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What do you call a proof of a claim inside a lemma? And that lemma is inside a theorem.
n_bourbaki said:The normal presentation for this would go something like:
Statement of Theorem
Comment that to prove the theorem we will use some simple lemmas
Statements and proofs of lemmas
Restatement of theorem, or just a statement that theorem X above now follows.
You should avoid a cascade of statements whose proofs depend on the following statements. Instead put the thing you prove first at the top, and perhaps precede with a comment such as 'we will use the following small result later', and then reference it when you do you use it.
HallsofIvy said:If I am reading the original post correctly, a "proof of a claim inside a lemma", if it is written as a separate proof, would, indeed, be a "sub-lemma".
HallsofIvy said:It certainly could, just as subroutines could be included in the computer program where they are called. A "lemma" is just a part of the main proof that is simpler to understand if it is done separately. The same could be true of a "sub-lemma".
If I am reading the original post correctly, a "proof of a claim inside a lemma", if it is written as a separate proof, would, indeed, be a "sub-lemma".
A proof of a result inside a lemma refers to the process of proving a smaller, intermediate result within a larger proof or argument. This intermediate result, called a lemma, is often used to prove a larger theorem or proposition.
Having a proof of a result inside a lemma can make a larger proof or argument more concise and easier to follow. It can also help to break down a complex problem into smaller, more manageable parts.
The process of constructing a proof of a result inside a lemma is similar to that of any other mathematical proof. It involves identifying the key assumptions, using logical reasoning and mathematical techniques to manipulate these assumptions, and arriving at a conclusion that proves the lemma.
A lemma is a smaller, intermediate result used to prove a larger theorem or proposition. A theorem, on the other hand, is a statement that has been proven to be true and is often of greater significance in the field of study. Lemmas are typically used to prove theorems and are not considered as significant in their own right.
Yes, a lemma can be used to prove multiple theorems. This is one of the reasons why lemmas are important in the field of mathematics - they can be applied to a variety of problems and help to simplify the proof process.