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Theorem dvheveccl 31302
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 31306 and dihpN 31526. (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
Dummy variable  f is distinct from all other variables.
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 30339 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  B )  e.  T )
72, 6syl 15 . . . 4  |-  ( ph  ->  (  _I  |`  B )  e.  T )
8 eqid 2283 . . . . . 6  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
94, 5, 8tendoidcl 30958 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  e.  ( ( TEndo `  K ) `  W
) )
102, 9syl 15 . . . 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 31278 . . . 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 1180 . . 3  |-  ( ph  -> 
<. (  _I  |`  B ) ,  (  _I  |`  T )
>.  e.  V )
15 eqid 2283 . . . . . 6  |-  ( f  e.  T  |->  (  _I  |`  B ) )  =  ( f  e.  T  |->  (  _I  |`  B ) )
163, 4, 5, 8, 15tendo1ne0 31017 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  (  _I  |`  T )  =/=  ( f  e.  T  |->  (  _I  |`  B ) ) )
172, 16syl 15 . . . 4  |-  ( ph  ->  (  _I  |`  T )  =/=  ( f  e.  T  |->  (  _I  |`  B ) ) )
18 dvheveccl.z . . . . . . . 8  |-  .0.  =  ( 0g `  U )
193, 4, 5, 11, 18, 15dvh0g 31301 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  .0.  =  <. (  _I  |`  B ) ,  ( f  e.  T  |->  (  _I  |`  B ) ) >. )
202, 19syl 15 . . . . . 6  |-  ( ph  ->  .0.  =  <. (  _I  |`  B ) ,  ( f  e.  T  |->  (  _I  |`  B ) ) >. )
21 eqtr 2300 . . . . . . 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 4246 . . . . . . . . 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 642 . . . . . . . 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 447 . . . . . . . 8  |-  ( ( (  _I  |`  B )  =  (  _I  |`  B )  /\  (  _I  |`  T )  =  ( f  e.  T  |->  (  _I  |`  B ) ) )  ->  (  _I  |`  T )  =  ( f  e.  T  |->  (  _I  |`  B ) ) )
2523, 24syl6bi 219 . . . . . . 7  |-  ( ph  ->  ( <. (  _I  |`  B ) ,  (  _I  |`  T )
>.  =  <. (  _I  |`  B ) ,  ( f  e.  T  |->  (  _I  |`  B )
) >.  ->  (  _I  |`  T )  =  ( f  e.  T  |->  (  _I  |`  B )
) ) )
2621, 25syl5 28 . . . . . 6  |-  ( ph  ->  ( ( <. (  _I  |`  B ) ,  (  _I  |`  T )
>.  =  .0.  /\  .0.  =  <. (  _I  |`  B ) ,  ( f  e.  T  |->  (  _I  |`  B ) ) >. )  ->  (  _I  |`  T )  =  ( f  e.  T  |->  (  _I  |`  B ) ) ) )
2720, 26mpan2d 655 . . . . 5  |-  ( ph  ->  ( <. (  _I  |`  B ) ,  (  _I  |`  T )
>.  =  .0.  ->  (  _I  |`  T )  =  ( f  e.  T  |->  (  _I  |`  B ) ) ) )
2827necon3d 2484 . . . 4  |-  ( ph  ->  ( (  _I  |`  T )  =/=  ( f  e.  T  |->  (  _I  |`  B ) )  ->  <. (  _I  |`  B ) ,  (  _I  |`  T ) >.  =/=  .0.  ) )
2917, 28mpd 14 . . 3  |-  ( ph  -> 
<. (  _I  |`  B ) ,  (  _I  |`  T )
>.  =/=  .0.  )
30 eldifsn 3749 . . 3  |-  ( <.
(  _I  |`  B ) ,  (  _I  |`  T )
>.  e.  ( V  \  {  .0.  } )  <->  ( <. (  _I  |`  B ) ,  (  _I  |`  T )
>.  e.  V  /\  <. (  _I  |`  B ) ,  (  _I  |`  T )
>.  =/=  .0.  ) )
3114, 29, 30sylanbrc 645 . 2  |-  ( ph  -> 
<. (  _I  |`  B ) ,  (  _I  |`  T )
>.  e.  ( V  \  {  .0.  } ) )
321, 31syl5eqel 2367 1  |-  ( ph  ->  E  e.  ( V 
\  {  .0.  }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149   {csn 3640   <.cop 3643    e. cmpt 4077    _I cid 4304    |` cres 4691   ` cfv 5255   Basecbs 13148   0gc0g 13400   HLchlt 29540   LHypclh 30173   LTrncltrn 30290   TEndoctendo 30941   DVecHcdvh 31268
This theorem is referenced by:  hdmapcl  32023  hdmapval2lem  32024  hdmapval0  32026  hdmapeveclem  32027  hdmapevec  32028  hdmapevec2  32029  hdmapval3lemN  32030  hdmapval3N  32031  hdmap10lem  32032  hdmap11lem1  32034  hdmap11lem2  32035  hdmapinvlem1  32111  hdmapinvlem2  32112  hdmapinvlem3  32113  hdmapinvlem4  32114  hdmapglem5  32115  hgmapvvlem3  32118  hdmapglem7a  32120  hdmapglem7b  32121  hdmapglem7  32122
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-undef 6298  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-0g 13404  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-mnd 14367  df-grp 14489  df-minusg 14490  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-drng 15514  df-lmod 15629  df-lvec 15856  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348  df-tendo 30944  df-edring 30946  df-dvech 31269
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