Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hdmap1l6k Unicode version

Theorem hdmap1l6k 32082
Description: Lemmma for hdmap1l6 32083. 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 451 . . 3  |-  ( (
ph  /\  Y  =  .0.  )  ->  ( K  e.  HL  /\  W  e.  H ) )
18 hdmap1l6.f . . . 4  |-  ( ph  ->  F  e.  D )
1918adantr 451 . . 3  |-  ( (
ph  /\  Y  =  .0.  )  ->  F  e.  D )
20 hdmap1l6cl.x . . . 4  |-  ( ph  ->  X  e.  ( V 
\  {  .0.  }
) )
2120adantr 451 . . 3  |-  ( (
ph  /\  Y  =  .0.  )  ->  X  e.  ( V  \  {  .0.  } ) )
22 hdmap1l6.mn . . . 4  |-  ( ph  ->  ( M `  ( N `  { X } ) )  =  ( L `  { F } ) )
2322adantr 451 . . 3  |-  ( (
ph  /\  Y  =  .0.  )  ->  ( M `
 ( N `  { X } ) )  =  ( L `  { F } ) )
24 simpr 447 . . 3  |-  ( (
ph  /\  Y  =  .0.  )  ->  Y  =  .0.  )
25 hdmap1l6k.z . . . 4  |-  ( ph  ->  Z  e.  V )
2625adantr 451 . . 3  |-  ( (
ph  /\  Y  =  .0.  )  ->  Z  e.  V )
27 hdmap1l6k.xn . . . 4  |-  ( ph  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
2827adantr 451 . . 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 32073 . 2  |-  ( (
ph  /\  Y  =  .0.  )  ->  ( I `
 <. X ,  F ,  ( Y  .+  Z ) >. )  =  ( ( I `
 <. X ,  F ,  Y >. )  .+b  (
I `  <. X ,  F ,  Z >. ) ) )
3016adantr 451 . . 3  |-  ( (
ph  /\  Z  =  .0.  )  ->  ( K  e.  HL  /\  W  e.  H ) )
3118adantr 451 . . 3  |-  ( (
ph  /\  Z  =  .0.  )  ->  F  e.  D )
3220adantr 451 . . 3  |-  ( (
ph  /\  Z  =  .0.  )  ->  X  e.  ( V  \  {  .0.  } ) )
3322adantr 451 . . 3  |-  ( (
ph  /\  Z  =  .0.  )  ->  ( M `
 ( N `  { X } ) )  =  ( L `  { F } ) )
34 hdmap1l6k.y . . . 4  |-  ( ph  ->  Y  e.  V )
3534adantr 451 . . 3  |-  ( (
ph  /\  Z  =  .0.  )  ->  Y  e.  V )
36 simpr 447 . . 3  |-  ( (
ph  /\  Z  =  .0.  )  ->  Z  =  .0.  )
3727adantr 451 . . 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 32074 . 2  |-  ( (
ph  /\  Z  =  .0.  )  ->  ( I `
 <. X ,  F ,  ( Y  .+  Z ) >. )  =  ( ( I `
 <. X ,  F ,  Y >. )  .+b  (
I `  <. X ,  F ,  Z >. ) ) )
3916adantr 451 . . 3  |-  ( (
ph  /\  ( Y  =/=  .0.  /\  Z  =/= 
.0.  ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
4018adantr 451 . . 3  |-  ( (
ph  /\  ( Y  =/=  .0.  /\  Z  =/= 
.0.  ) )  ->  F  e.  D )
4120adantr 451 . . 3  |-  ( (
ph  /\  ( Y  =/=  .0.  /\  Z  =/= 
.0.  ) )  ->  X  e.  ( V  \  {  .0.  } ) )
4222adantr 451 . . 3  |-  ( (
ph  /\  ( Y  =/=  .0.  /\  Z  =/= 
.0.  ) )  -> 
( M `  ( N `  { X } ) )  =  ( L `  { F } ) )
4327adantr 451 . . 3  |-  ( (
ph  /\  ( Y  =/=  .0.  /\  Z  =/= 
.0.  ) )  ->  -.  X  e.  ( N `  { Y ,  Z } ) )
4434adantr 451 . . . 4  |-  ( (
ph  /\  ( Y  =/=  .0.  /\  Z  =/= 
.0.  ) )  ->  Y  e.  V )
45 simprl 732 . . . 4  |-  ( (
ph  /\  ( Y  =/=  .0.  /\  Z  =/= 
.0.  ) )  ->  Y  =/=  .0.  )
46 eldifsn 3842 . . . 4  |-  ( Y  e.  ( V  \  {  .0.  } )  <->  ( Y  e.  V  /\  Y  =/= 
.0.  ) )
4744, 45, 46sylanbrc 645 . . 3  |-  ( (
ph  /\  ( Y  =/=  .0.  /\  Z  =/= 
.0.  ) )  ->  Y  e.  ( V  \  {  .0.  } ) )
4825adantr 451 . . . 4  |-  ( (
ph  /\  ( Y  =/=  .0.  /\  Z  =/= 
.0.  ) )  ->  Z  e.  V )
