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Theorem lspsneu 17216
Description: Nonzero vectors with equal singleton spans have a unique proportionality constant. (Contributed by NM, 31-May-2015.)
Hypotheses
Ref Expression
lspsneu.v  |-  V  =  ( Base `  W
)
lspsneu.s  |-  S  =  (Scalar `  W )
lspsneu.k  |-  K  =  ( Base `  S
)
lspsneu.o  |-  O  =  ( 0g `  S
)
lspsneu.t  |-  .x.  =  ( .s `  W )
lspsneu.z  |-  .0.  =  ( 0g `  W )
lspsneu.n  |-  N  =  ( LSpan `  W )
lspsneu.w  |-  ( ph  ->  W  e.  LVec )
lspsneu.x  |-  ( ph  ->  X  e.  V )
lspsneu.y  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
Assertion
Ref Expression
lspsneu  |-  ( ph  ->  ( ( N `  { X } )  =  ( N `  { Y } )  <->  E! k  e.  ( K  \  { O } ) X  =  ( k  .x.  Y
) ) )
Distinct variable groups:    k, K    k, O    .x. , k    k, X    k, Y
Allowed substitution hints:    ph( k)    S( k)    N( k)    V( k)    W( k)    .0. ( k)

Proof of Theorem lspsneu
Dummy variables  i 
j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lspsneu.v . . . . . . 7  |-  V  =  ( Base `  W
)
2 lspsneu.s . . . . . . 7  |-  S  =  (Scalar `  W )
3 lspsneu.k . . . . . . 7  |-  K  =  ( Base `  S
)
4 lspsneu.o . . . . . . 7  |-  O  =  ( 0g `  S
)
5 lspsneu.t . . . . . . 7  |-  .x.  =  ( .s `  W )
6 lspsneu.n . . . . . . 7  |-  N  =  ( LSpan `  W )
7 lspsneu.w . . . . . . 7  |-  ( ph  ->  W  e.  LVec )
8 lspsneu.x . . . . . . 7  |-  ( ph  ->  X  e.  V )
9 lspsneu.y . . . . . . . 8  |-  ( ph  ->  Y  e.  ( V 
\  {  .0.  }
) )
109eldifad 3352 . . . . . . 7  |-  ( ph  ->  Y  e.  V )
111, 2, 3, 4, 5, 6, 7, 8, 10lspsneq 17215 . . . . . 6  |-  ( ph  ->  ( ( N `  { X } )  =  ( N `  { Y } )  <->  E. j  e.  ( K  \  { O } ) X  =  ( j  .x.  Y
) ) )
1211biimpd 207 . . . . 5  |-  ( ph  ->  ( ( N `  { X } )  =  ( N `  { Y } )  ->  E. j  e.  ( K  \  { O } ) X  =  ( j  .x.  Y
) ) )
13 eqtr2 2461 . . . . . . . . . 10  |-  ( ( X  =  ( j 
.x.  Y )  /\  X  =  ( i  .x.  Y ) )  -> 
( j  .x.  Y
)  =  ( i 
.x.  Y ) )
14133ad2ant3 1011 . . . . . . . . 9  |-  ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  ( j  e.  ( K  \  { O } )  /\  i  e.  ( K  \  { O } ) )  /\  ( X  =  (
j  .x.  Y )  /\  X  =  (
i  .x.  Y )
) )  ->  (
j  .x.  Y )  =  ( i  .x.  Y ) )
15 lspsneu.z . . . . . . . . . 10  |-  .0.  =  ( 0g `  W )
16 simp1l 1012 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  ( j  e.  ( K  \  { O } )  /\  i  e.  ( K  \  { O } ) )  /\  ( X  =  (
j  .x.  Y )  /\  X  =  (
i  .x.  Y )
) )  ->  ph )
1716, 7syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  ( j  e.  ( K  \  { O } )  /\  i  e.  ( K  \  { O } ) )  /\  ( X  =  (
j  .x.  Y )  /\  X  =  (
i  .x.  Y )
) )  ->  W  e.  LVec )
18 simp2l 1014 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  ( j  e.  ( K  \  { O } )  /\  i  e.  ( K  \  { O } ) )  /\  ( X  =  (
j  .x.  Y )  /\  X  =  (
i  .x.  Y )
) )  ->  j  e.  ( K  \  { O } ) )
1918eldifad 3352 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  ( j  e.  ( K  \  { O } )  /\  i  e.  ( K  \  { O } ) )  /\  ( X  =  (
j  .x.  Y )  /\  X  =  (
i  .x.  Y )
) )  ->  j  e.  K )
20 simp2r 1015 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  ( j  e.  ( K  \  { O } )  /\  i  e.  ( K  \  { O } ) )  /\  ( X  =  (
j  .x.  Y )  /\  X  =  (
i  .x.  Y )
) )  ->  i  e.  ( K  \  { O } ) )
2120eldifad 3352 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  ( j  e.  ( K  \  { O } )  /\  i  e.  ( K  \  { O } ) )  /\  ( X  =  (
j  .x.  Y )  /\  X  =  (
i  .x.  Y )
) )  ->  i  e.  K )
2216, 10syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  ( j  e.  ( K  \  { O } )  /\  i  e.  ( K  \  { O } ) )  /\  ( X  =  (
j  .x.  Y )  /\  X  =  (
i  .x.  Y )
) )  ->  Y  e.  V )
23 eldifsni 4013 . . . . . . . . . . 11  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  Y  =/=  .0.  )
