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Theorem lspdisj2 16199
Description: Unequal spans are disjoint (share only the zero vector). (Contributed by NM, 22-Mar-2015.)
Hypotheses
Ref Expression
lspdisj2.v  |-  V  =  ( Base `  W
)
lspdisj2.o  |-  .0.  =  ( 0g `  W )
lspdisj2.n  |-  N  =  ( LSpan `  W )
lspdisj2.w  |-  ( ph  ->  W  e.  LVec )
lspdisj2.x  |-  ( ph  ->  X  e.  V )
lspdisj2.y  |-  ( ph  ->  Y  e.  V )
lspdisj2.q  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
Assertion
Ref Expression
lspdisj2  |-  ( ph  ->  ( ( N `  { X } )  i^i  ( N `  { Y } ) )  =  {  .0.  } )

Proof of Theorem lspdisj2
StepHypRef Expression
1 sneq 3825 . . . . . 6  |-  ( X  =  .0.  ->  { X }  =  {  .0.  } )
21fveq2d 5732 . . . . 5  |-  ( X  =  .0.  ->  ( N `  { X } )  =  ( N `  {  .0.  } ) )
3 lspdisj2.w . . . . . . 7  |-  ( ph  ->  W  e.  LVec )
4 lveclmod 16178 . . . . . . 7  |-  ( W  e.  LVec  ->  W  e. 
LMod )
53, 4syl 16 . . . . . 6  |-  ( ph  ->  W  e.  LMod )
6 lspdisj2.o . . . . . . 7  |-  .0.  =  ( 0g `  W )
7 lspdisj2.n . . . . . . 7  |-  N  =  ( LSpan `  W )
86, 7lspsn0 16084 . . . . . 6  |-  ( W  e.  LMod  ->  ( N `
 {  .0.  }
)  =  {  .0.  } )
95, 8syl 16 . . . . 5  |-  ( ph  ->  ( N `  {  .0.  } )  =  {  .0.  } )
102, 9sylan9eqr 2490 . . . 4  |-  ( (
ph  /\  X  =  .0.  )  ->  ( N `
 { X }
)  =  {  .0.  } )
1110ineq1d 3541 . . 3  |-  ( (
ph  /\  X  =  .0.  )  ->  ( ( N `  { X } )  i^i  ( N `  { Y } ) )  =  ( {  .0.  }  i^i  ( N `  { Y } ) ) )
12 lspdisj2.y . . . . . . 7  |-  ( ph  ->  Y  e.  V )
13 lspdisj2.v . . . . . . . 8  |-  V  =  ( Base `  W
)
14 eqid 2436 . . . . . . . 8  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
1513, 14, 7lspsncl 16053 . . . . . . 7  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  W ) )
165, 12, 15syl2anc 643 . . . . . 6  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  W ) )
176, 14lss0ss 16025 . . . . . 6  |-  ( ( W  e.  LMod  /\  ( N `  { Y } )  e.  (
LSubSp `  W ) )  ->  {  .0.  }  C_  ( N `  { Y } ) )
185, 16, 17syl2anc 643 . . . . 5  |-  ( ph  ->  {  .0.  }  C_  ( N `  { Y } ) )
19 df-ss 3334 . . . . 5  |-  ( {  .0.  }  C_  ( N `  { Y } )  <->  ( {  .0.  }  i^i  ( N `
 { Y }
) )  =  {  .0.  } )
2018, 19sylib 189 . . . 4  |-  ( ph  ->  ( {  .0.  }  i^i  ( N `  { Y } ) )  =  {  .0.  } )
2120adantr 452 . . 3  |-  ( (
ph  /\  X  =  .0.  )  ->  ( {  .0.  }  i^i  ( N `  { Y } ) )  =  {  .0.  } )
2211, 21eqtrd 2468 . 2  |-  ( (
ph  /\  X  =  .0.  )  ->  ( ( N `  { X } )  i^i  ( N `  { Y } ) )  =  {  .0.  } )
233adantr 452 . . 3  |-  ( (
ph  /\  X  =/=  .0.  )  ->  W  e. 
LVec )
2416adantr 452 . . 3  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( N `
 { Y }
)  e.  ( LSubSp `  W ) )
25 lspdisj2.x . . . 4  |-  ( ph  ->  X  e.  V )
2625adantr 452 . . 3  |-  ( (
ph  /\  X  =/=  .0.  )  ->  X  e.  V )
27 lspdisj2.q . . . . 5  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
2827adantr 452 . . . 4  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( N `
 { X }
)  =/=  ( N `
 { Y }
) )
2923adantr 452 . . . . . . 7  |-  ( ( ( ph  /\  X  =/=  .0.  )  /\  X  e.  ( N `  { Y } ) )  ->  W  e.  LVec )
3012adantr 452 . . . . . . . 8  |-  ( (
ph  /\  X  =/=  .0.  )  ->  Y  e.  V )
3130adantr 452 . . . . . . 7  |-  ( ( ( ph  /\  X  =/=  .0.  )  /\  X  e.  ( N `  { Y } ) )  ->  Y  e.  V )
32 simpr 448 . . . . . . 7  |-  ( ( ( ph  /\  X  =/=  .0.  )  /\  X  e.  ( N `  { Y } ) )  ->  X  e.  ( N `  { Y } ) )
33 simplr 732 . . . . . . 7  |-  ( ( ( ph  /\  X  =/=  .0.  )  /\  X  e.  ( N `  { Y } ) )  ->  X  =/=  .0.  )
3413, 6, 7, 29, 31, 32, 33lspsneleq 16187 . . . . . 6  |-  ( ( ( ph  /\  X  =/=  .0.  )  /\  X  e.  ( N `  { Y } ) )  -> 
( N `  { X } )  =  ( N `  { Y } ) )
3534ex 424 . . . . 5  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( X  e.  ( N `  { Y } )  -> 
( N `  { X } )  =  ( N `  { Y } ) ) )
3635necon3ad 2637 . . . 4  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  ->  -.  X  e.  ( N `  { Y } ) ) )
3728, 36mpd 15 . . 3  |-  ( (
ph  /\  X  =/=  .0.  )  ->  -.  X  e.  ( N `  { Y } ) )
3813, 6, 7, 14, 23, 24, 26, 37lspdisj 16197 . 2  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( ( N `  { X } )  i^i  ( N `  { Y } ) )  =  {  .0.  } )
3922, 38pm2.61dane 2682 1  |-  ( ph  ->  ( ( N `  { X } )  i^i  ( N `  { Y } ) )  =  {  .0.  } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599    i^i cin 3319    C_ wss 3320   {csn 3814   ` cfv 5454   Basecbs 13469   0gc0g 13723   LModclmod 15950   LSubSpclss 16008   LSpanclspn 16047   LVecclvec 16174
This theorem is referenced by:  lvecindp2  16211  hdmaprnlem9N  32658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-sbg 14814  df-mgp 15649  df-rng 15663  df-ur 15665  df-oppr 15728  df-dvdsr 15746  df-unit 15747  df-invr 15777  df-drng 15837  df-lmod 15952  df-lss 16009  df-lsp 16048  df-lvec 16175
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