Factorions Base 10

Theorem
The following positive integers are the only factorions base $10$:
 * $1, 2, 145, 40 \, 585$

Proof
From examples of factorials:

A computer search can verify solutions under $2540160 = 9! \times 7$ in seconds.

Let $n$ be a $7$-digit number with $n > 2540160$.

Then the sum of the factorials of its digits is not more than $9! \times 7 = 2540160$.

So $n$ cannot be a factorion base $10$.

Now let $n$ be a $k$-digit number, for $k \ge 8$.

Then the sum of the factorials of its digits is not more than $9! \times k$.

But we have:

So there are no more factorions base $10$.