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Theorem hdmap1l6d 32539
Description: Lemmma for hdmap1l6 32547. (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 32317 . . . . 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 31834 . . . . . . . . 9  |-  ( ph  ->  U  e.  LVec )
16 hdmap1l6d.w . . . . . . . . . 10  |-  ( ph  ->  w  e.  ( V 
\  {  .0.  }
) )
1716eldifad 3324 . . . . . . . . 9  |-  ( ph  ->  w  e.  V )
18 hdmap1l6cl.x . . . . . . . . . 10  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
1918eldifad 3324 . . . . . . . . 9  |-  ( ph  ->  X  e.  V )
20 hdmap1l6d.y . . . . . . . . . 10  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
2120eldifad 3324 . . . . . . . . 9  |-  ( ph  ->  Y  e.  V )
22 hdmap1l6d.wn . . . . . . . . 9  |-  ( ph  ->  -.  w  e.  ( N `  { X ,  Y } ) )
236, 8, 15, 17, 19, 21, 22lspindpi 16196 . . . . . . . 8  |-  ( ph  ->  ( ( N `  { w } )  =/=  ( N `  { X } )  /\  ( N `  { w } )  =/=  ( N `  { Y } ) ) )
2423simpld 446 . . . . . . 7  |-  ( ph  ->  ( N `  {
w } )  =/=  ( N `  { X } ) )
2524necomd 2681 . . . . . 6  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { w } ) )
261, 5, 6, 7, 8, 2, 9, 10, 11, 12, 3, 13, 14, 25, 18, 17hdmap1cl 32530 . . . . 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 15973 . . . . 5  |-  ( ( C  e.  LMod  /\  (
I `  <. X ,  F ,  w >. )  e.  D )  -> 
( ( I `  <. X ,  F ,  w >. )  .+b  Q
)  =  ( I `
 <. X ,  F ,  w >. ) )
304, 26, 29syl2anc 643 . . . 4  |-  ( ph  ->  ( ( I `  <. X ,  F ,  w >. )  .+b  Q
)  =  ( I `
 <. X ,  F ,  w >. ) )
3130adantr 452 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( (
I `  <. X ,  F ,  w >. ) 
.+b  Q )  =  ( I `  <. X ,  F ,  w >. ) )
32 oteq3 3987 . . . . . 6  |-  ( ( Y  .+  Z )  =  .0.  ->  <. X ,  F ,  ( Y  .+  Z ) >.  =  <. X ,  F ,  .0.  >.
)
3332fveq2d 5724 . . . . 5  |-  ( ( Y  .+  Z )  =  .0.  ->  (
I `  <. X ,  F ,  ( Y  .+  Z ) >. )  =  ( I `  <. X ,  F ,  .0.  >. ) )
341, 5, 6, 7, 2, 9, 28, 12, 3, 13, 19hdmap1val0 32525 . . . . 5  |-  ( ph  ->  ( I `  <. X ,  F ,  .0.  >.
)  =  Q )
3533, 34sylan9eqr 2489 . . . 4  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )  =  Q )
3635oveq2d 6089 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( (
I `  <. X ,  F ,  w >. ) 
.+b  ( I `  <. X ,  F , 
( Y  .+  Z
) >. ) )  =  ( ( I `  <. X ,  F ,  w >. )  .+b  Q
) )
37 oveq2 6081 . . . . . 6  |-  ( ( Y  .+  Z )  =  .0.  ->  (
w  .+  ( Y  .+  Z ) )  =  ( w  .+  .0.  ) )
381, 5, 3dvhlmod 31835 . . . . . . 7  |-  ( ph  ->  U  e.  LMod )
39 hdmap1l6.p . . . . . . . 8  |-  .+  =  ( +g  `  U )
406, 39, 7lmod0vrid 15973 . . . . . . 7  |-  ( ( U  e.  LMod  /\  w  e.  V )  ->  (
w  .+  .0.  )  =  w )
4138, 17, 40syl2anc 643 . . . . . 6  |-  ( ph  ->  ( w  .+  .0.  )  =  w )
4237, 41sylan9eqr 2489 . . . . 5  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( w  .+  ( Y  .+  Z
) )  =  w )
4342oteq3d 3990 . . . 4  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  <. X ,  F ,  ( w  .+  ( Y  .+  Z
) ) >.  =  <. X ,  F ,  w >. )
4443fveq2d 5724 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =  .0.  )  ->  ( I `  <. X ,  F ,  ( w  .+  ( Y  .+  Z ) ) >. )  =  ( I `  <. X ,  F ,  w >. ) )
