Difficulty proving congruence property(Modular Multiplicative Inverse).

Question

I watched some posts trying to discover why the modular multiplicative inverse of a given number was unique but i didnt understand why given:

$d_1*e≡ d_2*e ≡ 1 \mod(n)$ we can assume $d_1 ≡ d_1*d_2*e \mod(n)$.

I know it works cause I've tried out some examples like:

$3*5 ≡ 10*5 ≡ 1 \mod(7)$

$3 ≡ 3*10*5 \mod(7)$

$3 ≡ 150 \mod(7)$

$-147 \mod(7) = - 21$

This: $d_1 ≡ d_1*d_2*e \mod(n)$ is the step I'm missing. Can anybody supply me with a proof or the property I'm forgetting?

Answer 1

If $d_2 e \equiv 1$, then $d_1 \equiv d_1 1 \equiv d_1 (d_2 e)$.