A Convex Combination of a Viscosity Subsolution and a Classical Subsolution is a Viscosity Subsolution, for an Operator Convex in the Value, Gradient and Matrix Variables

In the setting of Second-Order Equations on Euclidean Open Sets, let $n\ge1$ be a natural number, let $U\subseteq\mathbb{R}^{n}$ be open, let $F$ be a second-order equation operator on $U$ that is convex in $(r,p,X)$, let $u:U\to\mathbb{R}$ be a viscosity subsolution of $F$ on $U$, let $\varphi:U\to\mathbb{R}$ be a classical subsolution of $F$ on $U$, and let $t\in\mathbb{R}$ satisfy $0\le t<1$.

Then the function $(1-t)u+t\varphi:U\to\mathbb{R}$, with value $(1-t)u(x)+t\varphi(x)$ at $x\in U$, is a viscosity subsolution of $F$ on $U$.