Fourier transform of distribution function 1

Question

I can't understand the third equation of the following equation series and I think I'm missing something pretty obvious. the $^\vee$ ist meant to represent the inverse fourier transform. $$\hat{1}[\phi]=1[\hat{\phi}]=\int_\mathbb{R^n}\hat{\phi(k)}dk=(2\pi)^n \hat{\phi}^\vee(0)=(2\pi)^n \phi(0) $$
I'd be very glad if someone could help me :)

Answer 1

Missing something simple indeed:

We know that $(\frac{1}{2\pi})^n\int_{\mathbb{R^n}}f(k)e^{ikx}dk=f^{\vee}(x) $

which implies $f^{\vee}(0)= (\frac{1}{2\pi})^n \int_{\mathbb{R^n}}f(k) dk $.
Replace f by $\hat{\phi}$ and you have the answer. Props to mathreadler for his tip.