Proof of 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

Each result cited is universally quantified over the data in its own statement. Write $\bar u=(1-t)u+t\varphi$ and $\sigma=1-t$. Since $t<1$, $\sigma$ is positive by claim 1 of Elementary Order Arithmetic in an Ordered Field, so $\sigma^{-1}$ exists and is positive by claim 7 there; and $0\le t$.

Upper semicontinuity. $u$ is upper semicontinuous on $U$ by Viscosity Subsolution and Supersolution of a Second-Order Equation. The function $\varphi$ is of class $C^{2}$ on $U$ by Classical Subsolution and Supersolution of a Second-Order Equation, hence continuous at every point of $U$ relative to $U$ by claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, and a continuous function is upper semicontinuous directly from Upper Semicontinuous Function on a Subset of a Metric Space (if $|\varphi(y)-\varphi(x)|<\varepsilon$ then $\varphi(y)<\varphi(x)+\varepsilon$ by claim 3 of Properties of the Absolute Value in an Ordered Field). By claims 2 and 1 of Sums and Nonnegative Multiples of Semicontinuous Functions, with the nonnegative multipliers $\sigma$ and $t$, $\bar u$ is upper semicontinuous on $U$.

The test. Let $\psi:U\to\mathbb{R}$ be of class $C^{2}$ on $U$ and let $x_{0}\in U$ be a point at which $\bar u-\psi$ has a local maximum relative to $U$, with radius $\rho$: $\bar u(y)-\psi(y)\le\bar u(x_{0})-\psi(x_{0})$ for every $y\in U$ with $d_{E}(x_{0},y)<\rho$. Define $\psi_{0}:U\to\mathbb{R}$ by

which is of class $C^{2}$ on $U$ by claim 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set. By claim 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, applied to first and then to second partial derivatives,

For every $y\in U$, the field axioms give $u(y)-\psi_{0}(y)=\sigma^{-1}\bigl(\sigma u(y)+t\varphi(y)-\psi(y)\bigr)=\sigma^{-1}\bigl(\bar u(y)-\psi(y)\bigr)$. Multiplying the local-maximum inequality by the positive $\sigma^{-1}$ (claim 5 of Elementary Arithmetic in an Ordered Field) shows that $u-\psi_{0}$ has a local maximum at $x_{0}$ relative to $U$, with the same radius $\rho$. Since $u$ is a viscosity subsolution of $F$ on $U$,

and since $\varphi$ is a classical subsolution of $F$ on $U$,

Convexity. In $\mathbb{R}^{n}$ and in $\mathcal{S}(n)$ (Second-Order Equations on Euclidean Open Sets §matrices) the vector-space axioms give

and $\sigma u(x_{0})+t\varphi(x_{0})=\bar u(x_{0})$. Hence the convexity of $F$ in $(r,p,X)$, applied at $x_{0}$ to the triples $(u(x_{0}),D\psi_{0}(x_{0}),D^{2}\psi_{0}(x_{0}))$ and $(\varphi(x_{0}),D\varphi(x_{0}),D^{2}\varphi(x_{0}))$ with weight $t$, gives

Multiplying (1) by $\sigma\ge0$ and (2) by $t\ge0$ (claim 5 of Elementary Arithmetic in an Ordered Field) shows that both summands on the right are at most $0$, so their sum is at most $0$ by claim 2 of Elementary Arithmetic in an Ordered Field applied to their additive inverses. Hence $F(x_{0},\bar u(x_{0}),D\psi(x_{0}),D^{2}\psi(x_{0}))\le0$. As $\psi$ and $x_{0}$ were arbitrary, $\bar u$ is a viscosity subsolution of $F$ on $U$.