# Simple property of DFAs

Do you know if this simple property of a DFA is stated as a property (or theorem) in some Automata theory book (possibly with a particular name)?

Property: Given a DFA $$A = \{ Q, \Sigma, \delta, q_0, F \}$$, if $$w = uv$$ and the state of $$A$$ on input $$w$$, after scanning $$u$$ is $$q_i$$ (when the head is at the beginning of subword $$v$$); and $$A' = \{ Q, \Sigma, \delta, q_i, F \}$$ then $$w \in L(A)$$ if and only if $$v \in L(A')$$

• It seems so obvious that I doubt anyone would name it or call it a theorem. – David Richerby Apr 20 '19 at 20:26

Actually I think I have spotted this property in a book. It uses the "extended" transition relation $$\delta^*$$, i.e., for strings rather than single letters.
Other properties you would expect $$\delta^*$$ to satisfy can be derived from our definition. For example, a natural generalization of the recursive statement in the definition is the formula $$\delta^*(q,xy) = \delta^*(\delta^*(q,x),y))$$ which is true for every state $$q$$ and every two strings $$x$$ and $$y$$ in $$\Sigma^*$$. The proof is by structural induction on $$y$$ and is similar to the proof [...] in Example 1.27.