Smallest Penholodigital Square

Theorem
The smallest square number which contains all the digits from $1$ to $9$ is:
 * $11 \, 826^2 = 139 \, 854 \, 276$

Proof
To streamline the argument, the term $9$-pan will be coined to mean $9$-digit pandigital number which excludes the $0$ digit.

Let $n$ be the smallest positive integer whose square is $9$-pan.

First it is noted that the smallest $9$-pan is $123 \, 456 \, 789$.

Hence any square $9$-pan must be at least as large as that.

Thus we can can say that:
 * $n \ge \ceiling {\sqrt {123 \, 456 \, 789} } = 11 \, 112$

where $\ceiling {\, \cdot \,}$ denotes the ceiling function.

It remains to be demonstrated that no positive integer between $11 \, 112$ and $11 \, 826$ has a $9$-pan square.