Prove : For all integers, m and n: -m=(-1)m Given : Axiom 1.1 If m, n, and p are

Question

Prove: For all integers, m and n: -m=(-1)m
Given: Axiom 1.1 If m, n, and p are integers, then:
1.1 i. m+n=n+m
1.1 ii. (m+n)+p=m+(n+p)
1.1 iii. m*(n+p)=mn+mp
1.1 iv. mn=nm
1.1 v. (m*n)*p=m*(n*p)
Axiom 1.2 There exists an integer 0 such that for every integer, m, m+0=m
Axiom 1.3 There exists an integer 1 such that 1 is not 0 and whenever m is integer, m*1=m.
Axiom 1.4 For each integer, m, there exists an integer, denoted by -m, such that m+(-m)=0
Axiom 1.5 Let m, n, and p be integers. In m*n=m*p and m does not equal 0, then n=p.

Explanation / Answer

From 1.4 : 1+(-1) = 0

From 1.1 iii : m*(1+(-1)) = m*1 + m*(-1)

Thus:

m*0 = m*1 + m*(-1)

0 = m*1 + m*(-1)

From 1.3 : m*1 = m

So we have:

0 = m + m*(-1)

Add -m :

-m + 0 = -m + m + m*(-1)

From 1.2 : -m+0 = -m

-m = -m + m + m*(-1)

From 1.4 : -m + m = 0

-m = 0 + m*(-1)

Again by 1.2 : 0 + m*(-1) = m*(-1)

Therefore:

-m = m*(-1)

and from 1.1 iv : m*(-1) = (-1)*m

So we have:

-m = (-1)m

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