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Theorem lspsnsubn0 16171
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 15962 . . . . 5  |-  ( ( W  e.  LMod  /\  X  e.  V  /\  Y  e.  V )  ->  (
( X  .-  Y
)  =  .0.  <->  X  =  Y ) )
92, 3, 4, 8syl3anc 1184 . . . 4  |-  ( ph  ->  ( ( X  .-  Y )  =  .0.  <->  X  =  Y ) )
10 sneq 3789 . . . . 5  |-  ( X  =  Y  ->  { X }  =  { Y } )
1110fveq2d 5695 . . . 4  |-  ( X  =  Y  ->  ( N `  { X } )  =  ( N `  { Y } ) )
129, 11syl6bi 220 . . 3  |-  ( ph  ->  ( ( X  .-  Y )  =  .0. 
->  ( N `  { X } )  =  ( N `  { Y } ) ) )
1312necon3d 2609 . 2  |-  ( ph  ->  ( ( N `  { X } )  =/=  ( N `  { Y } )  ->  ( X  .-  Y )  =/= 
.0.  ) )
141, 13mpd 15 1  |-  ( ph  ->  ( X  .-  Y
)  =/=  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    e. wcel 1721    =/= wne 2571   {csn 3778   ` cfv 5417  (class class class)co 6044   Basecbs 13428   0gc0g 13682   -gcsg 14647   LModclmod 15909
This theorem is referenced by:  mapdpglem4N  32163
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-riota 6512  df-0g 13686  df-mnd 14649  df-grp 14771  df-minusg 14772  df-sbg 14773  df-lmod 15911
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