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tl;dr version of the 2016 ACM Turing Award:

((g^a) mod p)^b mod p = (g^ab) mod p = ((g^b) mod p)^a mod p

-- The beauty of math.

This skips the most important part of the proof which explains why this is not easily invertible without the secret key.

What is this part? Isn't it just by assumption? (also, by secret key, I'm guessing you mean 'a', 'b')