Triumphant Navy vs Dioxazine Purple

How close are they?

ΔE 20.7, loose: different enough that you should pick on purpose. Dioxazine Purple is a little darker, more saturated, warmer than Triumphant Navy.

Can I use Dioxazine Purple instead of Triumphant Navy?

Not as a straight swap; see the closer alternatives below. Loose match (ΔE 21). Dioxazine Purple is a little darker, more saturated, warmer than Triumphant Navy.

Can I use Triumphant Navy instead of Dioxazine Purple?

The same difference applies the other way round: Triumphant Navy is a little lighter, more muted, cooler.

ΔE is the colour difference between our two swatch estimates: under about 6 is close, over about 15 is loose. Real paints also differ in finish, coverage and how they thin, so test before you commit.