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

Theorem lspdisj 15894
Description: The span of a vector not in a subspace is disjoint with the subspace. (Contributed by NM, 6-Apr-2015.)
Hypotheses
Ref Expression
lspdisj.v  |-  V  =  ( Base `  W
)
lspdisj.o  |-  .0.  =  ( 0g `  W )
lspdisj.n  |-  N  =  ( LSpan `  W )
lspdisj.s  |-  S  =  ( LSubSp `  W )
lspdisj.w  |-  ( ph  ->  W  e.  LVec )
lspdisj.u  |-  ( ph  ->  U  e.  S )
lspdisj.x  |-  ( ph  ->  X  e.  V )
lspdisj.e  |-  ( ph  ->  -.  X  e.  U
)
Assertion
Ref Expression
lspdisj  |-  ( ph  ->  ( ( N `  { X } )  i^i 
U )  =  {  .0.  } )

Proof of Theorem lspdisj
Dummy variables  v 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lspdisj.w . . . . . . . . . 10  |-  ( ph  ->  W  e.  LVec )
2 lveclmod 15875 . . . . . . . . . 10  |-  ( W  e.  LVec  ->  W  e. 
LMod )
31, 2syl 15 . . . . . . . . 9  |-  ( ph  ->  W  e.  LMod )
4 lspdisj.x . . . . . . . . 9  |-  ( ph  ->  X  e.  V )
5 eqid 2296 . . . . . . . . . 10  |-  (Scalar `  W )  =  (Scalar `  W )
6 eqid 2296 . . . . . . . . . 10  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
7 lspdisj.v . . . . . . . . . 10  |-  V  =  ( Base `  W
)
8 eqid 2296 . . . . . . . . . 10  |-  ( .s
`  W )  =  ( .s `  W
)
9 lspdisj.n . . . . . . . . . 10  |-  N  =  ( LSpan `  W )
105, 6, 7, 8, 9lspsnel 15776 . . . . . . . . 9  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
v  e.  ( N `
 { X }
)  <->  E. k  e.  (
Base `  (Scalar `  W
) ) v  =  ( k ( .s
`  W ) X ) ) )
113, 4, 10syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( v  e.  ( N `  { X } )  <->  E. k  e.  ( Base `  (Scalar `  W ) ) v  =  ( k ( .s `  W ) X ) ) )
1211biimpa 470 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( N `  { X } ) )  ->  E. k  e.  ( Base `  (Scalar `  W
) ) v  =  ( k ( .s
`  W ) X ) )
1312adantrr 697 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( N `  { X } )  /\  v  e.  U ) )  ->  E. k  e.  ( Base `  (Scalar `  W
) ) v  =  ( k ( .s
`  W ) X ) )
14 simprr 733 . . . . . . . . . 10  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  -> 
v  =  ( k ( .s `  W
) X ) )
15 lspdisj.e . . . . . . . . . . . . 13  |-  ( ph  ->  -.  X  e.  U
)
1615ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  ->  -.  X  e.  U
)
17 simplr 731 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  -> 
v  e.  U )
1814, 17eqeltrrd 2371 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  -> 
( k ( .s
`  W ) X )  e.  U )
19 eqid 2296 . . . . . . . . . . . . . . . 16  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
20 lspdisj.s . . . . . . . . . . . . . . . 16  |-  S  =  ( LSubSp `  W )
211ad2antrr 706 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  ->  W  e.  LVec )
22 lspdisj.u . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  U  e.  S )
2322ad2antrr 706 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  ->  U  e.  S )
244ad2antrr 706 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  ->  X  e.  V )
25 simprl 732 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  -> 
k  e.  ( Base `  (Scalar `  W )
) )
267, 8, 5, 6, 19, 20, 21, 23, 24, 25lssvs0or 15879 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  -> 
( ( k ( .s `  W ) X )  e.  U  <->  ( k  =  ( 0g
`  (Scalar `  W )
)  \/  X  e.  U ) ) )
2718, 26mpbid 201 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  -> 
