Two theorems in my linear algebra textbook state that if we have a basis $B=\{ b_1,...,b_n\} $ of vector space $V$, then there exists a unique one-to-one linear coordinate mapping $x \implies[x]_B$ from $V$ onto $\Bbb R^n$.
It further states that such a coordinate mapping is an isomorphism, which implies that "vector space calculations in $V$ are accurately reproduced in $\Bbb R^n$". Now, I know the term isomorphism from mathematical logic, and I've learned about the so called "Isomorphism Lemma", which states:
Isomorphism Lemma. If mathematical structures $\mathfrak A$ and $\mathfrak B$ are isomorphic, then all first-order $S$-sentences are true in $\mathfrak A$ iff they are true in $\mathfrak B$.
Where a map $\pi:A\implies B$ is called an isomorphism from $S$-structures $\mathfrak A$ to $\mathfrak B$, if all first-order functions and relations in the language $S$ are preserved. For functions: for any $n$-parameter function $f$ in $S$, $\pi(f^{\mathfrak A}(a_1,...,a_n)=f^\mathfrak > B(\pi(a_1),...,\pi(a_n)).$
My question is: What can we actually conclude about vector spaces from this Lemma, given that there is an isomorphism for all vector spaces of dimension $n$?
Sure, all theorems that are derivable from vector space axioms hold for all vector spaces, but we didn't need an isomorphism to conclude that. What does the fact that there is an isomorphism from an n-dimensional vector space onto $\Bbb R ^n$ actually imply in for us? Surely it doesn't mean "everything" that is true in $\Bbb R^n$ is also true in $n$-dimensional vector space $V$?
I understand the Isomorphism Lemma in the abstract, but I don't yet concretely see what this implies for a practical example like vector spaces. Perhaps some examples would help.