Quick question about Innate spells

[Syries]

The relevant rule in question wrote:

You’re always trained in spell attack rolls and spell DCs for your innate spells, even if you aren’t otherwise trained in spell attack rolls or spell DCs. If your proficiency in spell attack rolls or spell DCs is expert or better, apply that proficiency to your innate spells, too.

Say I'm a Gnome occult sorcerer with First World Magic to grab Electric Arc as an innate spell from the primal tradition. Once I become expert in Occult spell attacks/DC, considering the innate spell rules don't specify the innate spell must be from the same tradition to match proficiency, would my Electric Arc become expert as well?

It seems to be yes, but I just want to double check with the community.

full rules for Innate Spells here, for reference

[Blave]

That's correct. You are always at least trained in your innate spells. If you happen to have ANY spellcasting proficiency higher than trained, you can apply that proficiency to your innate spells.

[Syries]

Blave wrote:

That's correct. You are always at least trained in your innate spells. If you happen to have ANY spellcasting proficiency higher than trained, you can apply that proficiency to your innate spells.

Cool, that's what I thought. I just wanted to check if I didn't miss any obscure rule or anything. Thanks!