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Theorem hdmap1l6k 32308
Description: Lemmma for hdmap1l6 32309. Eliminate nonzero vector requirement. (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 } ) )
hdmap1l6k.y  |-  ( ph  ->  Y  e.  V )
hdmap1l6k.z  |-  ( ph  ->  Z  e.  V )
hdmap1l6k.xn  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
Assertion
Ref Expression
hdmap1l6k  |-  ( ph  ->  ( I `  <. X ,  F ,  ( Y  .+  Z )
>. )  =  (
( I `  <. X ,  F ,  Y >. )  .+b  ( I `  <. X ,  F ,  Z >. ) ) )

Proof of Theorem hdmap1l6k
StepHypRef Expression
1 hdmap1l6.h . . 3  |-  H  =  ( LHyp `  K
)
2 hdmap1l6.u . . 3  |-  U  =  ( ( DVecH `  K
) `  W )
3 hdmap1l6.v . . 3  |-  V  =  ( Base `  U
)
4 hdmap1l6.p . . 3  |-  .+  =  ( +g  `  U )
5 hdmap1l6.s . . 3  |-  .-  =  ( -g `  U )
6 hdmap1l6c.o . . 3  |-  .0.  =  ( 0g `  U )
7 hdmap1l6.n . . 3  |-  N  =  ( LSpan `  U )
8 hdmap1l6.c . . 3  |-  C  =  ( (LCDual `  K
) `  W )
9 hdmap1l6.d . . 3  |-  D  =  ( Base `  C
)
10 hdmap1l6.a . . 3  |-  .+b  =  ( +g  `  C )
11 hdmap1l6.r . . 3  |-  R  =  ( -g `  C
)
12 hdmap1l6.q . . 3  |-  Q  =  ( 0g `  C
)
13 hdmap1l6.l . . 3  |-  L  =  ( LSpan `  C )
14 hdmap1l6.m . . 3  |-  M  =  ( (mapd `  K
) `  W )
15 hdmap1l6.i . . 3  |-  I  =  ( (HDMap1 `  K
) `  W )
16 hdmap1l6.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
1716adantr 452 . . 3  |-  ( (
ph  /\  Y  =  .0.  )  ->  ( K  e.  HL  /\  W  e.  H ) )
18 hdmap1l6.f . . . 4  |-  ( ph  ->  F  e.  D )
1918adantr 452 . . 3  |-  ( (
ph  /\  Y  =  .0.  )  ->  F  e.  D )
20 hdmap1l6cl.x . . . 4  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
2120adantr 452 . . 3  |-  ( (
ph  /\  Y  =  .0.  )  ->  X  e.  ( V  \  {  .0.  } ) )
22 hdmap1l6.mn . . . 4  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( L `  { F } ) )
2322adantr 452 . . 3  |-  ( (
ph  /\  Y  =  .0.  )  ->  ( M `
 ( N `  { X } ) )  =  ( L `  { F } ) )
24 simpr 448 . . 3  |-  ( (
ph  /\  Y  =  .0.  )  ->  Y  =  .0.  )
25 hdmap1l6k.z . . . 4  |-  ( ph  ->  Z  e.  V )
2625adantr 452 . . 3  |-  ( (
ph  /\  Y  =  .0.  )  ->  Z  e.  V )
27 hdmap1l6k.xn . . . 4  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
2827adantr 452 . . 3  |-  ( (
ph  /\  Y  =  .0.  )  ->  -.  X  e.  ( N `  { Y ,  Z }
) )
291, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 19, 21, 23, 24, 26, 28hdmap1l6b 32299 . 2  |-  ( (
ph  /\  Y  =  .0.  )  ->  ( I `
 <. X ,  F ,  ( Y  .+  Z ) >. )  =  ( ( I `
 <. X ,  F ,  Y >. )  .+b  (
I `  <. X ,  F ,  Z >. ) ) )
3016adantr 452 . . 3  |-  ( (
ph  /\  Z  =  .0.  )  ->  ( K  e.  HL  /\  W  e.  H ) )
3118adantr 452 . . 3  |-  ( (
ph  /\  Z  =  .0.  )  ->  F  e.  D )
3220adantr 452 . . 3  |-  ( (
ph  /\  Z  =  .0.  )  ->  X  e.  ( V  \  {  .0.  } ) )
3322adantr 452 . . 3  |-  ( (
ph  /\  Z  =  .0.  )  ->  ( M `
 ( N `  { X } ) )  =  ( L `  { F } ) )
34 hdmap1l6k.y . . . 4  |-  ( ph  ->  Y  e.  V )
3534adantr 452 . . 3  |-  ( (
ph  /\  Z  =  .0.  )  ->  Y  e.  V )
36 simpr 448 . . 3  |-  ( (
ph  /\  Z  =  .0.  )  ->  Z  =  .0.  )
3727adantr 452 . . 3  |-  ( (
ph  /\  Z  =  .0.  )  ->  -.  X  e.  ( N `  { Y ,  Z }
) )
381, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 30, 31, 32, 33, 35, 36, 37hdmap1l6c 32300 . 2  |-  ( (
ph  /\  Z  =  .0.  )  ->  ( I `
 <. X ,  F ,  ( Y  .+  Z ) >. )  =  ( ( I `
 <. X ,  F ,  Y >. )  .+b  (
I `  <. X ,  F ,  Z >. ) ) )
3916adantr 452 . . 3  |-  ( (
ph  /\  ( Y  =/=  .0.  /\  Z  =/= 
.0.  ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
4018adantr 452 . . 3  |-  ( (
ph  /\  ( Y  =/=  .0.  /\  Z  =/= 
.0.  ) )  ->  F  e.  D )
4120adantr 452 . . 3  |-  ( (
ph  /\  ( Y  =/=  .0.  /\  Z  =/= 
.0.  ) )  ->  X  e.  ( V  \  {  .0.  } ) )
4222adantr 452 . . 3  |-  ( (
