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Theorem lspdisj2 15929
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 3685 . . . . . 6  |-  ( X  =  .0.  ->  { X }  =  {  .0.  } )
21fveq2d 5567 . . . . 5  |-  ( X  =  .0.  ->  ( N `  { X } )  =  ( N `  {  .0.  } ) )
3 lspdisj2.w . . . . . . 7  |-  ( ph  ->  W  e.  LVec )
4 lveclmod 15908 . . . . . . 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 15814 . . . . . 6  |-  ( W  e.  LMod  ->  ( N `
 {  .0.  }
)  =  {  .0.  } )
95, 8syl 15 . . . . 5  |-  ( ph  ->  ( N `  {  .0.  } )  =  {  .0.  } )
102, 9sylan9eqr 2370 . . . 4  |-  ( (
ph  /\  X  =  .0.  )  ->  ( N `
 { X }
)  =  {  .0.  } )
1110ineq1d 3403 . . 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 2316 . . . . . . . 8  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
1513, 14, 7lspsncl 15783 . . . . . . 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 15755 . . . . . 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 3200 . . . . 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 2348 . 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 15917 . . . . . 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 2515 . . . 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 15927 . 2  |-  ( (
ph  /\  X  =/=  .0.  )  ->  ( ( N `  { X } )  i^i  ( N `  { Y } ) )  =  {  .0.  } )
3922, 38pm2.61dane 2557 1  |-  ( ph  ->  ( ( N `  { X } )  i^i  ( N `  { Y } ) )  =  {  .0.  } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701    =/= wne 2479    i^i cin 3185    C_ wss 3186   {csn 3674   ` cfv 5292   Basecbs 13195   0gc0g 13449   LModclmod 15676   LSubSpclss 15738   LSpanclspn 15777   LVecclvec 15904
This theorem is referenced by:  lvecindp2  15941  hdmaprnlem9N  31868
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-tpos 6276  df-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-3 9850  df-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-ress 13202  df-plusg 13268  df-mulr 13269  df-0g 13453  df-mnd 14416  df-grp 14538  df-minusg 14539  df-sbg 14540  df-mgp 15375  df-rng 15389  df-ur 15391  df-oppr 15454  df-dvdsr 15472  df-unit 15473  df-invr 15503  df-drng 15563  df-lmod 15678  df-lss 15739  df-lsp 15778  df-lvec 15905
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