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Theorem lshpdisj 32632
Description: A hyperplane and the span in the hyperplane definition are disjoint. (Contributed by NM, 3-Jul-2014.)
Hypotheses
Ref Expression
lshpdisj.v  |-  V  =  ( Base `  W
)
lshpdisj.o  |-  .0.  =  ( 0g `  W )
lshpdisj.n  |-  N  =  ( LSpan `  W )
lshpdisj.p  |-  .(+)  =  (
LSSum `  W )
lshpdisj.h  |-  H  =  (LSHyp `  W )
lshpdisj.w  |-  ( ph  ->  W  e.  LVec )
lshpdisj.u  |-  ( ph  ->  U  e.  H )
lshpdisj.x  |-  ( ph  ->  X  e.  V )
lshpdisj.e  |-  ( ph  ->  ( U  .(+)  ( N `
 { X }
) )  =  V )
Assertion
Ref Expression
lshpdisj  |-  ( ph  ->  ( U  i^i  ( N `  { X } ) )  =  {  .0.  } )

Proof of Theorem lshpdisj
Dummy variables  v 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lshpdisj.w . . . . . . . . 9  |-  ( ph  ->  W  e.  LVec )
2 lveclmod 17187 . . . . . . . . 9  |-  ( W  e.  LVec  ->  W  e. 
LMod )
31, 2syl 16 . . . . . . . 8  |-  ( ph  ->  W  e.  LMod )
43adantr 465 . . . . . . 7  |-  ( (
ph  /\  v  e.  U )  ->  W  e.  LMod )
5 lshpdisj.x . . . . . . . 8  |-  ( ph  ->  X  e.  V )
65adantr 465 . . . . . . 7  |-  ( (
ph  /\  v  e.  U )  ->  X  e.  V )
7 eqid 2443 . . . . . . . 8  |-  (Scalar `  W )  =  (Scalar `  W )
8 eqid 2443 . . . . . . . 8  |-  ( Base `  (Scalar `  W )
)  =  ( Base `  (Scalar `  W )
)
9 lshpdisj.v . . . . . . . 8  |-  V  =  ( Base `  W
)
10 eqid 2443 . . . . . . . 8  |-  ( .s
`  W )  =  ( .s `  W
)
11 lshpdisj.n . . . . . . . 8  |-  N  =  ( LSpan `  W )
127, 8, 9, 10, 11lspsnel 17084 . . . . . . 7  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  (
v  e.  ( N `
 { X }
)  <->  E. k  e.  (
Base `  (Scalar `  W
) ) v  =  ( k ( .s
`  W ) X ) ) )
134, 6, 12syl2anc 661 . . . . . 6  |-  ( (
ph  /\  v  e.  U )  ->  (
v  e.  ( N `
 { X }
)  <->  E. k  e.  (
Base `  (Scalar `  W
) ) v  =  ( k ( .s
`  W ) X ) ) )
14 lshpdisj.p . . . . . . . . . . . . . . . . 17  |-  .(+)  =  (
LSSum `  W )
15 lshpdisj.h . . . . . . . . . . . . . . . . 17  |-  H  =  (LSHyp `  W )
16 lshpdisj.u . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  U  e.  H )
17 lshpdisj.e . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( U  .(+)  ( N `
 { X }
) )  =  V )
189, 11, 14, 15, 3, 16, 5, 17lshpnel 32628 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  -.  X  e.  U
)
1918ad2antrr 725 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  ( k ( .s `  W ) X )  =/=  .0.  )  ->  -.  X  e.  U )
20 lshpdisj.o . . . . . . . . . . . . . . . 16  |-  .0.  =  ( 0g `  W )
21 eqid 2443 . . . . . . . . . . . . . . . 16  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
221ad2antrr 725 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  ( k ( .s `  W ) X )  =/=  .0.  )  ->  W  e.  LVec )
2321, 15, 3, 16lshplss 32626 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  U  e.  ( LSubSp `  W ) )
2423ad2antrr 725 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  ( k ( .s `  W ) X )  =/=  .0.  )  ->  U  e.  (
LSubSp `  W ) )
255ad2antrr 725 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  ( k ( .s `  W ) X )  =/=  .0.  )  ->  X  e.  V
)
263adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) ) )  ->  W  e.  LMod )
27 simpr 461 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) ) )  -> 
k  e.  ( Base `  (Scalar `  W )
) )
285adantr 465 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) ) )  ->  X  e.  V )
299, 10, 7, 8, 11, 26, 27, 28lspsneli 17082 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) ) )  -> 
( k ( .s
`  W ) X )  e.  ( N `
 { X }
) )
3029adantr 465 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  ( k ( .s `  W ) X )  =/=  .0.  )  ->  ( k ( .s `  W ) X )  e.  ( N `  { X } ) )
31 simpr 461 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  ( k ( .s `  W ) X )  =/=  .0.  )  ->  ( k ( .s `  W ) X )  =/=  .0.  )
329, 20, 21, 11, 22, 24, 25, 30, 31lspsnel4 17205 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  ( k ( .s `  W ) X )  =/=  .0.  )  ->  ( X  e.  U  <->  ( k ( .s `  W ) X )  e.  U
) )
