I can't visualize what happens when we pump v and y in pumping lemma for $a^n b^n c^n$

Question

If you need some context-: https://www.andrew.cmu.edu/user/ko/pdfs/lecture-11.pdf around page 7.

Case 1-: Say vxy contains ab

So when I pump v and y, what will get pumped? And how the result would be. I can understand the case for when vxy is all a's, all b's or all c's.

Case 2-: Say vxy contains bc

When I pump v and y, what happens? Help me visualize this.

My try-:

I will take case 1.

take i=2 then

We let

$u=a^{n-k} $

$v=a^k $

x=$\in$

y=$b^l$ k+l<=n

z=$b^{n-l} c^n$

Now I find $u v^2 x y^2 z$=? whwich gives $a^{n+k} b^{n+l} c^n$ which is obviously not in L. Am I right here?

Answer 1

I answer for the case 1, the case 2 is similar.

Since $vxy\in a^*b^*$, and $vy\neq \varepsilon$, that means that $v$ and/or $y$ contains at least one $a$ or at least one $b$, and no $c$. That means that $uxz$ (pumping can also mean considering $uv^0xy^0z$) contains strictly more $c$'s than either $a$'s or $b$'s (or both), so $uxz\notin \{a^nb^nc^n\mid n\in\mathbb{N}\}$.