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Theorem lspsnsubn0 20899
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 20676 . . . . 5 ((𝑊 ∈ LMod ∧ 𝑋𝑉𝑌𝑉) → ((𝑋 𝑌) = 0𝑋 = 𝑌))
92, 3, 4, 8syl3anc 1370 . . . 4 (𝜑 → ((𝑋 𝑌) = 0𝑋 = 𝑌))
10 sneq 4638 . . . . 5 (𝑋 = 𝑌 → {𝑋} = {𝑌})
1110fveq2d 6895 . . . 4 (𝑋 = 𝑌 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))
129, 11syl6bi 253 . . 3 (𝜑 → ((𝑋 𝑌) = 0 → (𝑁‘{𝑋}) = (𝑁‘{𝑌})))
1312necon3d 2960 . 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 2939  {csn 4628  cfv 6543  (class class class)co 7412  Basecbs 17149  0gc0g 17390  -gcsg 18858  LModclmod 20615
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 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-0g 17392  df-mgm 18566  df-sgrp 18645  df-mnd 18661  df-grp 18859  df-minusg 18860  df-sbg 18861  df-lmod 20617
This theorem is referenced by:  mapdpglem4N  40851
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