Apple is not an internet service provider - Apple is an email account provider with a MobileMe account.
A MobileMe account is accessed as an IMAP account with the Mail.app on your Mac and with the iPhone's email client. All server stored mailboxes with an IMAP account are kept synchronized with the server automatically with each email client used to access the account.
If the other two email accounts are also IMAP accounts, the same applies.
If the other two email accounts are POP accounts, the same does not apply. New messages received at the incoming mail server with a POP account are downloaded by the email client used to access the account - the Inbox mailbox with a POP account is not synchronized with the server, and none of the other mailboxes for the account are stored on the server. When accessing a POP account with more than one email client and if you want the same new messages received at the incoming mail server downloaded with each email client used to access the account, you must select not removing the messages from the server when downloaded for the account preferences with each email client used to access the account. With a POP account, a new message downloaded and read by the Mail.app on your Mac first will be downloaded as a new and unread message when the account is checked for new messages at the server with the iPhone's mail client.
With an IMAP account, after reading a new message with the Mail.app on your Mac, the message will be marked as read automatically when checking the account's Inbox with the iPhone's email client, and vice versa since server stored mailboxes with an IMAP account are kept synchronized with the server automatically with each email client used to access the account.
With an IMAP account storing the account's Sent mailbox on the server with each email client used to access the account, send a message with the iPhone's mail client and the sent message will be available in the account's Sent mailbox with the Mail.app on your Mac automatically, and vice-versa.
This is not possible with a POP account.
