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We lay down an elementary yet fundamental lemma concerning a finitealgebraicness property of a smooth map from an Azumaya/matrix manifold with afundamental module to a smooth manifold. This gives us a starting point tobuild a synthetic (synonymously, $C^{\infty}$-algebraic) symplectic geometryand calibrated geometry that are both tailored to and guided by D-branephenomena in string theory and along the line of our previous works D(11.1)(arXiv:1406.0929 [math.DG]) and D(11.2) (arXiv:1412.0771 [hep-th]).

D-branes and synthetic/$C^{\infty}$-algebraic symplectic/calibrated  geometry, I: Lemma on a finite algebraicness property of smooth maps from  Azumaya/matrix manifolds