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Theorem lspsnsubn0 19909
 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 19690 . . . . 5 ((𝑊 ∈ LMod ∧ 𝑋𝑉𝑌𝑉) → ((𝑋 𝑌) = 0𝑋 = 𝑌))
92, 3, 4, 8syl3anc 1368 . . . 4 (𝜑 → ((𝑋 𝑌) = 0𝑋 = 𝑌))
10 sneq 4535 . . . . 5 (𝑋 = 𝑌 → {𝑋} = {𝑌})
1110fveq2d 6650 . . . 4 (𝑋 = 𝑌 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))
129, 11syl6bi 256 . . 3 (𝜑 → ((𝑋 𝑌) = 0 → (𝑁‘{𝑋}) = (𝑁‘{𝑌})))
1312necon3d 3008 . 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 209   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  {csn 4525  ‘cfv 6325  (class class class)co 7136  Basecbs 16478  0gc0g 16708  -gcsg 18100  LModclmod 19631 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-fv 6333  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-1st 7674  df-2nd 7675  df-0g 16710  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-grp 18101  df-minusg 18102  df-sbg 18103  df-lmod 19633 This theorem is referenced by:  mapdpglem4N  38991
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