Product of Affine Spaces is Affine Space
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Theorem
Let $\EE, \FF$ be affine spaces.
Let $\GG = \EE \times \FF$ be the product of $\EE$ and $\FF$.
Then $\GG$ is an affine space.
Proof
Let $G = \vec \GG$ be the difference space of $\GG$.
We are required to show that the following axioms are satisfied:
| \((1)\) | $:$ | \(\ds \forall p, q \in \GG:\) | \(\ds p + \paren {q - p} = q \) | ||||||
| \((2)\) | $:$ | \(\ds \forall p \in \GG: \forall u, v \in G:\) | \(\ds \paren {p + u} + v = p + \paren {u + v} \) | ||||||
| \((3)\) | $:$ | \(\ds \forall p, q \in \GG: \forall u \in G:\) | \(\ds \paren {p - q} + u = \paren {p + u} - q \) |
Proof of $(1)$:
Let $p = \tuple {p', p' '}, q = \tuple {q', q' '} \in \GG$.
We have:
| \(\ds p + \paren {q - p}\) | \(=\) | \(\ds \tuple {p', p' '} + \paren {\tuple {q', q' '} - \tuple {p', p' '} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \tuple {p', p' '} + \tuple {q' - p', q' ' - p' '}\) | Definition of $-$ in Product Space | |||||||||||
| \(\ds \) | \(=\) | \(\ds \tuple {p' + \paren {q' - p'}, p' ' + \paren {q' ' - p' '} }\) | Definition of $+$ in Product Space | |||||||||||
| \(\ds \) | \(=\) | \(\ds \tuple {q' , q' '}\) | Axiom $(1)$ in Affine Spaces $\EE$, $\FF$ |
$\Box$
Proof of $(2)$:
Let $p = \tuple {p', p' '} \in \GG$.
Let $u = \tuple {u', u' '}, v = \tuple {v', v' '} \in G$.
We have:
| \(\ds \paren {p + u} + v\) | \(=\) | \(\ds \paren {\tuple {p', p' '} + \tuple {u', u' '} } + \tuple {v', v' '}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \tuple {p' + u', p' ' + u' '} + \tuple {v', v' '}\) | Definition of $+$ in Product Space | |||||||||||
| \(\ds \) | \(=\) | \(\ds \tuple {\paren {p' + u'} + v', \paren {p' ' + u' '} + v' '}\) | Definition of $+$ in Product Space | |||||||||||
| \(\ds \) | \(=\) | \(\ds \tuple {p' + \paren {u' + v'}, p' ' + \paren {u' ' + v' '} }\) | Axiom $(2)$ in Affine Spaces $\EE$, $\FF$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \tuple {p', p' '} + \paren {\tuple {u', u' '} + \tuple {v', v' '} }\) | Definition of $+$ in Product Space | |||||||||||
| \(\ds \) | \(=\) | \(\ds p + \paren {u + v}\) |
$\Box$
Proof of $(3)$:
Let $p = \tuple {p', p' '}, q = \tuple {q', q' '} \in \GG$.
Let $u = \tuple {u', u' '} \in G$.
We have:
| \(\ds \paren {p - q} + u\) | \(=\) | \(\ds \paren {\tuple {p', p' '} - \tuple {q', q' '} } + \tuple {u', u' '}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \tuple {\paren {p' - q'} + u', \paren {p' ' - q' '} + u' '}\) | Definition of $+,-$ in Product Space | |||||||||||
| \(\ds \) | \(=\) | \(\ds \tuple {\paren {p' + u'} - q', \paren {p' ' + u' '} - q' '}\) | Axiom $(3)$ in Affine Spaces $\EE$, $\FF$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\tuple {p', p' '} + \tuple {u', u' '} } - \tuple {q', q' '}\) | Definition of $+,-$ in Product Space | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {p - q} + u\) |
$\blacksquare$