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Theorem lspdisj 15878
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 15859 . . . . . . . . . 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 2283 . . . . . . . . . 10  |-  (Scalar `  W )  =  (Scalar `  W )
6 eqid 2283 . . . . . . . . . 10  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
7 lspdisj.v . . . . . . . . . 10  |-  V  =  ( Base `  W
)
8 eqid 2283 . . . . . . . . . 10  |-  ( .s
`  W )  =  ( .s `  W
)
9 lspdisj.n . . . . . . . . . 10  |-  N  =  ( LSpan `  W )
105, 6, 7, 8, 9lspsnel 15760 . . . . . . . . 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 2358 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  -> 
( k ( .s
`  W ) X )  e.  U )
19 eqid 2283 . . . . . . . . . . . . . . . 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 15863 . . . . . . . . . . . . . . 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 5873 . . . . . . . . . 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 15663 . . . . . . . . . . 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 2319 . . . . . . . . 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 2666 . . . . . 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 3358 . . . 4  |-  ( v  e.  ( ( N `
 { X }
)  i^i  U )  <->  ( v  e.  ( N `
 { X }
)  /\  v  e.  U ) )
43 elsn 3655 . . . 4  |-  ( v  e.  {  .0.  }  <->  v  =  .0.  )
4441, 42, 433imtr4g 261 . . 3  |-  ( ph  ->  ( v  e.  ( ( N `  { X } )  i^i  U
)  ->  v  e.  {  .0.  } ) )
4544ssrdv 3185 . 2  |-  ( ph  ->  ( ( N `  { X } )  i^i 
U )  C_  {  .0.  } )
467, 20, 9lspsncl 15734 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  S
)
473, 4, 46syl2anc 642 . . . 4  |-  ( ph  ->  ( N `  { X } )  e.  S
)
4833, 20lss0ss 15706 . . . 4  |-  ( ( W  e.  LMod  /\  ( N `  { X } )  e.  S
)  ->  {  .0.  } 
C_  ( N `  { X } ) )
493, 47, 48syl2anc 642 . . 3  |-  ( ph  ->  {  .0.  }  C_  ( N `  { X } ) )
5033, 20lss0ss 15706 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  {  .0.  } 
C_  U )
513, 22, 50syl2anc 642 . . 3  |-  ( ph  ->  {  .0.  }  C_  U )
5249, 51ssind 3393 . 2  |-  ( ph  ->  {  .0.  }  C_  ( ( N `  { X } )  i^i 
U ) )
5345, 52eqssd 3196 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 1623    e. wcel 1684   E.wrex 2544    i^i cin 3151    C_ wss 3152   {csn 3640   ` cfv 5255  (class class class)co 5858   Basecbs 13148  Scalarcsca 13211   .scvsca 13212   0gc0g 13400   LModclmod 15627   LSubSpclss 15689   LSpanclspn 15728   LVecclvec 15855
This theorem is referenced by:  lspdisjb  15879  lspdisj2  15880  lvecindp  15891
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
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