JoshSmith
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Homework Statement
Prove \forall m \in \mathbb{Z} that -m=(-1)m
Homework Equations
The axioms of \mathbb{Z} are assumed, as are some of its propositions and corollaries. They would be too many to retype here, so assume that I have referenced the correct axioms and propositions if those are not stated outright.
The Attempt at a Solution
I believe, after working through it, that I got the solution. This has taken me hours, though, so please tell me if this works.
Proof. We'll show -m=(-1)m.
m=m by definition.
m+m=m+m by replacement.
m+m(1)=m+m by the multiplicative identity.
m+((-m)(-1))=m+m by some proposition not stated here.
(-m)(-1)=m by some proposition not stated here.
(-m)(-1)=m(1) by the multiplicative identity.
(-m)(-1)=m(-1)(-1) by some corollary not stated here.
-m=m(-1) by cancellation.
-m=(-1)m by associativity. Q.E.D.