# How to manipulate sequences of inequalities?

I am trying to write some symbolic inequality proofs in Mathematica. What would be a clean way to make the inference that a >= b and b > c implies a > c? Or alternatively, given an inequality b > c, how I could apply a function to the left hand side only performing the transformation f[b] = a? I would be able to code it, but it feels kind of clumsy, I would like to do something simple as 'ApplySides', but for one side of the inequality only. Is there a simple way?

Reduce[a >= b && b > c, b, Reals]

ApplyLeft[f_, rel_] := ReplacePart[rel, 1 -> f[rel[[1]]]]