Definition of Lie group isomorphism

Question

Recently I am studying the Lie group.

The definition of Lie group isomorphism is the following:
https://en.wikipedia.org/wiki/Lie_group#Homomorphisms_and_isomorphisms

Let $\phi: G\mapsto H$, where $G$ and $H$ are Lie goups. $\phi$ is Lie group isomorphism if

1. $\phi$ is bijective homomorphism.
2. $\phi^{-1}$ is a Lie group homomorphism.

My question is simply

Does 1. and 2. implies $\phi^{-1}$ is bijective?
Why not just say that $\phi$ and $\phi^{-1}$ are Lie group bijective homomorphism? (it seems much clear) Hope for a deep and clear explanation of both sentences.

Answer 1

Yes they imply that the inverse is a bijection. The inverse of a bijection is always a bijection. So stating one is a bijection is enough, the crux of it is that functions can be a homomorphism one way but not neccisserily the inverse. An example is homeomorphisms in topology. A function may be continuous but that does not mean that the inverse is continuous.

For isomorphisms of course we must have like with homeomorphisms that the inverse and original function are homomorphisms, and in both cases it must be a bijection.

Why it is not stated for the original function and inverse is as said, because they imply one another.

Answer 2

The existence of $\phi^{-1}$ implies that $\phi$ is bijective, and it is always true that $\phi^{-1}$ is bijective (since $(\phi^{-1})^{-1} = \phi$, so $\phi^{-1}$ is invertible). This is true of bijective maps in general, nothing specific to Lie groups.

Your (equivalent) definition of a Lie group isomorphism therefore has a redundant condition (that $\phi^{-1}$ is bijective) which should therefore be omitted. Then you simply have Wikipedia's definition.