49 simprr 733 . . . 4  |-  ( (
ph  /\  ( Y  =/=  .0.  /\  Z  =/= 
.0.  ) )  ->  Z  =/=  .0.  )
50 eldifsn 3842 . . . 4  |-  ( Z  e.  ( V  \  {  .0.  } )  <->  ( Z  e.  V  /\  Z  =/= 
.0.  ) )
5148, 49, 50sylanbrc 645 . . 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 32081 . 2  |-  ( (
ph  /\  ( Y  =/=  .0.  /\  Z  =/= 
.0.  ) )  -> 
( I `  <. X ,  F ,  ( Y  .+  Z )
>. )  =  (
( I `  <. X ,  F ,  Y >. )  .+b  ( I `  <. X ,  F ,  Z >. ) ) )
5329, 38, 52pm2.61da2ne 2608 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 358    = wceq 1647    e. wcel 1715    =/= wne 2529    \ cdif 3235   {csn 3729   {cpr 3730   <.cotp 3733   ` cfv 5358  (class class class)co 5981   Basecbs 13356   +g cplusg 13416   0gc0g 13610   -gcsg 14575   LSpanclspn 15938   HLchlt 29611   LHypclh 30244   DVecHcdvh 31339  LCDualclcd 31847  mapdcmpd 31885  HDMap1chdma1 32053
This theorem is referenced by:  hdmap1l6  32083
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960  ax-pre-mulgt0 8961
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-fal 1325  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rmo 2636  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-ot 3739  df-uni 3930  df-int 3965  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-of 6205  df-1st 6249  df-2nd 6250  df-tpos 6376  df-undef 6440  df-riota 6446  df-recs 6530  df-rdg 6565  df-1o 6621  df-oadd 6625  df-er 6802  df-map 6917  df-en 7007  df-dom 7008  df-sdom 7009  df-fin 7010  df-pnf 9016  df-mnf 9017  df-xr 9018  df-ltxr 9019  df-le 9020  df-sub 9186  df-neg 9187  df-nn 9894  df-2 9951  df-3 9952  df-4 9953  df-5 9954  df-6 9955  df-n0 10115  df-z 10176  df-uz 10382  df-fz 10936  df-struct 13358  df-ndx 13359  df-slot 13360  df-base 13361  df-sets 13362  df-ress 13363  df-plusg 13429  df-mulr 13430  df-sca 13432  df-vsca 13433  df-0g 13614  df-mre 13698  df-mrc 13699  df-acs 13701  df-poset 14290  df-plt 14302  df-lub 14318  df-glb 14319  df-join 14320  df-meet 14321  df-p0 14355  df-p1 14356  df-lat 14362  df-clat 14424  df-mnd 14577  df-submnd 14626  df-grp 14699  df-minusg 14700  df-sbg 14701  df-subg 14828  df-cntz 15003  df-oppg 15029  df-lsm 15157  df-cmn 15301  df-abl 15302  df-mgp 15536  df-rng 15550  df-ur 15552  df-oppr 15615  df-dvdsr 15633  df-unit 15634  df-invr 15664  df-dvr 15675  df-drng 15724  df-lmod 15839  df-lss 15900  df-lsp 15939  df-lvec 16066  df-lsatoms 29237  df-lshyp 29238  df-lcv 29280  df-lfl 29319  df-lkr 29347  df-ldual 29385  df-oposet 29437  df-ol 29439  df-oml 29440  df-covers 29527  df-ats 29528  df-atl 29559  df-cvlat 29583  df-hlat 29612  df-llines 29758  df-lplanes 29759  df-lvols 29760  df-lines 29761  df-psubsp 29763  df-pmap 29764  df-padd 30056  df-lhyp 30248  df-laut 30249  df-ldil 30364  df-ltrn 30365  df-trl 30419  df-tgrp 31003  df-tendo 31015  df-edring 31017  df-dveca 31263  df-disoa 31290  df-dvech 31340  df-dib 31400  df-dic 31434  df-dih 31490  df-doch 31609  df-djh 31656  df-lcdual 31848  df-mapd 31886  df-hdmap1 32055
  Copyright terms: Public domain W3C validator