( k  =  ( 0g `  (Scalar `  W ) )  \/  X  e.  U ) )
2827orcomd 377 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  -> 
( X  e.  U  \/  k  =  ( 0g `  (Scalar `  W
) ) ) )
2928ord 366 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  -> 
( -.  X  e.  U  ->  k  =  ( 0g `  (Scalar `  W ) ) ) )
3016, 29mpd 14 . . . . . . . . . . 11  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  -> 
k  =  ( 0g
`  (Scalar `  W )
) )
3130oveq1d 5889 . . . . . . . . . 10  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  -> 
( k ( .s
`  W ) X )  =  ( ( 0g `  (Scalar `  W ) ) ( .s `  W ) X ) )
323ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  ->  W  e.  LMod )
33 lspdisj.o . . . . . . . . . . . 12  |-  .0.  =  ( 0g `  W )
347, 5, 8, 19, 33lmod0vs 15679 . . . . . . . . . . 11  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( 0g `  (Scalar `  W ) ) ( .s `  W ) X )  =  .0.  )
3532, 24, 34syl2anc 642 . . . . . . . . . 10  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  -> 
( ( 0g `  (Scalar `  W ) ) ( .s `  W
) X )  =  .0.  )
3614, 31, 353eqtrd 2332 . . . . . . . . 9  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  -> 
v  =  .0.  )
3736exp32 588 . . . . . . . 8  |-  ( (
ph  /\  v  e.  U )  ->  (
k  e.  ( Base `  (Scalar `  W )
)  ->  ( v  =  ( k ( .s `  W ) X )  ->  v  =  .0.  ) ) )
3837adantrl 696 . . . . . . 7  |-  ( (
ph  /\  ( v  e.  ( N `  { X } )  /\  v  e.  U ) )  -> 
( k  e.  (
Base `  (Scalar `  W
) )  ->  (
v  =  ( k ( .s `  W
) X )  -> 
v  =  .0.  )
) )
3938rexlimdv 2679 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( N `  { X } )  /\  v  e.  U ) )  -> 
( E. k  e.  ( Base `  (Scalar `  W ) ) v  =  ( k ( .s `  W ) X )  ->  v  =  .0.  ) )
4013, 39mpd 14 . . . . 5  |-  ( (
ph  /\  ( v  e.  ( N `  { X } )  /\  v  e.  U ) )  -> 
v  =  .0.  )
4140ex 423 . . . 4  |-  ( ph  ->  ( ( v  e.  ( N `  { X } )  /\  v  e.  U )  ->  v  =  .0.  ) )
42 elin 3371 . . . 4  |-  ( v  e.  ( ( N `
 { X }
)  i^i  U )  <->  ( v  e.  ( N `
 { X }
)  /\  v  e.  U ) )
43 elsn 3668 . . . 4  |-  ( v  e.  {  .0.  }  <->  v  =  .0.  )
4441, 42, 433imtr4g 261 . . 3  |-  ( ph  ->  ( v  e.  ( ( N `  { X } )  i^i  U
)  ->  v  e.  {  .0.  } ) )
4544ssrdv 3198 . 2  |-  ( ph  ->  ( ( N `  { X } )  i^i 
U )  C_  {  .0.  } )
467, 20, 9lspsncl 15750 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  S
)
473, 4, 46syl2anc 642 . . . 4  |-  ( ph  ->  ( N `  { X } )  e.  S
)
4833, 20lss0ss 15722 . . . 4  |-  ( ( W  e.  LMod  /\  ( N `  { X } )  e.  S
)  ->  {  .0.  } 
C_  ( N `  { X } ) )
493, 47, 48syl2anc 642 . . 3  |-  ( ph  ->  {  .0.  }  C_  ( N `  { X } ) )
5033, 20lss0ss 15722 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  {  .0.  } 
C_  U )
513, 22, 50syl2anc 642 . . 3  |-  ( ph  ->  {  .0.  }  C_  U )
5249, 51ssind 3406 . 2  |-  ( ph  ->  {  .0.  }  C_  ( ( N `  { X } )  i^i 
U ) )
5345, 52eqssd 3209 1  |-  ( ph  ->  ( ( N `  { X } )  i^i 
U )  =  {  .0.  } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557    i^i cin 3164    C_ wss 3165   {csn 3653   ` cfv 5271  (class class class)co 5874   Basecbs 13164  Scalarcsca 13227   .scvsca 13228   0gc0g 13416   LModclmod 15643   LSubSpclss 15705   LSpanclspn 15744   LVecclvec 15871
This theorem is referenced by:  lspdisjb  15895  lspdisj2  15896  lvecindp  15907
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
  Copyright terms: Public domain W3C validator