Can't figure out logic proof

Question

I have this logic proof for a class that I can't get past: $A\to C\vdash A\to(E\to C)$.

Note: $\to$ means implies. I've already fiddled with material implication and transposition but it's not going anywhere.

Answer 1

By Natural Deduction: The premise is $A\to C$.   So if we assume $A$ and further assume $E$, then $C$ is derived by modus ponens (aka the conditional elimination rule).   Discharging those assumptions deduces (via the conditional introduction rule) $A\to(E\to C)$ , as was required.

By rules of replacement and implication: The basis is that anything will implie a truth (ie $X\to Y$ is true if $Y$ is true).   So $A\to C$ implies $E\to (A\to C)$, and from there you may use exportation and importation rules (if you know them, or by using the implication equivalence rule and association and commutation rules for disjunction).

Answer 2

Hint. "Implies" is a loaded term in English. For your problem, you can interpret $A \Rightarrow C$ as simply "$C$ is true whenever $A$ is; $C$ can only be false if $A$ is also."

If you were to look at it in graphical terms, $A$ would be a circle that is wholly within the circle representing $C$. In the conclusion, everything in the parentheses is operating in the domain inside the circle representing $A$. Is $C$ ever false in there?