MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lspsneu Structured version   Unicode version

Theorem lspsneu 17549
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 3488 . . . . . . 7  |-  ( ph  ->  Y  e.  V )
111, 2, 3, 4, 5, 6, 7, 8, 10lspsneq 17548 . . . . . 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 2494 . . . . . . . . . 10  |-  ( ( X  =  ( j 
.x.  Y )  /\  X  =  ( i  .x.  Y ) )  -> 
( j  .x.  Y
)  =  ( i 
.x.  Y ) )
14133ad2ant3 1019 . . . . . . . . 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 1020 . . . . . . . . . . 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 1022 . . . . . . . . . . 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 3488 . . . . . . . . . 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 1023 . . . . . . . . . . 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 3488 . . . . . . . . . 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 4153 . . . . . . . . . . 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 17538 . . . . . . . . 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 1195 . . . . . . 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 2887 . . . . 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 6289 . . . . . 6  |-  ( j  =  i  ->  (
j  .x.  Y )  =  ( i  .x.  Y ) )
3231eqeq2d 2481 . . . . 5  |-  ( j  =  i  ->  ( X  =  ( j  .x.  Y )  <->  X  =  ( i  .x.  Y
) ) )
3332reu4 3297 . . . 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 3078 . . . 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 6289 . . . 4  |-  ( j  =  k  ->  (
j  .x.  Y )  =  ( k  .x.  Y ) )
3938eqeq2d 2481 . . 3  |-  ( j  =  k  ->  ( X  =  ( j  .x.  Y )  <->  X  =  ( k  .x.  Y
) ) )
4039cbvreuv 3090 . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   E!wreu 2816    \ cdif 3473   {csn 4027   ` cfv 5586  (class class class)co 6282   Basecbs 14483  Scalarcsca 14551   .scvsca 14552   0gc0g 14688   LSpanclspn 17397   LVecclvec 17528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-tpos 6952  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-mulr 14562  df-0g 14690  df-mnd 15725  df-grp 15855  df-minusg 15856  df-sbg 15857  df-mgp 16929  df-ur 16941  df-rng 16985  df-oppr 17053  df-dvdsr 17071  df-unit 17072  df-invr 17102  df-drng 17178  df-lmod 17294  df-lss 17359  df-lsp 17398  df-lvec 17529
This theorem is referenced by:  hdmap14lem3  36670
  Copyright terms: Public domain W3C validator