Binomial coefficient congruence across a prime row ($\left(\genfrac{}{}{0ex}{}{p+k}{r}\right)\equiv \left(\genfrac{}{}{0ex}{}{k}{r}\right)(\mathrm{mod}p)(0\le r\le k<p)$)

In polynomial modular arithmetic, prime-row binomial coefficient divisibility gives $(1+X{)}^p\equiv 1+X^p$. Multiply by $(1+X{)}^k$ and compare coefficients of $X^r$ for $r<p$; the $X^p$ term cannot contribute.

// Target: foundations-of-mathematics.bigb