Quorum utramque viam cum summis vigiliis et summa providentia adnuente Deo perfecimus. et bellicos quidem sudores nostros barbaricae gentes sub iuga nostra deductae cognoscunt et tam Africa quam aliae innumerosae provinciae post tanta temporum spatia nostris victoriis a caelesti numine praestitis iterum dicioni Romanae nostroque additae imperio protestantur. omnes vero populi legibus iam a nobis vel promulgatis vel compositis reguntur.
This paragraph is the third paragraph of the Imperatoriam of the “Justiniani institutiones”, and I have three different kinds of translations at hand, and they are different from each other,so it puzzled me a lot, I didn’t know which one is right. The biggest difference between them is that the meaning of the phrase"bellicos quidem sudores nostros ", someone considered it as an accusitive while another thought it was ablative,is it the object to the verb of “protestantur”,or it is an adverbial phrase? 
Thank you !
I read the “bellicos quidem sudores nostros” as an the object of “cognoscunt;” “barbaricae gentes sub iuga nostra deductae” is an appositive description. But it makes sense to translate it into English with something like an ablative absolute, because “the barbarians, [having been/by being] lead under our yoke, know of our hard warlike labors.”
Here’s a quick translation of my own:
We completed both of these journies with the greatest vigilence and providence, with God approaching [?]. And the barbarian clans, controlled by our yoke, certainly know of our warlike labors; and both Africa and other innumerous provinces, after such lapses of time, and with us having excelled in our victories due to the power of heaven, again they testify as to Roman power as well as to our authority. For all people are now ruled by us either through declarations or truces.
I’m reading “adneunte” as “adeunte.” and I’m not possitive what to do with the “additae,” which I took as something like “in addition too/added to.”