Proofs are logical arguments establishing truth of statements using axioms and logical rules. Key terms: theorem (proven statement), axiom (assumed true), lemma (helper result), corollary (consequence), hypothesis and conclusion (if-then parts).
From your study of logical equivalences, you know how to manipulate statements symbolically — recognizing when "P → Q" is equivalent to "¬Q → ¬P," or when a conjunction is equivalent to a disjunction. A proof is what connects abstract logical structure to actual mathematical content: it is a logically valid argument that establishes the truth of a mathematical statement, starting from things we already know. Understanding the vocabulary of proof is the first step to reading and writing mathematics fluently.
The vocabulary of proof types organizes mathematical results by their role. A theorem is a statement that has been proved true — it's the main result you're trying to establish. An axiom (or postulate) is assumed true without proof; it's a foundational rule of the game, like "two distinct points determine a unique line" in geometry. A lemma is a helper result — a theorem proved specifically because it's needed to prove something larger. A corollary is a result that follows easily from a theorem just proved, almost as a free consequence. These distinctions reflect the architecture of mathematical knowledge: big theorems rest on lemmas, which rest on axioms, and corollaries extend the reach of theorems.
The structure of a theorem is almost always "if P, then Q," where P is the hypothesis (what we assume) and Q is the conclusion (what we want to prove). Reading a theorem carefully means separating these two parts before doing anything else. When you prove a theorem, your job is to start with P and arrive at Q by logically valid steps. A common beginner error is to assume what you're trying to prove — a circular argument — or to prove something slightly different from Q. Keeping hypothesis and conclusion clearly distinguished prevents both mistakes.
A written proof is not just a private chain of reasoning — it's a communication to a reader. The goal is to make the logical structure transparent: the reader should be able to check every step independently. This is why proof-writing requires precision in language. "It follows that" and "therefore" signal logical consequences; "assume" and "suppose" signal the beginning of a hypothesis; "we have shown" signals the conclusion has been reached. Learning this vocabulary lets you parse complex proofs written by others and structure your own proofs so they can be verified — which is ultimately what distinguishes mathematics from every other kind of argument.