Proof of Homomorphism: f(eG) = eH

[lockedup]

Homework Statement

If f:G\rightarrowH is a homomorphism, then f(e_G) = e_H.

Homework Equations

The proof from my professor's notes:
f(e_G) = f(e_G*_Ge_G) = f(e_G)*f(e_G)
f(e_G) = f(e_G)*e_H
f(e_G)*f(e_G) = f(e_G)*e_H
f(e_G) = e_H

The Attempt at a Solution

My question is, how do you get from the first line to the second. Because it looks like she's using the proposition to prove itself.

 

[Petek]

The second line doesn't follow from the first. It simply says that if you multiply any element of H by the identity element of H, you get ... (fill in the rest).

Petek

 

[lockedup]

The second line doesn't follow from the first. It simply says that if you multiply any element of H by the identity element of H, you get ... (fill in the rest).

Petek

I feel like an idiot. Thank you!