PCI DSS stands for Payment Card Industry Data Security Standard and contains a set of rules for processing credit card transactions.
The complete PCI DSS ruleset is pubslihed here: https://www.pcisecuritystandards.org/document_library
In principle, all those involved in a credit card payment (e.g. merchants and PSPs) are subject to these rules as soon as they process, transfer or save credit card data.
The compliance with these rules is checked as part of a PCI DSS certification. Your acquirer is responsible for verifying your PCI DSS certification.
Evidence is provided either by means of questionnaires which you must fill out. These questionnaires are referred to as SBF (Selbstbeurteilungsfragebogen) or SAQ (self-assessment questionnaire). The questionnaires are divided into different classes depending on how you process credit card data (SAQ A, SAQ A-EP, ... up to SAQ D).
Instead of the questionnaire, a personal PCI DSS audit may also be required. This decision is made by the acquirer or the PCI DSS auditor.
Computop Paygate supports you in adhering to these rules by relieving you of the direct processing of credit card data, if you use the "Hosted Payment Page" or the Credit Card Form (paySSL). In the simplest form, then a proof according to SAQ A is possible.
The PcNr (pseudo card number) is not subject to the PCI DSS rules, because it is not a credit card, but only a token that references a credit card.