4531, 36, 443eqtr4rd 2478 . 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 452 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( K  e.  HL  /\  W  e.  H ) )
4913adantr 452 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  F  e.  D
)
5018adantr 452 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  X  e.  ( V  \  {  .0.  } ) )
5114adantr 452 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( M `  ( N `  { X } ) )  =  ( L `  { F } ) )
5216adantr 452 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  w  e.  ( V  \  {  .0.  } ) )
53 hdmap1l6d.z . . . . . . 7  |-  ( ph  ->  Z  e.  ( V 
\  {  .0.  }
) )
5453eldifad 3324 . . . . . 6  |-  ( ph  ->  Z  e.  V )
556, 39lmodvacl 15956 . . . . . 6  |-  ( ( U  e.  LMod  /\  Y  e.  V  /\  Z  e.  V )  ->  ( Y  .+  Z )  e.  V )
5638, 21, 54, 55syl3anc 1184 . . . . 5  |-  ( ph  ->  ( Y  .+  Z
)  e.  V )
5756anim1i 552 . . . 4  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( ( Y 
.+  Z )  e.  V  /\  ( Y 
.+  Z )  =/= 
.0.  ) )
58 eldifsn 3919 . . . 4  |-  ( ( Y  .+  Z )  e.  ( V  \  {  .0.  } )  <->  ( ( Y  .+  Z )  e.  V  /\  ( Y 
.+  Z )  =/= 
.0.  ) )
5957, 58sylibr 204 . . 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 16196 . . . . . . . 8  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  /\  ( N `  { X } )  =/=  ( N `  { Z } ) ) )
6362simpld 446 . . . . . . 7  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
646, 39, 7, 8, 15, 18, 20, 53, 16, 60, 63, 22mapdindp1 32445 . . . . . 6  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { ( Y  .+  Z ) } ) )
656, 39, 7, 8, 15, 18, 20, 53, 16, 60, 63, 22mapdindp2 32446 . . . . . 6  |-  ( ph  ->  -.  w  e.  ( N `  { X ,  ( Y  .+  Z ) } ) )
666, 7, 8, 15, 18, 56, 17, 64, 65lspindp1 16197 . . . . 5  |-  ( ph  ->  ( ( N `  { w } )  =/=  ( N `  { ( Y  .+  Z ) } )  /\  -.  X  e.  ( N `  {
w ,  ( Y 
.+  Z ) } ) ) )
6766simprd 450 . . . 4  |-  ( ph  ->  -.  X  e.  ( N `  { w ,  ( Y  .+  Z ) } ) )
6867adantr 452 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  -.  X  e.  ( N `  { w ,  ( Y  .+  Z ) } ) )
6923simprd 450 . . . . . . . . 9  |-  ( ph  ->  ( N `  {
w } )  =/=  ( N `  { Y } ) )
706, 7, 8, 15, 16, 21, 69lspsnne1 16181 . . . . . . . 8  |-  ( ph  ->  -.  w  e.  ( N `  { Y } ) )
71 eqid 2435 . . . . . . . . . 10  |-  ( LSSum `  U )  =  (
LSSum `  U )
726, 8, 71, 38, 21, 54lsmpr 16153 . . . . . . . . 9  |-  ( ph  ->  ( N `  { Y ,  Z }
)  =  ( ( N `  { Y } ) ( LSSum `  U ) ( N `
 { Z }
) ) )
7360oveq2d 6089 . . . . . . . . 9  |-  ( ph  ->  ( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Y } ) )  =  ( ( N `  { Y } ) (
LSSum `  U ) ( N `  { Z } ) ) )
74 eqid 2435 . . . . . . . . . . . . 13  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
756, 74, 8lspsncl 16045 . . . . . . . . . . . 12  |-  ( ( U  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
7638, 21, 75syl2anc 643 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  U ) )
7774lsssubg 16025 . . . . . . . . . . 11  |-  ( ( U  e.  LMod  /\  ( N `  { Y } )  e.  (
LSubSp `  U ) )  ->  ( N `  { Y } )  e.  (SubGrp `  U )
)
7838, 76, 77syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  ( N `  { Y } )  e.  (SubGrp `  U ) )
