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Theorem lspsneu 16115
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 3268 . . . . . . 7  |-  ( ph  ->  Y  e.  V )
111, 2, 3, 4, 5, 6, 7, 8, 10lspsneq 16114 . . . . . 6  |-  ( ph  ->  ( ( N `  { X } )  =  ( N `  { Y } )  <->  E. j  e.  ( K  \  { O } ) X  =  ( j  .x.  Y
) ) )
1211biimpd 199 . . . . 5  |-  ( ph  ->  ( ( N `  { X } )  =  ( N `  { Y } )  ->  E. j  e.  ( K  \  { O } ) X  =  ( j  .x.  Y
) ) )
13 eqtr2 2398 . . . . . . . . . 10  |-  ( ( X  =  ( j 
.x.  Y )  /\  X  =  ( i  .x.  Y ) )  -> 
( j  .x.  Y
)  =  ( i 
.x.  Y ) )
14133ad2ant3 980 . . . . . . . . 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 981 . . . . . . . . . . 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 983 . . . . . . . . . . 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 3268 . . . . . . . . . 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 984 . . . . . . . . . . 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 3268 . . . . . . . . . 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 3864 . . . . . . . . . . 11  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  Y  =/=  .0.  )
2416, 9, 233syl 19 . . . . . . . . . 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 16104 . . . . . . . . 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 202 . . . . . . . 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 1152 . . . . . . 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 424 . . . . . 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 2736 . . . . 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 520 . . . 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 6020 . . . . . 6  |-  ( j  =  i  ->  (
j  .x.  Y )  =  ( i  .x.  Y ) )
3231eqeq2d 2391 . . . . 5  |-  ( j  =  i  ->  ( X  =  ( j  .x.  Y )  <->  X  =  ( i  .x.  Y
) ) )
3332reu4 3064 . . . 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 219 . . 3  |-  ( ph  ->  ( ( N `  { X } )  =  ( N `  { Y } )  ->  E! j  e.  ( K  \  { O } ) X  =  ( j 
.x.  Y ) ) )
35 reurex 2858 . . . 4  |-  ( E! j  e.  ( K 
\  { O }
) X  =  ( j  .x.  Y )  ->  E. j  e.  ( K  \  { O } ) X  =  ( j  .x.  Y
) )
3635, 11syl5ibr 213 . . 3  |-  ( ph  ->  ( E! j  e.  ( K  \  { O } ) X  =  ( j  .x.  Y
)  ->  ( N `  { X } )  =  ( N `  { Y } ) ) )
3734, 36impbid 184 . 2  |-  ( ph  ->  ( ( N `  { X } )  =  ( N `  { Y } )  <->  E! j  e.  ( K  \  { O } ) X  =  ( j  .x.  Y
) ) )
38 oveq1 6020 . . . 4  |-  ( j  =  k  ->  (
j  .x.  Y )  =  ( k  .x.  Y ) )
3938eqeq2d 2391 . . 3  |-  ( j  =  k  ->  ( X  =  ( j  .x.  Y )  <->  X  =  ( k  .x.  Y
) ) )
4039cbvreuv 2870 . 2  |-  ( E! j  e.  ( K 
\  { O }
) X  =  ( j  .x.  Y )  <-> 
E! k  e.  ( K  \  { O } ) X  =  ( k  .x.  Y
) )
4137, 40syl6bb 253 1  |-  ( ph  ->  ( ( N `  { X } )  =  ( N `  { Y } )  <->  E! k  e.  ( K  \  { O } ) X  =  ( k  .x.  Y
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2543   A.wral 2642   E.wrex 2643   E!wreu 2644    \ cdif 3253   {csn 3750   ` cfv 5387  (class class class)co 6013   Basecbs 13389  Scalarcsca 13452   .scvsca 13453   0gc0g 13643   LSpanclspn 15967   LVecclvec 16094
This theorem is referenced by:  hdmap14lem3  32039
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-tpos 6408  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-3 9984  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-ress 13396  df-plusg 13462  df-mulr 13463  df-0g 13647  df-mnd 14610  df-grp 14732  df-minusg 14733  df-sbg 14734  df-mgp 15569  df-rng 15583  df-ur 15585  df-oppr 15648  df-dvdsr 15666  df-unit 15667  df-invr 15697  df-drng 15757  df-lmod 15872  df-lss 15929  df-lsp 15968  df-lvec 16095
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