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Theorem lspdisj2 15880
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 3651 . . . . . 6  |-  ( X  =  .0.  ->  { X }  =  {  .0.  } )
21fveq2d 5529 . . . . 5  |-  ( X  =  .0.  ->  ( N `  { X } )  =  ( N `  {  .0.  } ) )
3 lspdisj2.w . . . . . . 7  |-  ( ph  ->  W  e.  LVec )
4 lveclmod 15859 . . . . . . 7  |-  ( W  e.  LVec  ->  W  e. 
LMod )
53, 4syl 15 . . . . . 6  |-  ( ph  ->  W  e.  LMod )
6 lspdisj2.o . . . . . . 7  |-  .0.  =  ( 0g `  W )
7 lspdisj2.n . . . . . . 7  |-  N  =  ( LSpan `  W )
86, 7lspsn0 15765 . . . . . 6  |-  ( W  e.  LMod  ->  ( N `
 {  .0.  }
)  =  {  .0.  } )
95, 8syl 15 . . . . 5  |-  ( ph  ->  ( N `  {  .0.  } )  =  {  .0.  } )
102, 9sylan9eqr 2337 . . . 4  |-  ( (
ph  /\  X  =  .0.  )  ->  ( N `
 { X }
)  =  {  .0.  } )
1110ineq1d 3369 . . 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 2283 . . . . . . . 8  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
1513, 14, 7lspsncl 15734 . . . . . . 7  |-  ( ( W  e.  LMod  /\  Y  e.  V )  ->  ( N `  { Y } )  e.  (
LSubSp `  W ) )
165, 12, 15syl2anc 642 . . . . . 6  |-  ( ph  ->  ( N `  { Y } )  e.  (
LSubSp `  W ) )
176, 14lss0ss 15706 . . . . . 6  |-  ( ( W  e.  LMod  /\  ( N `  { Y } )  e.  (
LSubSp `  W ) )  ->  {  .0.  }  C_  ( N `  { Y } ) )
185, 16, 17syl2anc 642 . . . . 5  |-  ( ph  ->  {  .0.  }  C_  ( N `  { Y } ) )
19 df-ss 3166 . . . . 5  |-  ( {  .0.  }  C_  ( N `  { Y } )  <->  ( {  .0.  }  i^i  ( N `
 { Y }
) )  =  {  .0.  } )
2018, 19sylib 188 . . . 4  |-  ( ph  ->  ( {  .0.  }  i^i  ( N `  { Y } ) )  =  {  .0.  } )
2120adantr 451 . . 3  |-  ( (
ph  /\  X  =  .0.  )  ->  ( {  .0.  }  i^i  ( N `  { Y } ) )  =  {  .0.  } )
2211, 21eqtrd 2315 . 2  |-  ( (
ph  /\  X  =  .0.  )  ->  ( ( N `  { X } )  i^i  ( N `  { Y } ) )  =  {  .0.  } )
233adantr 451 . . 3  |-  ( (
ph  /\  X  =/=  .0.  )  ->  W  e. 
LVec )
2416adantr 451 . . 3  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( N `
 { Y }
)  e.  ( LSubSp `  W ) )
25 lspdisj2.x . . . 4  |-  ( ph  ->  X  e.  V )
2625adantr 451 . . 3  |-  ( (
ph  /\  X  =/=  .0.  )  ->  X  e.  V )
27 lspdisj2.q . . . . 5  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
2827adantr 451 . . . 4  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( N `
 { X }
)  =/=  ( N `
 { Y }
) )
2923adantr 451 . . . . . . 7  |-  ( ( ( ph  /\  X  =/=  .0.  )  /\  X  e.  ( N `  { Y } ) )  ->  W  e.  LVec )
3012adantr 451 . . . . . . . 8  |-  ( (
ph  /\  X  =/=  .0.  )  ->  Y  e.  V )
3130adantr 451 . . . . . . 7  |-  ( ( ( ph  /\  X  =/=  .0.  )  /\  X  e.  ( N `  { Y } ) )  ->  Y  e.  V )
32 simpr 447 . . . . . . 7  |-  ( ( ( ph  /\  X  =/=  .0.  )  /\  X  e.  ( N `  { Y } ) )  ->  X  e.  ( N `  { Y } ) )
33 simplr 731 . . . . . . 7  |-  ( ( ( ph  /\  X  =/=  .0.  )  /\  X  e.  ( N `  { Y } ) )  ->  X  =/=  .0.  )
3413, 6, 7, 29, 31, 32, 33lspsneleq 15868 . . . . . 6  |-  ( ( ( ph  /\  X  =/=  .0.  )  /\  X  e.  ( N `  { Y } ) )  -> 
( N `  { X } )  =  ( N `  { Y } ) )
3534ex 423 . . . . 5  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( X  e.  ( N `  { Y } )  -> 
( N `  { X } )  =  ( N `  { Y } ) ) )
3635necon3ad 2482 . . . 4  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  ->  -.  X  e.  ( N `  { Y } ) ) )
3728, 36mpd 14 . . 3  |-  ( (
ph  /\  X  =/=  .0.  )  ->  -.  X  e.  ( N `  { Y } ) )
3813, 6, 7, 14, 23, 24, 26, 37lspdisj 15878 . 2  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( ( N `  { X } )  i^i  ( N `  { Y } ) )  =  {  .0.  } )
3922, 38pm2.61dane 2524 1  |-  ( ph  ->  ( ( N `  { X } )  i^i  ( N `  { Y } ) )  =  {  .0.  } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    i^i cin 3151    C_ wss 3152   {csn 3640   ` cfv 5255   Basecbs 13148   0gc0g 13400   LModclmod 15627   LSubSpclss 15689   LSpanclspn 15728   LVecclvec 15855
This theorem is referenced by:  lvecindp2  15892  hdmaprnlem9N  32050
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-sbg 14491  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856
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