How both definitions of a lie group are equivalent

Question

I can't proof that these definitions of lie groups are equivalent. Can anyone prove it directly?.

1. A Lie group is a set which carries the algebraic structure of a group and the differentiable structure of a smooth manifold such that the mapping $$ G\times G \rightarrow G \hspace{1cm} (a,b) \mapsto ab^{-1}$$ is smooth

2. A Lie group is a set which carries the algebraic structure of a group and the differentiable structure of a smooth manifold such that the mappings $$ G\times G \rightarrow G \hspace{1cm} (a,b) \mapsto ab$$ and $$ G \rightarrow G \hspace{1cm} g \mapsto g^{-1}$$ are smooth

Answer 1

1 implies 2.

Since $m:(a,b)\rightarrow ab^{-1}$ is smooth, and $i(a)=(e,a)$ is smooth, $i_v(a)=m\circ i(a)=a^{-1}$ is smooth.

$ f:(a,b)\rightarrow (a,i_v(b))=(a,b^{-1})$ is smooth and $m\circ f(a,b)=ab$ is smooth.

2. implies $a$,

let $g(a,b)=ab, m=g\circ (Id_G,i_v)$.