2416, 9, 233syl 20 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  ( j  e.  ( K  \  { O } )  /\  i  e.  ( K  \  { O } ) )  /\  ( X  =  (
j  .x.  Y )  /\  X  =  (
i  .x.  Y )
) )  ->  Y  =/=  .0.  )
251, 5, 2, 3, 15, 17, 19, 21, 22, 24lvecvscan2 17205 . . . . . . . . 9  |-  ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  ( j  e.  ( K  \  { O } )  /\  i  e.  ( K  \  { O } ) )  /\  ( X  =  (
j  .x.  Y )  /\  X  =  (
i  .x.  Y )
) )  ->  (
( j  .x.  Y
)  =  ( i 
.x.  Y )  <->  j  =  i ) )
2614, 25mpbid 210 . . . . . . . 8  |-  ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  ( j  e.  ( K  \  { O } )  /\  i  e.  ( K  \  { O } ) )  /\  ( X  =  (
j  .x.  Y )  /\  X  =  (
i  .x.  Y )
) )  ->  j  =  i )
27263exp 1186 . . . . . . 7  |-  ( (
ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  ->  ( ( j  e.  ( K  \  { O } )  /\  i  e.  ( K  \  { O } ) )  ->  ( ( X  =  ( j  .x.  Y )  /\  X  =  ( i  .x.  Y ) )  -> 
j  =  i ) ) )
2827ex 434 . . . . . 6  |-  ( ph  ->  ( ( N `  { X } )  =  ( N `  { Y } )  ->  (
( j  e.  ( K  \  { O } )  /\  i  e.  ( K  \  { O } ) )  -> 
( ( X  =  ( j  .x.  Y
)  /\  X  =  ( i  .x.  Y
) )  ->  j  =  i ) ) ) )
2928ralrimdvv 2822 . . . . 5  |-  ( ph  ->  ( ( N `  { X } )  =  ( N `  { Y } )  ->  A. j  e.  ( K  \  { O } ) A. i  e.  ( K  \  { O } ) ( ( X  =  ( j 
.x.  Y )  /\  X  =  ( i  .x.  Y ) )  -> 
j  =  i ) ) )
3012, 29jcad 533 . . . 4  |-  ( ph  ->  ( ( N `  { X } )  =  ( N `  { Y } )  ->  ( E. j  e.  ( K  \  { O }
) X  =  ( j  .x.  Y )  /\  A. j  e.  ( K  \  { O } ) A. i  e.  ( K  \  { O } ) ( ( X  =  ( j 
.x.  Y )  /\  X  =  ( i  .x.  Y ) )  -> 
j  =  i ) ) ) )
31 oveq1 6110 . . . . . 6  |-  ( j  =  i  ->  (
j  .x.  Y )  =  ( i  .x.  Y ) )
3231eqeq2d 2454 . . . . 5  |-  ( j  =  i  ->  ( X  =  ( j  .x.  Y )  <->  X  =  ( i  .x.  Y
) ) )
3332reu4 3165 . . . 4  |-  ( E! j  e.  ( K 
\  { O }
) X  =  ( j  .x.  Y )  <-> 
( E. j  e.  ( K  \  { O } ) X  =  ( j  .x.  Y
)  /\  A. j  e.  ( K  \  { O } ) A. i  e.  ( K  \  { O } ) ( ( X  =  ( j 
.x.  Y )  /\  X  =  ( i  .x.  Y ) )  -> 
j  =  i ) ) )
3430, 33syl6ibr 227 . . 3  |-  ( ph  ->  ( ( N `  { X } )  =  ( N `  { Y } )  ->  E! j  e.  ( K  \  { O } ) X  =  ( j 
.x.  Y ) ) )
35 reurex 2949 . . . 4  |-  ( E! j  e.  ( K 
\  { O }
) X  =  ( j  .x.  Y )  ->  E. j  e.  ( K  \  { O } ) X  =  ( j  .x.  Y
) )
3635, 11syl5ibr 221 . . 3  |-  ( ph  ->  ( E! j  e.  ( K  \  { O } ) X  =  ( j  .x.  Y
)  ->  ( N `  { X } )  =  ( N `  { Y } ) ) )
3734, 36impbid 191 . 2  |-  ( ph  ->  ( ( N `  { X } )  =  ( N `  { Y } )  <->  E! j  e.  ( K  \  { O } ) X  =  ( j  .x.  Y
) ) )
38 oveq1 6110 . . . 4  |-  ( j  =  k  ->  (
j  .x.  Y )  =  ( k  .x.  Y ) )
3938eqeq2d 2454 . . 3  |-  ( j  =  k  ->  ( X  =  ( j  .x.  Y )  <->  X  =  ( k  .x.  Y
) ) )
4039cbvreuv 2961 . 2  |-  ( E! j  e.  ( K 
\  { O }
) X  =  ( j  .x.  Y )  <-> 
E! k  e.  ( K  \  { O } ) X  =  ( k  .x.  Y
) )
4137, 40syl6bb 261 1  |-  ( ph  ->  ( ( N `  { X } )  =  ( N `  { Y } )  <->  E! k  e.  ( K  \  { O } ) X  =  ( k  .x.  Y
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2618   A.wral 2727   E.wrex 2728   E!wreu 2729    \ cdif 3337   {csn 3889   ` cfv 5430  (class class class)co 6103   Basecbs 14186  Scalarcsca 14253   .scvsca 14254   0gc0g 14390   LSpanclspn 17064   LVecclvec 17195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-tpos 6757  df-recs 6844  df-rdg 6878  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-2 10392  df-3 10393  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-ress 14193  df-plusg 14263  df-mulr 14264  df-0g 14392  df-mnd 15427  df-grp 15557  df-minusg 15558  df-sbg 15559  df-mgp 16604  df-ur 16616  df-rng 16659  df-oppr 16727  df-dvdsr 16745  df-unit 16746  df-invr 16776  df-drng 16846  df-lmod 16962  df-lss 17026  df-lsp 17065  df-lvec 17196
This theorem is referenced by:  hdmap14lem3  35530
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