Theorems of the form: "Let $A$ be ... . Then ..." or "Suppose $A$ ... . Then there exists ..."

Question

I often find in textbooks theorems of the form:

"Suppose that $A$… . Then there exists … ."

or

"Let $A$ be …. Then … ."

Are these kind of theorems implications of the form "if $A$ is … then …" (for the latter case for example)?

Answer 1

Instead of saying :

"If A then (if B then C)",

it is often more convenient to " export " the antecedent ( namely $A$) and to express it as an " hypothesis ", a "condition" , in the following way :

"Suppose A is true. In that case : if B then C.".

Also note that proving that ($B \rightarrow C$) is true under the assumption that A is true amounts to the same thing as proving that the implication ( $A \rightarrow (B\rightarrow C)$) is true.

This is known under the name : " conditional proof".

Being given the correspondence between semantics and syntax, saying that ( $A \rightarrow (B\rightarrow C)$) is a true conditional is equivalent to asserting that ($B\rightarrow C$) is derivable under the assumption $A$.