Properties of Affine Spaces
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Theorem
Let $\EE$ be an affine space with difference space $V$.
Let $0$ denote the zero element of $V$.
Then the following hold for all $p,q,r \in \EE$ and all $u, v \in V$:
- $(1): \quad p - p = 0$
- $(2): \quad p + 0 = p$
- $(3): \quad p + u = p + v \iff u = v$
- $(4): \quad q - p = r - p \iff q = r$
Proof
$(1): \quad p - p = 0$
We have:
| \(\ds \paren {p - p} + \paren {q - p}\) | \(=\) | \(\ds \paren {p + \paren {q - p} } - p\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds q - p\) |
From Zero Element is Unique:
- $p - p = 0$
$\Box$
$(2): \quad p + 0 = p$
Using $(1)$ we see that:
| \(\ds p + 0\) | \(=\) | \(\ds p + \paren {p - p}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds p\) |
$\Box$
$(3): \quad p + u = p + v \iff u = v$
Let $u = v$.
By definition a mapping has a unique image point on a given element.
It follows that:
- $p + u = p + v$
Let $p + u = p + v$.
We have:
| \(\ds p + u\) | \(=\) | \(\ds p + v\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {p + u} - p\) | \(=\) | \(\ds \paren {p + v} - p\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {p - p} + u\) | \(=\) | \(\ds \paren {p - p} + v\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds u\) | \(=\) | \(\ds v\) | by $(1)$ |
$\Box$
$(4): \quad q - p = r - p \iff q = r$
Let $q = r$.
By definition a mapping has a unique image point on a given element.
It follows that:
- $q - p = r - p$
Let $q - p = r - p \in V$.
Then:
| \(\ds q - p\) | \(=\) | \(\ds r - p\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds p + \paren {q - p}\) | \(=\) | \(\ds p + \paren {r - p}\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds q\) | \(=\) | \(\ds r\) | by hypothesis $q - p = r - p$ |
$\blacksquare$