Broadcast on Prime transmitted everywhere simultaneously (11)

I believe the answer is:

omnipresent

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'everywhere simultaneously' is the definition.
I can't tell whether this defines the answer.

'broadcast on prime transmitted' is the wordplay.
'broadcast' indicates an anagram (letters scattered or cast around).
'transmitted' becomes 'sent' (sending is a kind of transmitting).
'on'+'prime'='onprime'
'onprime' with letters rearranged gives 'omnipre'.
'omnipre'+'sent'='OMNIPRESENT'

Can you help me to learn more?

(Other definitions for omnipresent that I've seen before include "Everywhere at the same time" , "Found everywhere" , "Existing everywhere" , "Constantly encountered" , "Encountered everywhere" .)

I've seen this clue in The Independent.

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