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I am quite new to the subject, please comment if something does not make sense.

Let $M$ be a Lie group. $L_x$ be left multiplication by $x\in M$.
I want to show that vector space $X:M \to TM$ defined as $X_x = T_1 L_x(v)$ is differentiable. ($v\in T_1M$)

If I am not mistaken, I need to argue the map \begin{equation} a \mapsto \left(a,\ T_{\varphi^{-1}(a)}\varphi (T_1L_{\varphi^{-1}(a)}(v))\right) = \left(a,\ T_1(\varphi\ \circ L_{\varphi^{-1}(a)})(v)\right) \end{equation} is differentiable ($a\in \mathbb{R}^n$, $\varphi$ is a chart). I am not sure how to argue tangent map is differentiable.

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