Prove that $L_1 = \{\, 0^m 1^k 2^n \,\vert\, \lvert m - n \rvert = k \,\}$ is not regular using Pumping lemma

Question

I have been trying to understand Pumping lemmas for a while now, I am doing this exercise and I think that the pin has finally dropped. However, I am unsure if I am doing it even remotely correctly. I will briefly explain my attempt and thus my solution. Thereafter I would discuss a few points that are not as clear to me just yet. The exercise is the same as in the title. Here is my solution, but in-case you have forgotten:

Prove that $L_1 = \left\{\, 0^m 1^k 2^n \,\vert\, \lvert m - n \rvert = k \,\right\}$ is not regular using Pumping lemma

So, in order to show that a language is not regular, we first must assume it to be regular (in-order to be able to use pumping lemma). If $L_1$ is regular, then there exists a constant $n$ such that $ s \in L_1 $ and $ |s| \ge n$ implies that there is a string $xyz$ satisfying

1. $s = xyz$
2. $ |y| \ge 1 $ and $ |xy| \le n $
3. $xy^iz \in L1 $ for all $ i \ge 0 $

We'll try to extract a contradiction. This is how I did it
I first create a string such that $ m + k = n $, my string was $0^1 1^1 2^2$ or just $0 1 2 2 $ I then split this string into three parts $ x y z $ so that $x = 0 \\ y = 1 \\ z = 22$

I then selected $ i \Rightarrow 2 $ which gives us , by the third rule, $ xy^2z $. This gives us the string
$ 01122 \notin L1 $ because $1 + 2 \neq 2 $

That is my solution, I hope you could understand it. Now to my discussion points:
As you see, I have provided a contradiction, which should be enough. However, what I also want to know is that in other exercises , say $\{\, a^n b^n \,|\, n \ge 1 \,\}$ you select a Pumping length $P$ and then typically get a string that is $a^p b^p$ i.e say that $ P = 3 $ you'd get $aaabbb $ AND then you divide into strings, but since the powers are all different in my exercise, picking a $P$ would not be as helpful, right?

Is it possible to select a Pumping length $P$ here too? Do I need to check the other conditions as well? As I understand it, a language is proven non-regular if it can't satisfy all of the requirements at the same time.

To clarify the question, when I meant 'how do I apply $P$'? I meant, how do I apply to the already existing powers ($m$, $k$ and $n$). Do I simply attach P to each power which would make it $ 0^{PM} 1^{PK} 2^{PN} $ ? A follow-up question: When I select $P$, must I then also select $M$, $K$ and $N$? So say I select $ P = 3 $ , should I also select $M = 1$, $K = 1$ and $N = 2$?

Thanks to the guy down under I know understand how to select a string ($ 0^p 1^p 2^2p $) for instance. But, then I don't know how to split the string. Can do it like this:

$x= 0^r$ , $y=0^{p-r}1^t$ and $z = 1^{p-t}2^{2p} $? Would this string be in the language ? I assume it would be as $xyz = 0^p 1^p 2^{2p} $ BUT Idk, I might be doing it all wrong

Then if I select $i = 2$ so $xy^2z$ I would get $xy^2z = 0^r 0^{2p-2r} 1^{2t} 1^{p-t}2^{2p} = 0^{2p-r}1^{p + t}2^{2p} \notin L$ Thus our contradiction, is this correct thinking then?

To answer the guy below:
Assume $L$ is regular, then there exist a constant $p$ such that $ s \in L$ and $|s| \ge p$ implies that there exist string $xyz$ satisfying following:
$1) s = xyz$
$2) |y| \ge 1$ and $|xy| \le p $
$3) xy^iz \in L$ for very $i \ge 0$

(Which answers your first question)
We pick our string $s = o^p 1^p 2^{2p}$ as the property of a string in a language must be $m + k = n $ $ (p + p = 2p)$
We then split the string into three parts $xyz$ , I am going to split them differently to you. $x = 0^p$ $ y = 1^p $ and $z = 2^{2p} $ if we then select $ i = 2$ we get $xy^2z = 0^p 1^{2p} 2^{2p} \notin L $ BECAUSE $ 1 + 2 \neq 2$

Answer 1

For your choice of $s=0^11^12^2$, the pumping lemma might not apply. Remember, assuming that the language is regular means that there is an (unknown) integer $p$ such that the pumping lemma applies to every string of length $p$ or more. You don't know what $p$ is: if it were, say, 6 then since $|s|=4$ the PL simply wouldn't apply to your string and you could draw no conclusion.

That's why the strings chosen for pumping almost invariably involve $p$: it's there to assure that no matter what $p$ was, the pumped string would be at least $p$ characters long. For your example, you could choose $0^p1^p2^{2p}$ or $0^p1\,2^{p+1}$ or $0^p2^p$ even something like $0^{p^2}1^{2p+1}2^{(p+1)^2}$.

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Here's what a PL proof of this result looks like, with some bits for you to fill in: Assume the language $L$ is regular. Choose $s=0^p1^p2^{2p}\in L$, where $p$ is the integer of the pumping lemma (why can we say the PL applies to $s$?). Then if we write $s=xyz$ as in the PL, we'll have $y=0^t$ for some $0<t\le p$ (why?) and then $xy^2z\notin L$ (why?), a contradiction (why?).

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The proof without the PL is an example of a more or less common idiom: using closure properties. If $L$ were regular, then $L\cap 0^*2^*$ would be also, since regular languages are closed under intersection. However $$ L\cap 0^*2^*=\{0^n2^n\mid n\ge 0\} $$ (since we set $k=0$ when doing the intersection) but we know that that language isn't regular, so we have a contradiction and hence $L$ must not be regular.