 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( (
I `  <. X ,  F ,  w >. ) 
.+b  Q )  =  ( I `  <. X ,  F ,  w >. ) )
32 oteq3 4201 . . . . . 6  |-  ( ( Y  .+  Z )  =  .0.  ->  <. X ,  F ,  ( Y  .+  Z ) >.  =  <. X ,  F ,  .0.  >.
)
3332fveq2d 5885 . . . . 5  |-  ( ( Y  .+  Z )  =  .0.  ->  (
I `  <. X ,  F ,  ( Y  .+  Z ) >. )  =  ( I `  <. X ,  F ,  .0.  >. ) )
341, 5, 6, 7, 2, 9, 28, 12, 3, 13, 19hdmap1val0 35077 . . . . 5  |-  ( ph  ->  ( I `  <. X ,  F ,  .0.  >.
)  =  Q )
3533, 34sylan9eqr 2492 . . . 4  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )  =  Q )
3635oveq2d 6321 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( (
I `  <. X ,  F ,  w >. ) 
.+b  ( I `  <. X ,  F , 
( Y  .+  Z
) >. ) )  =  ( ( I `  <. X ,  F ,  w >. )  .+b  Q
) )
37 oveq2 6313 . . . . . 6  |-  ( ( Y  .+  Z )  =  .0.  ->  (
w  .+  ( Y  .+  Z ) )  =  ( w  .+  .0.  ) )
381, 5, 3dvhlmod 34387 . . . . . . 7  |-  ( ph  ->  U  e.  LMod )
39 hdmap1l6.p . . . . . . . 8  |-  .+  =  ( +g  `  U )
406, 39, 7lmod0vrid 18057 . . . . . . 7  |-  ( ( U  e.  LMod  /\  w  e.  V )  ->  (
w  .+  .0.  )  =  w )
4138, 17, 40syl2anc 665 . . . . . 6  |-  ( ph  ->  ( w  .+  .0.  )  =  w )
4237, 41sylan9eqr 2492 . . . . 5  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( w  .+  ( Y  .+  Z
) )  =  w )
4342oteq3d 4204 . . . 4  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  <. X ,  F ,  ( w  .+  ( Y  .+  Z
) ) >.  =  <. X ,  F ,  w >. )
4443fveq2d 5885 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( I `  <. X ,  F ,  ( w  .+  ( Y  .+  Z ) ) >. )  =  ( I `  <. X ,  F ,  w >. ) )
4531, 36, 443eqtr4rd 2481 . 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 466 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( K  e.  HL  /\  W  e.  H ) )
4913adantr 466 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  F  e.  D
)
5018adantr 466 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  X  e.  ( V  \  {  .0.  } ) )
5114adantr 466 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( M `  ( N `  { X } ) )  =  ( L `  { F } ) )
5216adantr 466 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  w  e.  ( V  \  {  .0.  } ) )
53 hdmap1l6d.z . . . . . . 7  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
5453eldifad 3454 . . . . . 6  |-  ( ph  ->  Z  e.  V )
556, 39lmodvacl 18040 . . . . . 6  |-  ( ( U  e.  LMod  /\  Y  e.  V  /\  Z  e.  V )  ->  ( Y  .+  Z )  e.  V )
5638, 21, 54, 55syl3anc 1264 . . . . 5  |-  ( ph  ->  ( Y  .+  Z
)  e.  V )
5756anim1i 570 . . . 4  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( ( Y 
.+  Z )  e.  V  /\  ( Y 
.+  Z )  =/= 
.0.  ) )
58 eldifsn 4128 . . . 4  |-  ( ( Y  .+  Z )  e.  ( V  \  {  .0.  } )  <->  ( ( Y  .+  Z )  e.  V  /\  ( Y 
.+  Z )  =/= 
.0.  ) )
5957, 58sylibr 215 . . 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 18290 . . . . . . . 8  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  /\  ( N `  { X } )  =/=  ( N `  { Z } ) ) )
6362simpld 460 . . . . . . 7  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
646, 39, 7, 8, 15, 18, 20, 53, 16, 60, 63, 22mapdindp1 34997 . . . . . 6  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { ( Y  .+  Z ) } ) )
656, 39, 7, 8, 15, 18, 20, 53, 16, 60, 63, 22mapdindp2 34998 . . . . . 6  |-  ( ph  ->  -.  w  e.  ( N `  { X ,  ( Y  .+  Z ) } ) )
666, 7, 8, 15, 18, 56, 17, 64, 65lspindp1 18291 . . . . 5  |-  ( ph  ->  ( ( N `  { w } )  =/=  ( N `  { ( Y  .+  Z ) } )  /\  -.  X  e.  ( N `  {
w ,  ( Y 
.+  Z ) } ) ) )
6766simprd 464 . . . 4  |-  ( ph  ->  -.  X  e.  ( N `  { w ,  ( Y  .+  Z ) } ) )
6867adantr 466 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  -.  X  e.  ( N `  { w ,  ( Y  .+  Z ) } ) )
6923simprd 464 . . . . . . . . 9  |-  ( ph  ->  ( N `  {
w } )  =/=  ( N `  { Y } ) )
706, 7, 8, 15, 16, 21, 69lspsnne1 18275 . . . . . . . 8  |-  ( ph  ->  -.  w  e.  ( N `  { Y } ) )
71 eqid 2429 . . . . . . . . . 10  |-  ( LSSum `  U )  =  (
LSSum `  U )
726, 8, 71, 38, 21, 54lsmpr 18247 . . . . . . . . 9  |-  ( ph  ->  ( N `  { Y ,  Z }
)  =  ( ( N `  { Y } ) ( LSSum `  U ) ( N `
 { Z }
) ) )
