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The phrase "we first prove" is grammatically correct and commonly used in written English.
It is typically used to introduce a step or process in a logical argument or proof. Example: "In order to understand the effects of climate change, we first prove that human activities have contributed to the rise in global temperatures."
Exact(60)
We first prove ergodicity.
We first prove (a).
We first prove (1).
(a) We first prove that.
We first prove the case.
We first prove part (i).
We first prove (i) ⇒ (ii).
We first prove Lemma 1.11.
We first prove Theorem 1.2.
We first prove assertion (I).
We first prove the necessity.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com