Multiplication by Negative Real Number

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Theorem

$\forall x, y \in \R: x \times \paren {-y} = -\paren {x \times y} = \paren {-x} \times y$


Corollary

$\forall x \in \R: \paren {-1} \times x = -x$


Proof

\(\displaystyle x \times \paren {\paren {-y} + y}\) \(=\) \(\displaystyle x \times 0\) Real Number Axioms: $\R A 4$: Inverse for Addition
\(\displaystyle \) \(=\) \(\displaystyle 0\) Real Zero is Zero Element
\(\displaystyle \leadsto \ \ \) \(\displaystyle \paren {x \times \paren {-y} } + \paren {x \times y}\) \(=\) \(\displaystyle 0\) Real Number Axioms: $\R D$: Distributivity
\(\displaystyle \leadsto \ \ \) \(\displaystyle \paren {\paren {x \times \paren {-y} } + \paren {x \times y} } + \paren {-\paren {x \times y} }\) \(=\) \(\displaystyle 0 + \paren {-\paren {x \times y} }\) adding $-\paren {x \times y}$ to both sides
\(\displaystyle \leadsto \ \ \) \(\displaystyle \paren {x \times \paren {-y} } + \paren {\paren {x \times y} + \paren {-\paren {x \times y} } }\) \(=\) \(\displaystyle 0 + \paren {-\paren {x \times y} }\) Real Number Axioms: $\R A 1$: Associativity
\(\displaystyle \leadsto \ \ \) \(\displaystyle \paren {x \times \paren {-y} } + 0\) \(=\) \(\displaystyle 0 + \paren {-\paren {x \times y} }\) Real Number Axioms: $\R A 4$: Inverse for Addition
\(\displaystyle \leadsto \ \ \) \(\displaystyle x \times \paren {-y}\) \(=\) \(\displaystyle -\paren {x \times y}\) Real Number Axioms: $\R A 3$: Identity for Addition


Similarly:

\(\displaystyle \paren {x + \paren {-x} } \times y\) \(=\) \(\displaystyle 0 \times y\) Real Number Axioms: $\R A 4$: Inverse for Addition
\(\displaystyle \) \(=\) \(\displaystyle 0\) Real Zero is Zero Element
\(\displaystyle \leadsto \ \ \) \(\displaystyle \paren {x \times y} + \paren {\paren {-x} \times y}\) \(=\) \(\displaystyle 0\) Real Number Axioms: $\R D$: Distributivity
\(\displaystyle \leadsto \ \ \) \(\displaystyle -\paren {x \times y} + \paren {\paren {x \times y} + \paren {\paren {-x} \times y} }\) \(=\) \(\displaystyle -\left({x \times y}\right) + 0\) adding $-\left({x \times y}\right)$ to both sides
\(\displaystyle \leadsto \ \ \) \(\displaystyle \paren {-\paren {x \times y} + \paren {x \times y} } + \paren {\paren {-x} \times y}\) \(=\) \(\displaystyle -\paren {x \times y} + 0\) Real Number Axioms: $\R A 1$: Associativity
\(\displaystyle \leadsto \ \ \) \(\displaystyle 0 + \paren {\paren {-x} \times y}\) \(=\) \(\displaystyle -\paren {x \times y} + 0\) Real Number Axioms: $\R A 4$: Inverse for Addition
\(\displaystyle \leadsto \ \ \) \(\displaystyle \paren {-x} \times y\) \(=\) \(\displaystyle -\paren {x \times y}\) Real Number Axioms: $\R A 3$: Identity for Addition

$\blacksquare$


Sources