7360oveq2d 6321 . . . . . . . . 9  |-  ( ph  ->  ( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Y } ) )  =  ( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Z } ) ) )
74 eqid 2429 . . . . . . . . . . . . 13  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
756, 74, 8lspsncl 18135 . . . . . . . . . . . 12  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
7638, 21, 75syl2anc 665 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
7774lsssubg 18115 . . . . . . . . . . 11  |-  ( ( U  e.  LMod  /\  ( N `  { Y } )  e.  (
LSubSp `  U ) )  ->  ( N `  { Y } )  e.  (SubGrp `  U )
)
7838, 76, 77syl2anc 665 . . . . . . . . . 10  |-  ( ph  ->  ( N `  { Y } )  e.  (SubGrp `  U ) )
7971lsmidm 17249 . . . . . . . . . 10  |-  ( ( N `  { Y } )  e.  (SubGrp `  U )  ->  (
( N `  { Y } ) ( LSSum `  U ) ( N `
 { Y }
) )  =  ( N `  { Y } ) )
8078, 79syl 17 . . . . . . . . 9  |-  ( ph  ->  ( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Y } ) )  =  ( N `  { Y } ) )
8172, 73, 803eqtr2d 2476 . . . . . . . 8  |-  ( ph  ->  ( N `  { Y ,  Z }
)  =  ( N `
 { Y }
) )
8270, 81neleqtrrd 2542 . . . . . . 7  |-  ( ph  ->  -.  w  e.  ( N `  { Y ,  Z } ) )
836, 39, 8, 38, 21, 54, 17, 82lspindp4 18295 . . . . . 6  |-  ( ph  ->  -.  w  e.  ( N `  { Y ,  ( Y  .+  Z ) } ) )
846, 8, 15, 17, 21, 56, 83lspindpi 18290 . . . . 5  |-  ( ph  ->  ( ( N `  { w } )  =/=  ( N `  { Y } )  /\  ( N `  { w } )  =/=  ( N `  { ( Y  .+  Z ) } ) ) )
8584simprd 464 . . . 4  |-  ( ph  ->  ( N `  {
w } )  =/=  ( N `  {
( Y  .+  Z
) } ) )
8685adantr 466 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( N `  { w } )  =/=  ( N `  { ( Y  .+  Z ) } ) )
87 eqidd 2430 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( I `  <. X ,  F ,  w >. )  =  ( I `  <. X ,  F ,  w >. ) )
88 eqidd 2430 . . 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 35087 . 2  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( I `  <. X ,  F , 
( w  .+  ( Y  .+  Z ) )
>. )  =  (
( I `  <. X ,  F ,  w >. )  .+b  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )
) )
9045, 89pm2.61dane 2749 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 370    = wceq 1437    e. wcel 1870    =/= wne 2625    \ cdif 3439   {csn 4002   {cpr 4004   <.cotp 4010   ` cfv 5601  (class class class)co 6305   Basecbs 15084   +g cplusg 15152   0gc0g 15297   -gcsg 16622  SubGrpcsubg 16762   LSSumclsm 17221   LModclmod 18026   LSubSpclss 18090   LSpanclspn 18129   HLchlt 32625   LHypclh 33258   DVecHcdvh 34355  LCDualclcd 34863  mapdcmpd 34901  HDMap1chdma1 35069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-riotaBAD 32234
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 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-ot 4011  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-tpos 6981  df-undef 7028  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11783  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-sca 15168  df-vsca 15169  df-0g 15299  df-mre 15443  df-mrc 15444  df-acs 15446  df-preset 16124  df-poset 16142  df-plt 16155  df-lub 16171  df-glb 16172  df-join 16173  df-meet 16174  df-p0 16236  df-p1 16237  df-lat 16243  df-clat 16305  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-grp 16624  df-minusg 16625  df-sbg 16626  df-subg 16765  df-cntz 16922  df-oppg 16948  df-lsm 17223  df-cmn 17367  df-abl 17368  df-mgp 17659  df-ur 17671  df-ring 17717  df-oppr 17786  df-dvdsr 17804  df-unit 17805  df-invr 17835  df-dvr 17846  df-drng 17912  df-lmod 18028  df-lss 18091  df-lsp 18130  df-lvec 18261  df-lsatoms 32251  df-lshyp 32252  df-lcv 32294  df-lfl 32333  df-lkr 32361  df-ldual 32399  df-oposet 32451  df-ol 32453  df-oml 32454  df-covers 32541  df-ats 32542  df-atl 32573  df-cvlat 32597  df-hlat 32626  df-llines 32772  df-lplanes 32773  df-lvols 32774  df-lines 32775  df-psubsp 32777  df-pmap 32778  df-padd 33070  df-lhyp 33262  df-laut 33263  df-ldil 33378  df-ltrn 33379  df-trl 33434  df-tgrp 34019  df-tendo 34031  df-edring 34033  df-dveca 34279  df-disoa 34306  df-dvech 34356  df-dib 34416  df-dic 34450  df-dih 34506  df-doch 34625  df-djh 34672  df-lcdual 34864  df-mapd 34902  df-hdmap1 35071
This theorem is referenced by:  hdmap1l6g  35094
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