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

Theorem dvheveccl 30453
Description: Properties of a unit vector that we will use later as a convenient reference vector. This vector is called "e" in the remark after Lemma M of [Crawley] p. 121. line 17. See also dvhopN 30457 and dihpN 30677. (Contributed by NM, 27-Mar-2015.)
Hypotheses
Ref Expression
dvheveccl.h  |-  H  =  ( LHyp `  K
)
dvheveccl.b  |-  B  =  ( Base `  K
)
dvheveccl.t  |-  T  =  ( ( LTrn `  K
) `  W )
dvheveccl.u  |-  U  =  ( ( DVecH `  K
) `  W )
dvheveccl.v  |-  V  =  ( Base `  U
)
dvheveccl.z  |-  .0.  =  ( 0g `  U )
dvheveccl.e  |-  E  = 
<. (  _I  |`  B ) ,  (  _I  |`  T )
>.
dvheveccl.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
Assertion
Ref Expression
dvheveccl  |-  ( ph  ->  E  e.  ( V 
\  {  .0.  }
) )

Proof of Theorem dvheveccl
StepHypRef Expression
1 dvheveccl.e . 2  |-  E  = 
<. (  _I  |`  B ) ,  (  _I  |`  T )
>.
2 dvheveccl.k . . . 4  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
3 dvheveccl.b . . . . . 6  |-  B  =  ( Base `  K
)
4 dvheveccl.h . . . . . 6  |-  H  =  ( LHyp `  K
)
5 dvheveccl.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
63, 4, 5idltrn 29490 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  T )
72, 6syl 17 . . . 4  |-  ( ph  ->  (  _I  |`  B )  e.  T )
8 eqid 2256 . . . . . 6  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
94, 5, 8tendoidcl 30109 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  e.  ( ( TEndo `  K ) `  W
) )
102, 9syl 17 . . . 4  |-  ( ph  ->  (  _I  |`  T )  e.  ( ( TEndo `  K ) `  W
) )
11 dvheveccl.u . . . . 5  |-  U  =  ( ( DVecH `  K
) `  W )
12 dvheveccl.v . . . . 5  |-  V  =  ( Base `  U
)
134, 5, 8, 11, 12dvhelvbasei 30429 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( (  _I  |`  B )  e.  T  /\  (  _I  |`  T )  e.  ( ( TEndo `  K ) `  W
) ) )  ->  <. (  _I  |`  B ) ,  (  _I  |`  T )
>.  e.  V )
142, 7, 10, 13syl12anc 1185 . . 3  |-  ( ph  -> 
<. (  _I  |`  B ) ,  (  _I  |`  T )
>.  e.  V )
15 eqid 2256 . . . . . 6  |-  ( f  e.  T  |->  (  _I  |`  B ) )  =  ( f  e.  T  |->  (  _I  |`  B ) )
163, 4, 5, 8, 15tendo1ne0 30168 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  =/=  ( f  e.  T  |->  (  _I  |`  B ) ) )
172, 16syl 17 . . . 4  |-  ( ph  ->  (  _I  |`  T )  =/=  ( f  e.  T  |->  (  _I  |`  B ) ) )
18 dvheveccl.z . . . . . . . 8  |-  .0.  =  ( 0g `  U )
193, 4, 5, 11, 18, 15dvh0g 30452 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  =  <. (  _I  |`  B ) ,  ( f  e.  T  |->  (  _I  |`  B ) ) >. )
202, 19syl 17 . . . . . 6  |-  ( ph  ->  .0.  =  <. (  _I  |`  B ) ,  ( f  e.  T  |->  (  _I  |`  B ) ) >. )
21 eqtr 2273 . . . . . . 7  |-  ( (
<. (  _I  |`  B ) ,  (  _I  |`  T )
>.  =  .0.  /\  .0.  =  <. (  _I  |`  B ) ,  ( f  e.  T  |->  (  _I  |`  B ) ) >. )  ->  <. (  _I  |`  B ) ,  (  _I  |`  T )
>.  =  <. (  _I  |`  B ) ,  ( f  e.  T  |->  (  _I  |`  B )
) >. )
22 opthg 4204 . . . . . . . . 9  |-  ( ( (  _I  |`  B )  e.  T  /\  (  _I  |`  T )  e.  ( ( TEndo `  K
) `  W )
)  ->  ( <. (  _I  |`  B ) ,  (  _I  |`  T )
>.  =  <. (  _I  |`  B ) ,  ( f  e.  T  |->  (  _I  |`  B )
) >. 
<->  ( (  _I  |`  B )  =  (  _I  |`  B )  /\  (  _I  |`  T )  =  ( f  e.  T  |->  (  _I  |`  B ) ) ) ) )
237, 10, 22syl2anc 645 . . . . . . . 8  |-  ( ph  ->  ( <. (  _I  |`  B ) ,  (  _I  |`  T )
>.  =  <. (  _I  |`  B ) ,  ( f  e.  T  |->  (  _I  |`  B )
) >. 
