Dragon DNA v Downer Material

[SuicidalPikachu]

ok, so there was a question at the BR i was at. Does Dragon DNA from Gyarados get affected by Azelf's Downer material which makes basic's attacks cost 1 more to use. I argued that because downer material only affects basic pokemon, Gyarados would be able to attack normally. however, there was no precedence if you will to back this up. would i be correct in my arguement?

 

[PokePop]

Gyarados is doing the attack. Azelf can't "see" the Magikarp. It only sees Gyarados.
The attack does not cost any more because it is Gyarados using the attack.