ph  /\  ( Y  =/=  .0.  /\  Z  =/= 
.0.  ) )  -> 
( M `  ( N `  { X } ) )  =  ( L `  { F } ) )
4327adantr 452 . . 3  |-  ( (
ph  /\  ( Y  =/=  .0.  /\  Z  =/= 
.0.  ) )  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
4434adantr 452 . . . 4  |-  ( (
ph  /\  ( Y  =/=  .0.  /\  Z  =/= 
.0.  ) )  ->  Y  e.  V )
45 simprl 733 . . . 4  |-  ( (
ph  /\  ( Y  =/=  .0.  /\  Z  =/= 
.0.  ) )  ->  Y  =/=  .0.  )
46 eldifsn 3891 . . . 4  |-  ( Y  e.  ( V  \  {  .0.  } )  <->  ( Y  e.  V  /\  Y  =/= 
.0.  ) )
4744, 45, 46sylanbrc 646 . . 3  |-  ( (
ph  /\  ( Y  =/=  .0.  /\  Z  =/= 
.0.  ) )  ->  Y  e.  ( V  \  {  .0.  } ) )
4825adantr 452 . . . 4  |-  ( (
ph  /\  ( Y  =/=  .0.  /\  Z  =/= 
.0.  ) )  ->  Z  e.  V )
49 simprr 734 . . . 4  |-  ( (
ph  /\  ( Y  =/=  .0.  /\  Z  =/= 
.0.  ) )  ->  Z  =/=  .0.  )
50 eldifsn 3891 . . . 4  |-  ( Z  e.  ( V  \  {  .0.  } )  <->  ( Z  e.  V  /\  Z  =/= 
.0.  ) )
5148, 49, 50sylanbrc 646 . . 3  |-  ( (
ph  /\  ( Y  =/=  .0.  /\  Z  =/= 
.0.  ) )  ->  Z  e.  ( V  \  {  .0.  } ) )
521, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 39, 40, 41, 42, 43, 47, 51hdmap1l6j 32307 . 2  |-  ( (
ph  /\  ( Y  =/=  .0.  /\  Z  =/= 
.0.  ) )  -> 
( I `  <. X ,  F ,  ( Y  .+  Z )
>. )  =  (
( I `  <. X ,  F ,  Y >. )  .+b  ( I `  <. X ,  F ,  Z >. ) ) )
5329, 38, 52pm2.61da2ne 2650 1  |-  ( ph  ->  ( I `  <. X ,  F ,  ( Y  .+  Z )
>. )  =  (
( I `  <. X ,  F ,  Y >. )  .+b  ( I `  <. X ,  F ,  Z >. ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2571    \ cdif 3281   {csn 3778   {cpr 3779   <.cotp 3782   ` cfv 5417  (class class class)co 6044   Basecbs 13428   +g cplusg 13488   0gc0g 13682   -gcsg 14647   LSpanclspn 16006   HLchlt 29837   LHypclh 30470   DVecHcdvh 31565  LCDualclcd 32073  mapdcmpd 32111  HDMap1chdma1 32279
This theorem is referenced by:  hdmap1l6  32309
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-fal 1326  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-ot 3788  df-uni 3980  df-int 4015  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-of 6268  df-1st 6312  df-2nd 6313  df-tpos 6442  df-undef 6506  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-er 6868  df-map 6983  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-nn 9961  df-2 10018  df-3 10019  df-4 10020  df-5 10021  df-6 10022  df-n0 10182  df-z 10243  df-uz 10449  df-fz 11004  df-struct 13430  df-ndx 13431  df-slot 13432  df-base 13433  df-sets 13434  df-ress 13435  df-plusg 13501  df-mulr 13502  df-sca 13504  df-vsca 13505  df-0g 13686  df-mre 13770  df-mrc 13771  df-acs 13773  df-poset 14362  df-plt 14374  df-lub 14390  df-glb 14391  df-join 14392  df-meet 14393  df-p0 14427  df-p1 14428  df-lat 14434  df-clat 14496  df-mnd 14649  df-submnd 14698  df-grp 14771  df-minusg 14772  df-sbg 14773  df-subg 14900  df-cntz 15075  df-oppg 15101  df-lsm 15229  df-cmn 15373  df-abl 15374  df-mgp 15608  df-rng 15622  df-ur 15624  df-oppr 15687  df-dvdsr 15705  df-unit 15706  df-invr 15736  df-dvr 15747  df-drng 15796  df-lmod 15911  df-lss 15968  df-lsp 16007  df-lvec 16134  df-lsatoms 29463  df-lshyp 29464  df-lcv 29506  df-lfl 29545  df-lkr 29573  df-ldual 29611  df-oposet 29663  df-ol 29665  df-oml 29666  df-covers 29753  df-ats 29754  df-atl 29785  df-cvlat 29809  df-hlat 29838  df-llines 29984  df-lplanes 29985  df-lvols 29986  df-lines 29987  df-psubsp 29989  df-pmap 29990  df-padd 30282  df-lhyp 30474  df-laut 30475  df-ldil 30590  df-ltrn 30591  df-trl 30645  df-tgrp 31229  df-tendo 31241  df-edring 31243  df-dveca 31489  df-disoa 31516  df-dvech 31566  df-dib 31626  df-dic 31660  df-dih 31716  df-doch 31835  df-djh 31882  df-lcdual 32074  df-mapd 32112  df-hdmap1 32281
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