Abstract Algebra/Group Theory/Homomorphism/Homomorphism Maps Identity to Identity

Theorem

Let f be a homomorphism from group G to group K.

Let e_G and e_K be identities of G and K.

f(e_G) = e_K

Proof

0.   $f(e_G)\in K$	f maps to K
1.   $[f(e_G){]}^{-1}$	inverse in K
.
2.   $f(e_G\ast e_G)=f(e_G)⊛f(e_G)$	f is a homomorphism
3.   $f(e_G)=f(e_G)⊛f(e_G)$	identity eG
.
4.   $[f(e_G){]}^{-1}⊛f(e_G)=[f(e_G){]}^{-1}⊛f(e_G)⊛f(e_G)$	1.
.
5.   $e_K=e_K⊛f(e_G)$	identity eK, definition of inverse
6.   $e_K=f(e_G)$	identity eK