# What are all the gauge symmetries & derivatives of the QED lagrangian?

I find that the gauge symmetries of the lagrangian are a topic that gets obfuscated quite a bit. I'm trying to understand the big picture of this in QED. My understanding is that:

1. Gauge derives its name ultimately from that thing that I call the
classical gauge transformation of classical electromagnetism: $$\mathbf{A} \rightarrow \mathbf{A}' + \nabla f$$.
2. In QED, the spacetime equivalent of this is $$A_\mu \rightarrow A_\mu + \frac{1}{g}(\partial_\mu \Lambda)$$.
3. Basic spacetime symmetry requires us to have global symmetry, for which we have $$\phi \rightarrow e^{i \theta} \phi$$.
4. Other spacetime symmetries require us to have local symmetry, for which we have $$\phi \rightarrow e^{i \theta (x)} \phi$$.
5. You have to have a gauge covariant derivative, thus we have $$\partial_\mu \rightarrow \partial_\mu - igA_\mu$$.

Now my question is this: are any of the above gauge symmetries & derivatives (that is, 2 through 5) redundant? For example, does rule 5 above actually the same as rule 2?

The 5) is a way in which one has to modify derivative in order to make the theory gauge invariant. For the simple case of complex scalar field $$\phi$$ with $$U(1)$$ gauge symmetry, the lagrangian looks as follows: $$\mathcal{L} = \partial_\mu \phi^{*} \partial^{\mu} \phi + \ldots$$ Applying an infinitesimal position-dependent transformation $$\phi \rightarrow e^{i \theta}\phi$$, one gets in the lowest order: $$\delta \mathcal{L} = i \partial_\mu \theta (\phi^{*} \partial^{\mu} \phi - \phi \partial^{\mu} \phi^{*})$$ So to cancel this change in the Lagrangian one needs to promote $$\partial_\mu$$ to $$\partial_\mu + i A_\mu$$, with the aforementioned transformation properties : $$A_\mu \rightarrow A_\mu + \partial_\mu \theta$$. So that the simultaneous transformation of both fields would leave the Lagranigian invariant.