Prove that $1^\text{nan} = 1.00$
The fact that $1^{\text{nan}}= 1.0$ is by definition, and not by proof, more specifically, it is by definition of the standard IEEE 754, IEEE Standard for Floating-Point Arithmetic.
From the definition of floating-point arithmetic, as specified in IEEE 754 revision 2008, 9.2.1 Special values:

pow(+1, y) is 1 for any y (even a quiet NaN).

Here is the full list of special values for pow:

For the pow function (integral exponents get special treatment):

pow(x, ±0) is 1 for any x (even a zero, quiet NaN, or infinity)
pow(±0, y) is ±∞s and signals the divideByZero exception for y an odd integer &lt; 0
pow(±0, −∞) is +∞ with no exception
pow(±0, +∞) is +0 with no exception
pow(±0, y) is +∞ and signals the divideByZero exception for finite y &lt; 0 and not an odd integer
pow(±0, y) is ±0 for finite y &gt; 0 an odd integer
pow(±0, y) is +0 for finite y &gt; 0 and not an odd integer
pow(−1, ±∞) is 1 with no exception
pow(+1, y) is 1 for any y (even a quiet NaN)
pow(x, y) signals the invalid operation exception for finite x &lt; 0 and finite non-integer y.