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I'm trying to remember the Gemara word for the equivalent of QED, when an argument would be settled between competing opinions and nothing more would be said about the issue.

You might be thinking of the word תיובתא as, for example, in Berachot 10b:

תיובתא דרב חסדא תיובתא

Is not this a refutation of R. Hisda? It is [indeed] a refutation. (Soncino translation)

As explained by several rishonim, e.g. Rashbam to Bava Batra 52b:

כל היכא דאמר בגמ' תיובתא דפלוני תיובתא בטלו דברי מי שהתיובתא עליו לגמרי

Anywhere that the Talmud says "Is not this a refutation of So-And-So? It is [indeed] a refutation", the words of he whom the refutation is upon are entirely rejected.

While this would be an official end to the discussion, it does not necessarily carry the same connotation of a rigorous logical/mathematical proof that QED would.

I believe the word you are looking for is שמע מינה (we can [learn] from here). שמע מינה is often used when the Gemara asks for a proof to certain statement. If the Gemara brings a sastifying proof to something, it may end off by saying

.שמע מינה [like this statement] ,שמע מינה

It is basically saying we can learn out this statement from this proof, much like Q.E.D., it has been proven.

However, שמע מינה isn't only used to prove statements. It is also used in an attempt to prove a statement, before the actual proof is said. In that case it is more like saying let us try to learn from here. However, overall I would say that's definitely the word you are looking for.

I suspect that the phrase you're thinking of is קא משמע לן (ka mashma lan). It's not exactly QED, but it's often used similarly.

A typical usage that's most similar to Q.E.D. would be: "We would have thought X but Y therefore Z. Ka mashma lan."