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

Theorem lspsneu 15876
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 3298 . . . . . . . 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 15875 . . . . . 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 2301 . . . . . . . . . 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 3298 . . . . . . . . . . 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 3298 . . . . . . . . . . 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 3750 . . . . . . . . . . . 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 15865 . . . . . . . . 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 2637 . . . . 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 5865 . . . . . 6  |-  ( j  =  i  ->  (
j  .x.  Y )  =  ( i  .x.  Y ) )
3635eqeq2d 2294 . . . . 5  |-  ( j  =  i  ->  ( X  =  ( j  .x.  Y )  <->  X  =  ( i  .x.  Y
) ) )
3736reu4 2959 . . . 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 2754 . . . 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 5865 . . . 4  |-  ( j  =  k  ->  (
j  .x.  Y )  =  ( k  .x.  Y ) )
4342eqeq2d 2294 . . 3  |-  ( j  =  k  ->  ( X  =  ( j  .x.  Y )  <->  X  =  ( k  .x.  Y
) ) )
4443cbvreuv 2766 . 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 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   E!wreu 2545    \ cdif 3149   {csn 3640   ` cfv 5255  (class class class)co 5858   Basecbs 13148  Scalarcsca 13211   .scvsca 13212   0gc0g 13400   LSpanclspn 15728   LVecclvec 15855
This theorem is referenced by:  hdmap14lem3  32063
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-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-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-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  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-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856
  Copyright terms: Public domain W3C validator