How to negate a conditional statement with the term "either"

Consider this statement for example:

If $mn$ is even, then either $m$ is even or $n$ is even.

If you were to negate this statement, could you let P = "$mn$ is even", Q = "$m$ is even", and R ="$n$ is even" and set up the conditional statement like this:

$P \implies Q \lor R$ and thus the negation would be $P \implies \lnot Q \land \lnot R$? If so, would this read "If $mn$ is even, then $m$ is odd and $n$ is odd."? I'm unsure of how the word "either" would affect the logic in the original statement.

Since $\neg (A \lor B) \equiv \neg A \land \neg B$, the negation of the conclusion "$m$ is even or $n$ is even" is "$m$ is odd and $n$ is odd".

The negation of an implication is given as follows: $\neg (A \implies B) \equiv A \land \neg B$. Thus, the negation of the entire implication is "$mn$ is even, $m$ is odd, and $n$ is odd".

the negation of $A \Rightarrow B$ is : $A \land \neg B$

which is not the same than $A \Rightarrow \neg B$

An example :

a thief shouts to the police who are approaching him :

"If you step forward, then I shoot"

Which is equivalent to :

"Don't step forward, or I shoot"

whose negation is :

"Step forward, and I don't shoot".