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Theorem lspsnsubn0 20898
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 20675 . . . . 5 ((𝑊 ∈ LMod ∧ 𝑋𝑉𝑌𝑉) → ((𝑋 𝑌) = 0𝑋 = 𝑌))
92, 3, 4, 8syl3anc 1369 . . . 4 (𝜑 → ((𝑋 𝑌) = 0𝑋 = 𝑌))
10 sneq 4637 . . . . 5 (𝑋 = 𝑌 → {𝑋} = {𝑌})
1110fveq2d 6894 . . . 4 (𝑋 = 𝑌 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))
129, 11syl6bi 252 . . 3 (𝜑 → ((𝑋 𝑌) = 0 → (𝑁‘{𝑋}) = (𝑁‘{𝑌})))
1312necon3d 2959 . 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 1539  wcel 2104  wne 2938  {csn 4627  cfv 6542  (class class class)co 7411  Basecbs 17148  0gc0g 17389  -gcsg 18857  LModclmod 20614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-0g 17391  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18858  df-minusg 18859  df-sbg 18860  df-lmod 20616
This theorem is referenced by:  mapdpglem4N  40850
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