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Theorem lspdisj 17898
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 17879 . . . . . . . . . 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 2457 . . . . . . . . . 10  |-  (Scalar `  W )  =  (Scalar `  W )
6 eqid 2457 . . . . . . . . . 10  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
7 lspdisj.v . . . . . . . . . 10  |-  V  =  ( Base `  W
)
8 eqid 2457 . . . . . . . . . 10  |-  ( .s
`  W )  =  ( .s `  W
)
9 lspdisj.n . . . . . . . . . 10  |-  N  =  ( LSpan `  W )
105, 6, 7, 8, 9lspsnel 17776 . . . . . . . . 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 661 . . . . . . . 8  |-  ( ph  ->  ( v  e.  ( N `  { X } )  <->  E. k  e.  ( Base `  (Scalar `  W ) ) v  =  ( k ( .s `  W ) X ) ) )
1211biimpa 484 . . . . . . 7  |-  ( (
ph  /\  v  e.  ( N `  { X } ) )  ->  E. k  e.  ( Base `  (Scalar `  W
) ) v  =  ( k ( .s
`  W ) X ) )
1312adantrr 716 . . . . . 6  |-  ( (
ph  /\  ( v  e.  ( N `  { X } )  /\  v  e.  U ) )  ->  E. k  e.  ( Base `  (Scalar `  W
) ) v  =  ( k ( .s
`  W ) X ) )
14 simprr 757 . . . . . . . . . 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 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  ->  -.  X  e.  U
)
17 simplr 755 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  -> 
v  e.  U )
1814, 17eqeltrrd 2546 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  -> 
( k ( .s
`  W ) X )  e.  U )
19 eqid 2457 . . . . . . . . . . . . . . . 16  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
20 lspdisj.s . . . . . . . . . . . . . . . 16  |-  S  =  ( LSubSp `  W )
211ad2antrr 725 . . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  ->  U  e.  S )
244ad2antrr 725 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  ->  X  e.  V )
25 simprl 756 . . . . . . . . . . . . . . . 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 17883 . . . . . . . . . . . . . . 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 210 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  -> 
( k  =  ( 0g `  (Scalar `  W ) )  \/  X  e.  U ) )
2827orcomd 388 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  -> 
( X  e.  U  \/  k  =  ( 0g `  (Scalar `  W
) ) ) )
2928ord 377 . . . . . . . . . . . 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 6311 . . . . . . . . . 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 725 . . . . . . . . . . 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 17672 . . . . . . . . . . 11  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
( 0g `  (Scalar `  W ) ) ( .s `  W ) X )  =  .0.  )
3532, 24, 34syl2anc 661 . . . . . . . . . 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 2502 . . . . . . . . 9  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  -> 
v  =  .0.  )
3736exp32 605 . . . . . . . 8  |-  ( (
ph  /\  v  e.  U )  ->  (
k  e.  ( Base `  (Scalar `  W )
)  ->  ( v  =  ( k ( .s `  W ) X )  ->  v  =  .0.  ) ) )
3837adantrl 715 . . . . . . 7  |-  ( (
ph  /\  ( v  e.  ( N `  { X } )  /\  v  e.  U ) )  -> 
( k  e.  (
Base `  (Scalar `  W
) )  ->  (
v  =  ( k ( .s `  W
) X )  -> 
v  =  .0.  )
) )
3938rexlimdv 2947 . . . . . 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 434 . . . 4  |-  ( ph  ->  ( ( v  e.  ( N `  { X } )  /\  v  e.  U )  ->  v  =  .0.  ) )
42 elin 3683 . . . 4  |-  ( v  e.  ( ( N `
 { X }
)  i^i  U )  <->  ( v  e.  ( N `
 { X }
)  /\  v  e.  U ) )
43 elsn 4046 . . . 4  |-  ( v  e.  {  .0.  }  <->  v  =  .0.  )
4441, 42, 433imtr4g 270 . . 3  |-  ( ph  ->  ( v  e.  ( ( N `  { X } )  i^i  U
)  ->  v  e.  {  .0.  } ) )
4544ssrdv 3505 . 2  |-  ( ph  ->  ( ( N `  { X } )  i^i 
U )  C_  {  .0.  } )
467, 20, 9lspsncl 17750 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  S
)
473, 4, 46syl2anc 661 . . . 4  |-  ( ph  ->  ( N `  { X } )  e.  S
)
4833, 20lss0ss 17722 . . . 4  |-  ( ( W  e.  LMod  /\  ( N `  { X } )  e.  S
)  ->  {  .0.  } 
C_  ( N `  { X } ) )
493, 47, 48syl2anc 661 . . 3  |-  ( ph  ->  {  .0.  }  C_  ( N `  { X } ) )
5033, 20lss0ss 17722 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  {  .0.  } 
C_  U )
513, 22, 50syl2anc 661 . . 3  |-  ( ph  ->  {  .0.  }  C_  U )
5249, 51ssind 3718 . 2  |-  ( ph  ->  {  .0.  }  C_  ( ( N `  { X } )  i^i 
U ) )
5345, 52eqssd 3516 1  |-  ( ph  ->  ( ( N `  { X } )  i^i 
U )  =  {  .0.  } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1395    e. wcel 1819   E.wrex 2808    i^i cin 3470    C_ wss 3471   {csn 4032   ` cfv 5594  (class class class)co 6296   Basecbs 14644  Scalarcsca 14715   .scvsca 14716   0gc0g 14857   LModclmod 17639   LSubSpclss 17705   LSpanclspn 17744   LVecclvec 17875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-tpos 6973  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-ress 14651  df-plusg 14725  df-mulr 14726  df-0g 14859  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-grp 16184  df-minusg 16185  df-sbg 16186  df-mgp 17269  df-ur 17281  df-ring 17327  df-oppr 17399  df-dvdsr 17417  df-unit 17418  df-invr 17448  df-drng 17525  df-lmod 17641  df-lss 17706  df-lsp 17745  df-lvec 17876
This theorem is referenced by:  lspdisjb  17899  lspdisj2  17900  lvecindp  17911
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