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Theorem lspsneu 15892
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.  }
) )
10 eldifi 3311 . . . . . . . 8  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  Y  e.  V )
119, 10syl 15 . . . . . . 7  |-  ( ph  ->  Y  e.  V )
121, 2, 3, 4, 5, 6, 7, 8, 11lspsneq 15891 . . . . . 6  |-  ( ph  ->  ( ( N `  { X } )  =  ( N `  { Y } )  <->  E. j  e.  ( K  \  { O } ) X  =  ( j  .x.  Y
) ) )
1312biimpd 198 . . . . 5  |-  ( ph  ->  ( ( N `  { X } )  =  ( N `  { Y } )  ->  E. j  e.  ( K  \  { O } ) X  =  ( j  .x.  Y
) ) )
14 eqtr2 2314 . . . . . . . . . 10  |-  ( ( X  =  ( j 
.x.  Y )  /\  X  =  ( i  .x.  Y ) )  -> 
( j  .x.  Y
)  =  ( i 
.x.  Y ) )
15143ad2ant3 978 . . . . . . . . 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 ) )
16 lspsneu.z . . . . . . . . . 10  |-  .0.  =  ( 0g `  W )
17 simp1l 979 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  ( j  e.  ( K  \  { O } )  /\  i  e.  ( K  \  { O } ) )  /\  ( X  =  (
j  .x.  Y )  /\  X  =  (
i  .x.  Y )
) )  ->  ph )
1817, 7syl 15 . . . . . . . . . 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 )
19 simp2l 981 . . . . . . . . . . 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 } ) )
20 eldifi 3311 . . . . . . . . . . 11  |-  ( j  e.  ( K  \  { O } )  -> 
j  e.  K )
2119, 20syl 15 . . . . . . . . . 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 )
22 simp2r 982 . . . . . . . . . . 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 } ) )
23 eldifi 3311 . . . . . . . . . . 11  |-  ( i  e.  ( K  \  { O } )  -> 
i  e.  K )
2422, 23syl 15 . . . . . . . . . 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 )
2517, 11syl 15 . . . . . . . . . 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 )
26 eldifsni 3763 . . . . . . . . . . . 12  |-  ( Y  e.  ( V  \  {  .0.  } )  ->  Y  =/=  .0.  )
279, 26syl 15 . . . . . . . . . . 11  |-  ( ph  ->  Y  =/=  .0.  )
2817, 27syl 15 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  ( j  e.  ( K  \  { O } )  /\  i  e.  ( K  \  { O } ) )  /\  ( X  =  (
j  .x.  Y )  /\  X  =  (
i  .x.  Y )
) )  ->  Y  =/=  .0.  )
291, 5, 2, 3, 16, 18, 21, 24, 25, 28lvecvscan2 15881 . . . . . . . . 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 ) )
3015, 29mpbid 201 . . . . . . . 8  |-  ( ( ( ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  /\  ( j  e.  ( K  \  { O } )  /\  i  e.  ( K  \  { O } ) )  /\  ( X  =  (
j  .x.  Y )  /\  X  =  (
i  .x.  Y )
) )  ->  j  =  i )
31303exp 1150 . . . . . . 7  |-  ( (
ph  /\  ( N `  { X } )  =  ( N `  { Y } ) )  ->  ( ( j  e.  ( K  \  { O } )  /\  i  e.  ( K  \  { O } ) )  ->  ( ( X  =  ( j  .x.  Y )  /\  X  =  ( i  .x.  Y ) )  -> 
j  =  i ) ) )
3231ex 423 . . . . . 6  |-  ( ph  ->  ( ( N `  { X } )  =  ( N `  { Y } )  ->  (
( j  e.  ( K  \  { O } )  /\  i  e.  ( K  \  { O } ) )  -> 
( ( X  =  ( j  .x.  Y
)  /\  X  =  ( i  .x.  Y
) )  ->  j  =  i ) ) ) )
3332ralrimdvv 2650 . . . . 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 ) ) )
3413, 33jcad 519 . . . 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 ) ) ) )
35 oveq1 5881 . . . . . 6  |-  ( j  =  i  ->  (
j  .x.  Y )  =  ( i  .x.  Y ) )
3635eqeq2d 2307 . . . . 5  |-  ( j  =  i  ->  ( X  =  ( j  .x.  Y )  <->  X  =  ( i  .x.  Y
) ) )
3736reu4 2972 . . . 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 ) ) )
3834, 37syl6ibr 218 . . 3  |-  ( ph  ->  ( ( N `  { X } )  =  ( N `  { Y } )  ->  E! j  e.  ( K  \  { O } ) X  =  ( j 
.x.  Y ) ) )
39 reurex 2767 . . . 4  |-  ( E! j  e.  ( K 
\  { O }
) X  =  ( j  .x.  Y )  ->  E. j  e.  ( K  \  { O } ) X  =  ( j  .x.  Y
) )
4039, 12syl5ibr 212 . . 3  |-  ( ph  ->  ( E! j  e.  ( K  \  { O } ) X  =  ( j  .x.  Y
)  ->  ( N `  { X } )  =  ( N `  { Y } ) ) )
4138, 40impbid 183 . 2  |-  ( ph  ->  ( ( N `  { X } )  =  ( N `  { Y } )  <->  E! j  e.  ( K  \  { O } ) X  =  ( j  .x.  Y
) ) )
42 oveq1 5881 . . . 4  |-  ( j  =  k  ->  (
j  .x.  Y )  =  ( k  .x.  Y ) )
4342eqeq2d 2307 . . 3  |-  ( j  =  k  ->  ( X  =  ( j  .x.  Y )  <->  X  =  ( k  .x.  Y
) ) )
4443cbvreuv 2779 . 2  |-  ( E! j  e.  ( K 
\  { O }
) X  =  ( j  .x.  Y )  <-> 
E! k  e.  ( K  \  { O } ) X  =  ( k  .x.  Y
) )
4541, 44syl6bb 252 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   E!wreu 2558    \ cdif 3162   {csn 3653   ` cfv 5271  (class class class)co 5874   Basecbs 13164  Scalarcsca 13227   .scvsca 13228   0gc0g 13416   LSpanclspn 15744   LVecclvec 15871
This theorem is referenced by:  hdmap14lem3  32685
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-drng 15530  df-lmod 15645  df-lss 15706  df-lsp 15745  df-lvec 15872
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