Sub-theorem

A Sub-theorem is a statement that is a logical consequence of a larger, more fundamental theorem within a mathematical or scientific system. It provides a specific, often more focused, result derived from a broader principle. Essentially, it functions as a supporting theorem, building upon established facts to establish further conclusions. The validity of a Sub-theorem is entirely dependent on the validity of the parent theorem, the axioms of the system, and the logical deduction used to derive it. It can be used to simplify proofs or offer deeper insights into the broader concepts. The term helps distinguish a specific proven statement with its connections to a major theorem.

Sub-theorem meaning with examples

• In number theory, if the prime number theorem is the main theorem, a Sub-theorem might prove a specific distribution property of primes within a defined interval. This Sub-theorem simplifies analysis of prime gaps. The proof's validity is dependent on the veracity of the prime number theorem, demonstrating a hierarchical relationship.
• When using the Pythagorean theorem, a Sub-theorem could define relationships about a triangle's area when it is a certain type, a 45-45-90 or a 30-60-90 triangle. This will allow the user to know certain information before going through the whole theorem. This adds efficiency in any computation.
• Within the domain of graph theory, if a major theorem proves a property of all connected graphs, a Sub-theorem may address the same property but restricted to the subset of acyclic graphs. This allows us to understand the general relationship with its component parts.
• In Euclidean geometry, if Euclid's fifth postulate is assumed, many other statements emerge. One such derived statement concerning the equality of alternate interior angles would function as a Sub-theorem, supporting more complex geometrical proofs. Such supporting statements are integral.