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Theorem lspsnsubn0 20508
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 = (0g𝑊)
lspsnsubn0.m = (-g𝑊)
lspsnsubn0.w (𝜑𝑊 ∈ LMod)
lspsnsubn0.x (𝜑𝑋𝑉)
lspsnsubn0.y (𝜑𝑌𝑉)
lspsnsubn0.e (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
Assertion
Ref Expression
lspsnsubn0 (𝜑 → (𝑋 𝑌) ≠ 0 )

Proof of Theorem lspsnsubn0
StepHypRef Expression
1 lspsnsubn0.e . 2 (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
2 lspsnsubn0.w . . . . 5 (𝜑𝑊 ∈ LMod)
3 lspsnsubn0.x . . . . 5 (𝜑𝑋𝑉)
4 lspsnsubn0.y . . . . 5 (𝜑𝑌𝑉)
5 lspsnsubn0.v . . . . . 6 𝑉 = (Base‘𝑊)
6 lspsnsubn0.o . . . . . 6 0 = (0g𝑊)
7 lspsnsubn0.m . . . . . 6 = (-g𝑊)
85, 6, 7lmodsubeq0 20288 . . . . 5 ((𝑊 ∈ LMod ∧ 𝑋𝑉𝑌𝑉) → ((𝑋 𝑌) = 0𝑋 = 𝑌))
92, 3, 4, 8syl3anc 1370 . . . 4 (𝜑 → ((𝑋 𝑌) = 0𝑋 = 𝑌))
10 sneq 4583 . . . . 5 (𝑋 = 𝑌 → {𝑋} = {𝑌})
1110fveq2d 6829 . . . 4 (𝑋 = 𝑌 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))
129, 11syl6bi 252 . . 3 (𝜑 → ((𝑋 𝑌) = 0 → (𝑁‘{𝑋}) = (𝑁‘{𝑌})))
1312necon3d 2961 . 2 (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) → (𝑋 𝑌) ≠ 0 ))
141, 13mpd 15 1 (𝜑 → (𝑋 𝑌) ≠ 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1540  wcel 2105  wne 2940  {csn 4573  cfv 6479  (class class class)co 7337  Basecbs 17009  0gc0g 17247  -gcsg 18675  LModclmod 20229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-fv 6487  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-1st 7899  df-2nd 7900  df-0g 17249  df-mgm 18423  df-sgrp 18472  df-mnd 18483  df-grp 18676  df-minusg 18677  df-sbg 18678  df-lmod 20231
This theorem is referenced by:  mapdpglem4N  39952
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