7971lsmidm 15288 . . . . . . . . . 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 2473 . . . . . . . 8  |-  ( ph  ->  ( N `  { Y ,  Z }
)  =  ( N `
 { Y }
) )
8270, 81neleqtrrd 2531 . . . . . . 7  |-  ( ph  ->  -.  w  e.  ( N `  { Y ,  Z } ) )
836, 39, 8, 38, 21, 54, 17, 82lspindp4 16201 . . . . . 6  |-  ( ph  ->  -.  w  e.  ( N `  { Y ,  ( Y  .+  Z ) } ) )
846, 8, 15, 17, 21, 56, 83lspindpi 16196 . . . . 5  |-  ( ph  ->  ( ( N `  { w } )  =/=  ( N `  { Y } )  /\  ( N `  { w } )  =/=  ( N `  { ( Y  .+  Z ) } ) ) )
8584simprd 450 . . . 4  |-  ( ph  ->  ( N `  {
w } )  =/=  ( N `  {
( Y  .+  Z
) } ) )
8685adantr 452 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( N `  { w } )  =/=  ( N `  { ( Y  .+  Z ) } ) )
87 eqidd 2436 . . 3  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( I `  <. X ,  F ,  w >. )  =  ( I `  <. X ,  F ,  w >. ) )
88 eqidd 2436 . . 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 32535 . 2  |-  ( (
ph  /\  ( Y  .+  Z )  =/=  .0.  )  ->  ( I `  <. X ,  F , 
( w  .+  ( Y  .+  Z ) )
>. )  =  (
( I `  <. X ,  F ,  w >. )  .+b  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )
) )
9045, 89pm2.61dane 2676 1  |-  ( ph  ->  ( I `  <. X ,  F ,  ( w  .+  ( Y 
.+  Z ) )
>. )  =  (
( I `  <. X ,  F ,  w >. )  .+b  ( I `  <. X ,  F ,  ( Y  .+  Z ) >. )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598    \ cdif 3309   {csn 3806   {cpr 3807   <.cotp 3810   ` cfv 5446  (class class class)co 6073   Basecbs 13461   +g cplusg 13521   0gc0g 13715   -gcsg 14680  SubGrpcsubg 14930   LSSumclsm 15260   LModclmod 15942   LSubSpclss 16000   LSpanclspn 16039   HLchlt 30075   LHypclh 30708   DVecHcdvh 31803  LCDualclcd 32311  mapdcmpd 32349  HDMap1chdma1 32517
This theorem is referenced by:  hdmap1l6g  32542
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-fal 1329  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-ot 3816  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-tpos 6471  df-undef 6535  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-sca 13537  df-vsca 13538  df-0g 13719  df-mre 13803  df-mrc 13804  df-acs 13806  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-mnd 14682  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-subg 14933  df-cntz 15108  df-oppg 15134  df-lsm 15262  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-ur 15657  df-oppr 15720  df-dvdsr 15738  df-unit 15739  df-invr 15769  df-dvr 15780  df-drng 15829  df-lmod 15944  df-lss 16001  df-lsp 16040  df-lvec 16167  df-lsatoms 29701  df-lshyp 29702  df-lcv 29744  df-lfl 29783  df-lkr 29811  df-ldual 29849  df-oposet 29901  df-ol 29903  df-oml 29904  df-covers 29991  df-ats 29992  df-atl 30023  df-cvlat 30047  df-hlat 30076  df-llines 30222  df-lplanes 30223  df-lvols 30224  df-lines 30225  df-psubsp 30227  df-pmap 30228  df-padd 30520  df-lhyp 30712  df-laut 30713  df-ldil 30828  df-ltrn 30829  df-trl 30883  df-tgrp 31467  df-tendo 31479  df-edring 31481  df-dveca 31727  df-disoa 31754  df-dvech 31804  df-dib 31864  df-dic 31898  df-dih 31954  df-doch 32073  df-djh 32120  df-lcdual 32312  df-mapd 32350  df-hdmap1 32519
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