Totožnost prokážeme občanským průkazem.

Because prokážeme comes from the perfective verb prokázat. In Czech, perfective verbs do not normally describe an action happening right now; their “present” forms usually refer to a completed action in the future.

So:

• prokážeme = we will prove / we will show
• imperfective partner: prokazujeme = we are proving / we prove habitually

In formal instructions, this future form can also sound like a procedural statement: this is how identity will be shown.

They are an aspect pair:

• prokázat = perfective: to prove/show successfully, as a completed act
• prokazovat = imperfective: to be proving, to prove repeatedly, to show regularly

So Totožnost prokážeme... focuses on the result: identity gets successfully proved.
If you used prokazujeme, it would sound more like a general/habitual process.

Czech usually leaves out subject pronouns unless they are needed for emphasis or contrast. The ending -me in prokážeme already tells you the subject is we.

So:

• prokážeme = we will prove
• my prokážeme = we will prove with extra emphasis, like we (not someone else)

This omission is very normal in Czech.

Yes. Totožnost is the direct object, so it is in the accusative case.

The reason it looks unchanged is that totožnost is a feminine noun of the -ost / -nost type, and in the singular its nominative and accusative have the same form.

So here:

• nominative: totožnost
• accusative: totožnost

That is completely normal for this noun type.

Because Czech uses the instrumental to express the means or instrument by which something is done. With prokázat, a common pattern is:

• prokázat něco něčím = to prove/show something by means of something

So:

• totožnost = what is being proved
• občanským průkazem = the means used to prove it

In English this often becomes with or by means of.

Because Czech does not normally use s + instrumental for a tool or means in this kind of sentence. The plain instrumental by itself already expresses by/with the help of.

So:

• občanským průkazem = with/by means of an ID card

Using s would more naturally suggest together with or in the company of, not the normal idea of an instrument here.

Občanský průkaz is the standard Czech term for a national identity card / ID card.

Literally:

• občanský = civic, citizen-related
• průkaz = card, pass, document

In the sentence, both words change because they must agree in case, number, and gender:

• base form: občanský průkaz
• instrumental singular: občanským průkazem

Yes, and that is often very natural. Svou totožnost means our own identity here.

Czech often leaves the possessor unspoken when it is obvious from context, so Totožnost prokážeme... can already be understood as We will prove our identity...
If you want to make it explicit, svou is a good choice.

Also note: when the possessor is the same as the subject, Czech usually prefers svůj rather than náš.

Yes, and that is a good observation. In formal or procedural Czech, 1st person plural can be used in an inclusive or impersonal way, a bit like let us / we do X in instructions.

Depending on context, the sentence may feel like:

• We will prove identity with an ID card
• or a formal procedural statement: Identity is proved with an ID card

If you wanted to address one person directly, Czech could also say:

• Totožnost prokážete občanským průkazem. = You will prove your identity with an ID card.
• Prokažte totožnost občanským průkazem. = Prove your identity with an ID card.

Yes. Czech word order is flexible because the endings show the grammatical roles.

The given sentence is perfectly fine, but other orders are possible:

• Totožnost prokážeme občanským průkazem.
• Občanským průkazem prokážeme totožnost.
• Totožnost občanským průkazem prokážeme.

These versions have slightly different emphasis or information structure, but the basic meaning stays the same. English is much more rigid here than Czech.