ThmDex – An index of mathematical definitions, results, and conjectures.
Result R1540 on D175: Convex set
Affine map preserves convex sets in images
Formulation 0
Let $\mathbb{R}$ form the D4707: 3-structure of basic real numbers.
Let $V$ and $W$ form a D29: Vector space over $\mathbb{R}$.
Let $f : V \to W$ be a D1494: Affine-lin map with respect to $V$ and $W$.
Let $C \subseteq V$ be a D175: Convex set in $V$.
Then $f(C) \subseteq W$ is a D175: Convex set in $W$.
Proofs
Proof 0
Let $\mathbb{R}$ form the D4707: 3-structure of basic real numbers.
Let $V$ and $W$ form a D29: Vector space over $\mathbb{R}$.
Let $f : V \to W$ be a D1494: Affine-lin map with respect to $V$ and $W$.
Let $C \subseteq V$ be a D175: Convex set in $V$.
This result is a consequence of the results
(i) R1537: Linear map preserves convex sets in images
(ii) R1538: Translation preserves convex sets in images

$\square$