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I need to prove $(-\vec{v})+\vec{v}=\vec{0}$ using only the axioms of vector space addition and multiplication, except the commutativity axiom.

I have tried using this approach $\vec{0}=\vec{v}+(-\vec{v})$ but I get stuck and can't find a way to get the $(-\vec{v})$ first.

DoubleFishFire
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  • @sahibaArora The "duplicate" uses the theorem $-1(\vec{v})=-\vec{v}$ which I can't use. – DoubleFishFire May 25 '20 at 14:24
  • Please see Benjamin Dickman's answer. – Sahiba Arora May 25 '20 at 14:26
  • @SahibaArora The proof of proposition one is using what we are trying to prove in the proof. The proof starts with $y+x$ and then just goes back to $y+x$. – DoubleFishFire May 25 '20 at 14:37
  • @MattSamuel no it is the same link. I can't use $-\vec{v}=-1\vec{v}$, I can only use the axioms of vector space addition and multiplication. Namely, $\vec{v}-\vec{v}=0$. – DoubleFishFire May 25 '20 at 15:15
  • @DoubleFishFire I wasn't really asking, that was an auto-comment. I'm asserting that Benjamin Dickson's answer is sufficient to answer your question. It happens when one votes to close a question as a duplicate. – Matt Samuel May 25 '20 at 15:17
  • @DoubleFishFire One of the axioms of vector space addition is commutativity... – Rick May 25 '20 at 16:37
  • @Rick I am proving commutativity and so can't use it – DoubleFishFire May 25 '20 at 20:13
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    @DoubleFishFire Then I think you should state clearly what axioms you are allowed to use, because if you say axioms of vector space addition everyone will understand that you include commutativity, as this is the most common practice. – Rick May 25 '20 at 20:16

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