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

Theorem lspdisj 17321
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 17302 . . . . . . . . . 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 2451 . . . . . . . . . 10  |-  (Scalar `  W )  =  (Scalar `  W )
6 eqid 2451 . . . . . . . . . 10  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
7 lspdisj.v . . . . . . . . . 10  |-  V  =  ( Base `  W
)
8 eqid 2451 . . . . . . . . . 10  |-  ( .s
`  W )  =  ( .s `  W
)
9 lspdisj.n . . . . . . . . . 10  |-  N  =  ( LSpan `  W )
105, 6, 7, 8, 9lspsnel 17199 . . . . . . . . 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 756 . . . . . . . . . 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 754 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  -> 
v  e.  U )
1814, 17eqeltrrd 2540 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  v  e.  U )  /\  (
k  e.  ( Base `  (Scalar `  W )
)  /\  v  =  ( k ( .s
`  W ) X ) ) )  -> 
( k ( .s
`  W ) X )  e.  U )
19 eqid 2451 . . . . . . . . . . . . . . . 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 755 . . . . . . . . . . . . . . . 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 17306 . . . . . . . . . . . . . . 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 6208 . . . . . . . . . 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 17096 . . . . . . . . . . 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 2496 . . . . . . . . 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 2939 . . . . . 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 3640 . . . 4  |-  ( v  e.  ( ( N `
 { X }
)  i^i  U )  <->  ( v  e.  ( N `
 { X }
)  /\  v  e.  U ) )
43 elsn 3992 . . . 4  |-  ( v  e.  {  .0.  }  <->  v  =  .0.  )
4441, 42, 433imtr4g 270 . . 3  |-  ( ph  ->  ( v  e.  ( ( N `  { X } )  i^i  U
)  ->  v  e.  {  .0.  } ) )
4544ssrdv 3463 . 2  |-  ( ph  ->  ( ( N `  { X } )  i^i 
U )  C_  {  .0.  } )
467, 20, 9lspsncl 17173 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  S
)
473, 4, 46syl2anc 661 . . . 4  |-  ( ph  ->  ( N `  { X } )  e.  S
)
4833, 20lss0ss 17145 . . . 4  |-  ( ( W  e.  LMod  /\  ( N `  { X } )  e.  S
)  ->  {  .0.  } 
C_  ( N `  { X } ) )
493, 47, 48syl2anc 661 . . 3  |-  ( ph  ->  {  .0.  }  C_  ( N `  { X } ) )
5033, 20lss0ss 17145 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  S )  ->  {  .0.  } 
C_  U )
513, 22, 50syl2anc 661 . . 3  |-  ( ph  ->  {  .0.  }  C_  U )
5249, 51ssind 3675 . 2  |-  ( ph  ->  {  .0.  }  C_  ( ( N `  { X } )  i^i 
U ) )
5345, 52eqssd 3474 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 1370    e. wcel 1758   E.wrex 2796    i^i cin 3428    C_ wss 3429   {csn 3978   ` cfv 5519  (class class class)co 6193   Basecbs 14285  Scalarcsca 14352   .scvsca 14353   0gc0g 14489   LModclmod 17063   LSubSpclss 17128   LSpanclspn 17167   LVecclvec 17298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-tpos 6848  df-recs 6935  df-rdg 6969  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-2 10484  df-3 10485  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-mulr 14363  df-0g 14491  df-mnd 15526  df-grp 15656  df-minusg 15657  df-sbg 15658  df-mgp 16706  df-ur 16718  df-rng 16762  df-oppr 16830  df-dvdsr 16848  df-unit 16849  df-invr 16879  df-drng 16949  df-lmod 17065  df-lss 17129  df-lsp 17168  df-lvec 17299
This theorem is referenced by:  lspdisjb  17322  lspdisj2  17323  lvecindp  17334
  Copyright terms: Public domain W3C validator