(...) In my own experience, in high school (second year) we defined real numbers by Dedekind cuts (a construction easily forgotten and immediately replaced by intuitive use of the usual properties! But we were told exactly what $\sqrt{2}$ is in that framework). Then in the fifth year of high school real analysis was done pretty rigorously including epsilon-deltas, continuity, derivatives, fundamental theorem of calculus (but with Riemann integral done with no proofs and with only semi-rigorous definition; and no mention of Cauchy sequences).

@Steven Landsburg : we're probably using different definitions of "freshman calculus". In some countries this not-yet-rigorous version of elementary real analysis simply doesn't exist: people learn to deal with all the rigorous definitions and all the proofs from the get go. There, every freshman is supposed to get that knowledge of epsilon-deltas etc during "calculus" because "calculus" is done that way. So the question is probably location-dependent. (...)

@Ziad H. Muhammad: you're right. In fact this was pointed out in comments to my answer below. I didn't realize there was still that comment of mine here (that I've now deleted).

@Dave L Renfro: maybe he would've said: "I see your Schwartz is as big as mine"...

@Angelo: you weren't thinking of Ax-Grothendieck as the easy step, were you?

I must be missing something. Are you saying $f(X)$ has to be a projective variety? But $X$ isn't nec. projective

I figured ZMT (stacks.math.columbia.edu/tag/05K0) implies $f$ is an open embedding. How can one easily see that $f$ is also surjective?

Yes, of course you're right: I meant $x\mapsto 2x$.

I wouldn't count Abelian varieties as interesting counterexamples in this context, because they always admit the $x\mapsto -x$ isogeny, so they trivially do not satisfy (RP) or (JC).

[I changed "if" to "since" in the title. Of course I'm ok with you reverting back the edit, if you deem appropriate]

By "dimension" is it meant the dimension of $A$ as a vector space over the base field, or the dimension of the corresponding "quantum group" (if the latter makes sense..)?