$$\begin{array}{rcl}
(1) & \partial_KC_t(T,K) & \leq 0 \\
(2) & \partial^2_KKC_t(T,K) & > 0 \\
(3) & \partial_T C_t(T,K) & \geq 0 \\
\end{array}$$
If $(1)$ doesnot hold, it exists $K_1<K_2$ such that $C_t(T,K_1)<C_t(T,K_2)$. Then as barrycarter said in his comment, you sell $C_t(T,K_2)$ and you buy $C_t(T,K_1)$, so your cash position is $C_t(T,K_2)-C_t(T,K_1)>0$, at maturity you receive $(S_T-K_1)^+-(S_T-K_2)^+\geq 0$. There is an arbitrage.
If $(2)$ doesnot hold, it exists $\epsilon>0$ and $K>\epsilon$ such that $C_t(T,K-\epsilon)+C_t(T,K+\epsilon)\leq 2 C_t(T,K)$. Then you buy $C_t(T,K-\epsilon)$ and $C_t(T,K+\epsilon)$ and you sell $2C_t(T,K)$, your cash position is $2 C_t(T,K) - C_t(T,K-\epsilon)+C_t(T,K+\epsilon)\geq 0$. at maturity you get $(S_T-(K+\epsilon))^++(S_T-(K-\epsilon))^+-2(S_T-K)^+\geq 0$ which is the butterfly spread you mention. Note that with non-null probability, this payoff is positive. There is again an arbitrage.
Assuming you talk about american call options. If $(3)$ doesnot hold, it exists $T_1<T_2$ such that $C_t(T_1,K)>C_t(T_2,K)$. Then you buy $C_t(T_2,K)$, you sell $C_t(T_1,K)$, your cash position is $C_t(T_1,K)-C_t(T_2,K)>0$. At any time $\tau\leq T_1$, the buyer of $C_t(T_1,K)$ can exercise its right, and then you owe him $(S_\tau-K)^+$, but since you buy an american option $C_t(T_2,K)$, you can also exercise your right at time $\tau$ and you net position. Again since you build a positive cash position leading to a non-negative positive (here equal to $0$), you build an arbitrage.
Note that this third relationship will hold with any american options. And note that the two first relationships will also work with american calls by adapting the $T$ to $\tau$ being the exercise time of the agent who buys call options.
