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Theorem lspdisj 16126
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 16107 . . . . . . . . . 10  |-  ( W  e.  LVec  ->  W  e. 
LMod )
31, 2syl 16 . . . . . . . . 9  |-  ( ph  ->  W  e.  LMod )
4 lspdisj.x . . . . . . . . 9  |-  ( ph  ->  X  e.  V )
5 eqid 2389 . . . . . . . . . 10  |-  (Scalar `  W )  =  (Scalar `  W )
6 eqid 2389 . . . . . . . . . 10  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
7 lspdisj.v . . . . . . . . . 10  |-  V  =  ( Base `  W
)
8 eqid 2389 . . . . . . . . . 10  |-  ( .s
`  W )  =  ( .s `  W
)
9 lspdisj.n . . . . . . . . . 10  |-  N  =  ( LSpan `  W )
105, 6, 7, 8, 9lspsnel 16008 . . . . . . . . 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 643 . . . . . . . 8  |-  ( ph  ->  ( v  e.  ( N `  { X } )  <->  E. k  e.  ( Base `  (Scalar `  W ) ) v  =  ( k ( .s `  W ) X ) ) )
1211biimpa 471 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( N `  { X } ) )  ->  E. k  e.  ( Base `  (Scalar `  W
) ) v  =  ( k ( .s
`  W ) X ) )
1312adantrr 698 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( N `  { X } )  /\  v  e.  U ) )  ->  E. k  e.  ( Base `  (Scalar `  W
) ) v  =  ( k ( .s
`  W ) X ) )
14 simprr 734 . . . . . . . . . 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 707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  ->  -.  X  e.  U
)
17 simplr 732 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  -> 
v  e.  U )
1814, 17eqeltrrd 2464 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  -> 
( k ( .s
`  W ) X )  e.  U )
19 eqid 2389 . . . . . . . . . . . . . . . 16  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
20 lspdisj.s . . . . . . . . . . . . . . . 16  |-  S  =  ( LSubSp `  W )
211ad2antrr 707 . . . . . . . . . . . . . . . 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 707 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  ->  U  e.  S )
244ad2antrr 707 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  ->  X  e.  V )
25 simprl 733 . . . . . . . . . . . . . . . 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 16111 . . . . . . . . . . . . . . 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 202 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  -> 
( k  =  ( 0g `  (Scalar `  W ) )  \/  X  e.  U ) )
2827orcomd 378 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  -> 
( X  e.  U  \/  k  =  ( 0g `  (Scalar `  W
) ) ) )
2928ord 367 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  -> 
( -.  X  e.  U  ->  k  =  ( 0g `  (Scalar `  W ) ) ) )
3016, 29mpd 15 . . . . . . . . . . 11  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  -> 
k  =  ( 0g
`  (Scalar `  W )
) )
3130oveq1d 6037 . . . . . . . . . 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 707 . . . . . . . . . . 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 15912 . . . . . . . . . . 11  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( 0g `  (Scalar `  W ) ) ( .s `  W ) X )  =  .0.  )
3532, 24, 34syl2anc 643 . . . . . . . . . 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 2425 . . . . . . . . 9  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  -> 
v  =  .0.  )
3736exp32 589 . . . . . . . 8  |-  ( (
ph  /\  v  e.  U )  ->  (
k  e.  ( Base `  (Scalar `  W )
)  ->  ( v  =  ( k ( .s `  W ) X )  ->  v  =  .0.  ) ) )
3837adantrl 697 . . . . . . 7  |-  ( (
ph  /\  ( v  e.  ( N `  { X } )  /\  v  e.  U ) )  -> 
( k  e.  (
Base `  (Scalar `  W
) )  ->  (
v  =  ( k ( .s `  W
) X )  -> 
v  =  .0.  )
) )
3938rexlimdv 2774 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( N `  { X } )  /\  v  e.  U ) )  -> 
( E. k  e.  ( Base `  (Scalar `  W ) ) v  =  ( k ( .s `  W ) X )  ->  v  =  .0.  ) )
4013, 39mpd 15 . . . . 5  |-  ( (
ph  /\  ( v  e.  ( N `  { X } )  /\  v  e.  U ) )  -> 
v  =  .0.  )
4140ex 424 . . . 4  |-  ( ph  ->  ( ( v  e.  ( N `  { X } )  /\  v  e.  U )  ->  v  =  .0.  ) )
42 elin 3475 . . . 4  |-  ( v  e.  ( ( N `
 { X }
)  i^i  U )  <->  ( v  e.  ( N `
 { X }
)  /\  v  e.  U ) )
43 elsn 3774 . . . 4  |-  ( v  e.  {  .0.  }  <->  v  =  .0.  )
4441, 42, 433imtr4g 262 . . 3  |-  ( ph  ->  ( v  e.  ( ( N `  { X } )  i^i  U
)  ->  v  e.  {  .0.  } ) )
4544ssrdv 3299 . 2  |-  ( ph  ->  ( ( N `  { X } )  i^i 
U )  C_  {  .0.  } )
467, 20, 9lspsncl 15982 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  S
)
473, 4, 46syl2anc 643 . . . 4  |-  ( ph  ->  ( N `  { X } )  e.  S
)
4833, 20lss0ss 15954 . . . 4  |-  ( ( W  e.  LMod  /\  ( N `  { X } )  e.  S
)  ->  {  .0.  } 
C_  ( N `  { X } ) )
493, 47, 48syl2anc 643 . . 3  |-  ( ph  ->  {  .0.  }  C_  ( N `  { X } ) )
5033, 20lss0ss 15954 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  {  .0.  } 
C_  U )
513, 22, 50syl2anc 643 . . 3  |-  ( ph  ->  {  .0.  }  C_  U )
5249, 51ssind 3510 . 2  |-  ( ph  ->  {  .0.  }  C_  ( ( N `  { X } )  i^i 
U ) )
5345, 52eqssd 3310 1  |-  ( ph  ->  ( ( N `  { X } )  i^i 
U )  =  {  .0.  } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717   E.wrex 2652    i^i cin 3264    C_ wss 3265   {csn 3759   ` cfv 5396  (class class class)co 6022   Basecbs 13398  Scalarcsca 13461   .scvsca 13462   0gc0g 13652   LModclmod 15879   LSubSpclss 15937   LSpanclspn 15976   LVecclvec 16103
This theorem is referenced by:  lspdisjb  16127  lspdisj2  16128  lvecindp  16139
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-tpos 6417  df-riota 6487  df-recs 6571  df-rdg 6606  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-3 9993  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-mulr 13472  df-0g 13656  df-mnd 14619  df-grp 14741  df-minusg 14742  df-sbg 14743  df-mgp 15578  df-rng 15592  df-ur 15594  df-oppr 15657  df-dvdsr 15675  df-unit 15676  df-invr 15706  df-drng 15766  df-lmod 15881  df-lss 15938  df-lsp 15977  df-lvec 16104
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