The mana generated by Renowned Weaponsmith’s first ability can’t be spent to activate abilities of artifact sources that aren’t on the battlefield.

Activated abilities contain a colon. They’re generally written “[Cost]: [Effect].” Some keyword abilities, such as equip, are activated abilities and will have colons in their reminder text. You can’t spend mana added by this ability to pay costs of triggered abilities of artifacts such as the first ability of Chandra’s Regulator.