This is a misunderstanding of Kant. Although he speaks of '5 + 7 = 12' and 'space has three dimensions' as synthetic a priori statements which might suggest this simplistic idea, Kant aims at something different than proving that there we can determine truth of certain non-trivial statements a priori (of course his examples are ridiculous, neither of these statements is either synthetic or analytic, a posteriori or a priori, because a single statement doesn't even have a meaning of its own - in contexts of applied mathematics 5 + 7 = 12 might be "synthetic a posteriori" if you want to... doesn't really matter).
An interesting feature of Kant's philosophy is that Kant, strictly speaking, doesn't need the notion of analycity in the modern sense - even though he was the one to invent it (Leibniz and Hume had similar ideas but nevertheless). Kant in the preface to the 1787 edition of the Critique of Pure Reason speaks of the intellect constraining itself only by what it itself posits and then in Postulates of Empirical thought in General his explanation of modality doesn't reference thinkability without contradictions like Leibniz's notion of a possible world (which was taken up by the early Wittgenstein in the Tractatus). If we deem all true statements analytic, then thinkability without contradictions in Leibniz's sense constraints what is possible to what is actual. If we follow Quine by claiming that there's no distinction, then only obvious contradictions like "p and ~p" would exclude a world from being possible in the virtue of non-contradictoriness. Leibnizian modality is dependent on analycity.
But Kant isn't satisfied by that because he knows, 200 years before Kripke and thanks to Hume's skepticism, that statements of physics, for example, are not analytic but are anyways necessary. Kant's 'synthetic a priori' is partly due to an apparent confusion of necessity with a priority - it is clear that Kant identifies the two - but for Kant a priority rather should be understood in terms of necessity (Kant's other achievement in this regard is noticing that notwendigkeit exists in the sphere of deontic and aletheic modality, as moral duty and laws of, i.e. physics, although Leibniz apparently noted that earlier in his logical writings, which weren't published until the end of the nineteenth century).
Kant's Copernican Revolution then follows due to the need to explain necessity in terms of the intellect positing laws as conditions of experience which is then understood as being equally an objective constraint on the objects of experience: "the conditions of the possibility of experience in general are likewise conditions of the possibility of the objects of experience" (A158/B197). Postulates of Empirical Thought in General is where Kant very clearly outlines this intent. This system of subjective but equally objective posits which is elaborated in the Transcendental Analytic can be equally understood as an answer to the question of How is determinate (contentful) use of logical notions (in judging) possible? which is nothing but How are synthetic a priori judgements possible?... or, in other words, How do apparent mere-thought-determinations of logic relate to the world?.
That's why Kant claims the synthetic a priori question unifies his whole critical project - because it does, the originality of the question is not due to the fact that the pre-Kantians didn't think substantial a priori knowledge was possible (Kant himself thought synthetic a priori is merely formal... its not some magical intuition) but rather because it comprises of numerous questions which weren't asked before Kant. Kant's own answer - given fully in the schematism chapter of the Analytic - relies, of course, on time, which Heidegger unsurprisingly greatly appreciated.
The American philosopher Wilfrid Sellars's account of modality is similar to Kant's. It makes no reference to possible worlds but understands necessity and possibility as metalinguistic constraints on our application of concepts - this is a Carnapian idea... which is interesting because Carnap was taught by neo-Kantians.