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Theorem lspsnsubn0 17324
Description: Unequal singleton spans imply nonzero vector subtraction. (Contributed by NM, 19-Mar-2015.)
Hypotheses
Ref Expression
lspsnsubn0.v  |-  V  =  ( Base `  W
)
lspsnsubn0.o  |-  .0.  =  ( 0g `  W )
lspsnsubn0.m  |-  .-  =  ( -g `  W )
lspsnsubn0.w  |-  ( ph  ->  W  e.  LMod )
lspsnsubn0.x  |-  ( ph  ->  X  e.  V )
lspsnsubn0.y  |-  ( ph  ->  Y  e.  V )
lspsnsubn0.e  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
Assertion
Ref Expression
lspsnsubn0  |-  ( ph  ->  ( X  .-  Y
)  =/=  .0.  )

Proof of Theorem lspsnsubn0
StepHypRef Expression
1 lspsnsubn0.e . 2  |-  ( ph  ->  ( N `  { X } )  =/=  ( N `  { Y } ) )
2 lspsnsubn0.w . . . . 5  |-  ( ph  ->  W  e.  LMod )
3 lspsnsubn0.x . . . . 5  |-  ( ph  ->  X  e.  V )
4 lspsnsubn0.y . . . . 5  |-  ( ph  ->  Y  e.  V )
5 lspsnsubn0.v . . . . . 6  |-  V  =  ( Base `  W
)
6 lspsnsubn0.o . . . . . 6  |-  .0.  =  ( 0g `  W )
7 lspsnsubn0.m . . . . . 6  |-  .-  =  ( -g `  W )
85, 6, 7lmodsubeq0 17107 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V  /\  Y  e.  V )  ->  (
( X  .-  Y
)  =  .0.  <->  X  =  Y ) )
92, 3, 4, 8syl3anc 1219 . . . 4  |-  ( ph  ->  ( ( X  .-  Y )  =  .0.  <->  X  =  Y ) )
10 sneq 3982 . . . . 5  |-  ( X  =  Y  ->  { X }  =  { Y } )
1110fveq2d 5790 . . . 4  |-  ( X  =  Y  ->  ( N `  { X } )  =  ( N `  { Y } ) )
129, 11syl6bi 228 . . 3  |-  ( ph  ->  ( ( X  .-  Y )  =  .0. 
->  ( N `  { X } )  =  ( N `  { Y } ) ) )
1312necon3d 2670 . 2  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  ->  ( X  .-  Y )  =/= 
.0.  ) )
141, 13mpd 15 1  |-  ( ph  ->  ( X  .-  Y
)  =/=  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1370    e. wcel 1758    =/= wne 2642   {csn 3972   ` cfv 5513  (class class class)co 6187   Basecbs 14273   0gc0g 14477   -gcsg 15512   LModclmod 17051
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 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-1st 6674  df-2nd 6675  df-0g 14479  df-mnd 15514  df-grp 15644  df-minusg 15645  df-sbg 15646  df-lmod 17053
This theorem is referenced by:  mapdpglem4N  35624
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