3319, 32mtbid 300 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  k  e.  ( Base `  (Scalar `  W ) ) )  /\  ( k ( .s `  W ) X )  =/=  .0.  )  ->  -.  ( k
( .s `  W
) X )  e.  U )
3433ex 434 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) ) )  -> 
( ( k ( .s `  W ) X )  =/=  .0.  ->  -.  ( k ( .s `  W ) X )  e.  U
) )
3534necon4ad 2672 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) ) )  -> 
( ( k ( .s `  W ) X )  e.  U  ->  ( k ( .s
`  W ) X )  =  .0.  )
)
36 eleq1 2503 . . . . . . . . . . . . 13  |-  ( v  =  ( k ( .s `  W ) X )  ->  (
v  e.  U  <->  ( k
( .s `  W
) X )  e.  U ) )
37 eqeq1 2449 . . . . . . . . . . . . 13  |-  ( v  =  ( k ( .s `  W ) X )  ->  (
v  =  .0.  <->  ( k
( .s `  W
) X )  =  .0.  ) )
3836, 37imbi12d 320 . . . . . . . . . . . 12  |-  ( v  =  ( k ( .s `  W ) X )  ->  (
( v  e.  U  ->  v  =  .0.  )  <->  ( ( k ( .s
`  W ) X )  e.  U  -> 
( k ( .s
`  W ) X )  =  .0.  )
) )
3935, 38syl5ibrcom 222 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ( Base `  (Scalar `  W
) ) )  -> 
( v  =  ( k ( .s `  W ) X )  ->  ( v  e.  U  ->  v  =  .0.  ) ) )
4039ex 434 . . . . . . . . . 10  |-  ( ph  ->  ( k  e.  (
Base `  (Scalar `  W
) )  ->  (
v  =  ( k ( .s `  W
) X )  -> 
( v  e.  U  ->  v  =  .0.  )
) ) )
4140com23 78 . . . . . . . . 9  |-  ( ph  ->  ( v  =  ( k ( .s `  W ) X )  ->  ( k  e.  ( Base `  (Scalar `  W ) )  -> 
( v  e.  U  ->  v  =  .0.  )
) ) )
4241com24 87 . . . . . . . 8  |-  ( ph  ->  ( v  e.  U  ->  ( k  e.  (
Base `  (Scalar `  W
) )  ->  (
v  =  ( k ( .s `  W
) X )  -> 
v  =  .0.  )
) ) )
4342imp31 432 . . . . . . 7  |-  ( ( ( ph  /\  v  e.  U )  /\  k  e.  ( Base `  (Scalar `  W ) ) )  ->  ( v  =  ( k ( .s
`  W ) X )  ->  v  =  .0.  ) )
4443rexlimdva 2841 . . . . . 6  |-  ( (
ph  /\  v  e.  U )  ->  ( E. k  e.  ( Base `  (Scalar `  W
) ) v  =  ( k ( .s
`  W ) X )  ->  v  =  .0.  ) )
4513, 44sylbid 215 . . . . 5  |-  ( (
ph  /\  v  e.  U )  ->  (
v  e.  ( N `
 { X }
)  ->  v  =  .0.  ) )
4645expimpd 603 . . . 4  |-  ( ph  ->  ( ( v  e.  U  /\  v  e.  ( N `  { X } ) )  -> 
v  =  .0.  )
)
47 elin 3539 . . . 4  |-  ( v  e.  ( U  i^i  ( N `  { X } ) )  <->  ( v  e.  U  /\  v  e.  ( N `  { X } ) ) )
48 elsn 3891 . . . 4  |-  ( v  e.  {  .0.  }  <->  v  =  .0.  )
4946, 47, 483imtr4g 270 . . 3  |-  ( ph  ->  ( v  e.  ( U  i^i  ( N `
 { X }
) )  ->  v  e.  {  .0.  } ) )
5049ssrdv 3362 . 2  |-  ( ph  ->  ( U  i^i  ( N `  { X } ) )  C_  {  .0.  } )
519, 21, 11lspsncl 17058 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V )  ->  ( N `  { X } )  e.  (
LSubSp `  W ) )
523, 5, 51syl2anc 661 . . . 4  |-  ( ph  ->  ( N `  { X } )  e.  (
LSubSp `  W ) )
5321lssincl 17046 . . . 4  |-  ( ( W  e.  LMod  /\  U  e.  ( LSubSp `  W )  /\  ( N `  { X } )  e.  (
LSubSp `  W ) )  ->  ( U  i^i  ( N `  { X } ) )  e.  ( LSubSp `  W )
)
543, 23, 52, 53syl3anc 1218 . . 3  |-  ( ph  ->  ( U  i^i  ( N `  { X } ) )  e.  ( LSubSp `  W )
)
5520, 21lss0ss 17030 . . 3  |-  ( ( W  e.  LMod  /\  ( U  i^i  ( N `  { X } ) )  e.  ( LSubSp `  W
) )  ->  {  .0.  } 
C_  ( U  i^i  ( N `  { X } ) ) )
563, 54, 55syl2anc 661 . 2  |-  ( ph  ->  {  .0.  }  C_  ( U  i^i  ( N `  { X } ) ) )
5750, 56eqssd 3373 1  |-  ( ph  ->  ( U  i^i  ( N `  { X } ) )  =  {  .0.  } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2606   E.wrex 2716    i^i cin 3327    C_ wss 3328   {csn 3877   ` cfv 5418  (class class class)co 6091   Basecbs 14174  Scalarcsca 14241   .scvsca 14242   0gc0g 14378   LSSumclsm 16133   LModclmod 16948   LSubSpclss 17013   LSpanclspn 17052   LVecclvec 17183  LSHypclsh 32620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-tpos 6745  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-0g 14380  df-mnd 15415  df-submnd 15465  df-grp 15545  df-minusg 15546  df-sbg 15547  df-subg 15678  df-lsm 16135  df-mgp 16592  df-ur 16604  df-rng 16647  df-oppr 16715  df-dvdsr 16733  df-unit 16734  df-invr 16764  df-drng 16834  df-lmod 16950  df-lss 17014  df-lsp 17053  df-lvec 17184  df-lshyp 32622
This theorem is referenced by:  lshpsmreu  32754  lshpkrlem5  32759
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