Prove transitivity of partial ordered set in lattice.

Question

Prove transitivity of partial ordered set in lattice.

I am given this

If $(A, \le_{A})$ is a lattice and C is a set, $([C \rightarrow A], \le)$ is also a lattice.

And $\rightarrow$ is defined as follows: $f \le g$ if and only if for any $c \in C$, $f(c) \le_{A} g(c)$

Such lattices are defined as point wise.

And for $([C \rightarrow A], \le)$ as point wise lattice, we can drop $_{A}$.

Then I need to prove that $\le$ of $([C \rightarrow A], \le)$ is transitive.

And here's my try that I am told wrong. How do I fix it? I am not satisfied with my proof either. It looks like just redefining the question, not proving it.

1. Let's say that we have $f, g, h \in [C \rightarrow A]$.
2. And let's say that we have $f \le g$ and $f \le h$
3. For any $c \in C$, $f(c) \le_{A} g(c)$. So we have $f \le g$
4. Then for any $c \in C$, $g(c) \le_{A} h(c)$. So we have $g \le h$
5. Then by definition of $\le$, for any $c \in C$, $f \le h$
6. Therefore, $\le$ of $([C \rightarrow A], \le)$ is transitive

I corrected my try. Sorry about that.

Answer 1

Let $f, g, h \in [C \to A]$ be arbitrary such that $f \leq g$ and $g \leq h$. Then by definition $f(c) \leq_A g(c)$, and $g(c) \leq_A h(c)$ for any $c \in C$. Since $A$ is a lattice, it is a partially ordered set, therefore transitive. We know $f(c), g(c), h(c) \in A$ and $f(c) \leq_A g(c)$ and $g(c) \leq_A h(c)$, thus $f(c) \leq_A h(c)$ for any $c \in C$ (here we use transitivity of $A$). The latter implies $f \leq h$ (hence transitivity of $[C \to A]$).