<->  ( (  _I  |`  B )  =  (  _I  |`  B )  /\  (  _I  |`  T )  =  ( f  e.  T  |->  (  _I  |`  B ) ) ) ) )
24 simpr 449 . . . . . . . 8  |-  ( ( (  _I  |`  B )  =  (  _I  |`  B )  /\  (  _I  |`  T )  =  ( f  e.  T  |->  (  _I  |`  B ) ) )  ->  (  _I  |`  T )  =  ( f  e.  T  |->  (  _I  |`  B ) ) )
2523, 24syl6bi 221 . . . . . . 7  |-  ( ph  ->  ( <. (  _I  |`  B ) ,  (  _I  |`  T )
>.  =  <. (  _I  |`  B ) ,  ( f  e.  T  |->  (  _I  |`  B )
) >.  ->  (  _I  |`  T )  =  ( f  e.  T  |->  (  _I  |`  B )
) ) )
2621, 25syl5 30 . . . . . 6  |-  ( ph  ->  ( ( <. (  _I  |`  B ) ,  (  _I  |`  T )
>.  =  .0.  /\  .0.  =  <. (  _I  |`  B ) ,  ( f  e.  T  |->  (  _I  |`  B ) ) >. )  ->  (  _I  |`  T )  =  ( f  e.  T  |->  (  _I  |`  B ) ) ) )
2720, 26mpan2d 658 . . . . 5  |-  ( ph  ->  ( <. (  _I  |`  B ) ,  (  _I  |`  T )
>.  =  .0.  ->  (  _I  |`  T )  =  ( f  e.  T  |->  (  _I  |`  B ) ) ) )
2827necon3d 2457 . . . 4  |-  ( ph  ->  ( (  _I  |`  T )  =/=  ( f  e.  T  |->  (  _I  |`  B ) )  ->  <. (  _I  |`  B ) ,  (  _I  |`  T ) >.  =/=  .0.  ) )
2917, 28mpd 16 . . 3  |-  ( ph  -> 
<. (  _I  |`  B ) ,  (  _I  |`  T )
>.  =/=  .0.  )
30 eldifsn 3709 . . 3  |-  ( <.
(  _I  |`  B ) ,  (  _I  |`  T )
>.  e.  ( V  \  {  .0.  } )  <->  ( <. (  _I  |`  B ) ,  (  _I  |`  T )
>.  e.  V  /\  <. (  _I  |`  B ) ,  (  _I  |`  T )
>.  =/=  .0.  ) )
3114, 29, 30sylanbrc 648 . 2  |-  ( ph  -> 
<. (  _I  |`  B ) ,  (  _I  |`  T )
>.  e.  ( V  \  {  .0.  } ) )
321, 31syl5eqel 2340 1  |-  ( ph  ->  E  e.  ( V 
\  {  .0.  }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2419    \ cdif 3110   {csn 3600   <.cop 3603    e. cmpt 4037    _I cid 4262    |` cres 4649   ` cfv 4659   Basecbs 13096   0gc0g 13348   HLchlt 28691   LHypclh 29324   LTrncltrn 29441   TEndoctendo 30092   DVecHcdvh 30419
This theorem is referenced by:  hdmapcl  31174  hdmapval2lem  31175  hdmapval0  31177  hdmapeveclem  31178  hdmapevec  31179  hdmapevec2  31180  hdmapval3lemN  31181  hdmapval3N  31182  hdmap10lem  31183  hdmap11lem1  31185  hdmap11lem2  31186  hdmapinvlem1  31262  hdmapinvlem2  31263  hdmapinvlem3  31264  hdmapinvlem4  31265  hdmapglem5  31266  hgmapvvlem3  31269  hdmapglem7a  31271  hdmapglem7b  31272  hdmapglem7  31273
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-fal 1316  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-iin 3868  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-tpos 6154  df-iota 6211  df-undef 6250  df-riota 6258  df-recs 6342  df-rdg 6377  df-1o 6433  df-oadd 6437  df-er 6614  df-map 6728  df-en 6818  df-dom 6819  df-sdom 6820  df-fin 6821  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-n 9701  df-2 9758  df-3 9759  df-4 9760  df-5 9761  df-6 9762  df-n0 9919  df-z 9978  df-uz 10184  df-fz 10735  df-struct 13098  df-ndx 13099  df-slot 13100  df-base 13101  df-sets 13102  df-ress 13103  df-plusg 13169  df-mulr 13170  df-sca 13172  df-vsca 13173  df-0g 13352  df-poset 14028  df-plt 14040  df-lub 14056  df-glb 14057  df-join 14058  df-meet 14059  df-p0 14093  df-p1 14094  df-lat 14100  df-clat 14162  df-mnd 14315  df-grp 14437  df-minusg 14438  df-mgp 15274  df-ring 15288  df-ur 15290  df-oppr 15353  df-dvdsr 15371  df-unit 15372  df-invr 15402  df-dvr 15413  df-drng 15462  df-lmod 15577  df-lvec 15804  df-oposet 28517  df-ol 28519  df-oml 28520  df-covers 28607  df-ats 28608  df-atl 28639  df-cvlat 28663  df-hlat 28692  df-llines 28838  df-lplanes 28839  df-lvols 28840  df-lines 28841  df-psubsp 28843  df-pmap 28844  df-padd 29136  df-lhyp 29328  df-laut 29329  df-ldil 29444  df-ltrn 29445  df-trl 29499  df-tendo 30095  df-edring 30097  df-dvech 30420
  Copyright terms: Public domain W3C validator