I would suggest a different source for aerodynamics. Langwiesche is the best for basic flying technique even almost 80 years later, but he's deliberately a bit fuzzy on the theory (he makes it clear in his intro that the book won't be about that).
The EAS stays constant. The CAS and IAS will start to underread noticeably above 10,000 ft and/or 200 kt because of compressibility errors — that's why higher-performance aircraft have lower Vne (in IAS) specified for higher altitudes.
But for the most part (with variations for specific aircraft) the answer to the OP's question is yes, IAS will typically remain close to constant for the same power setting under the same flight conditions from sea level to 10,000 ft. If in doubt, check your POH for any model-specific quirks (mine has CAS constant within ½ kt).
Caveats:
- Holding a constant altitude in a subsiding airmass (eg under a higher pressure system) your IAS will be a few knots slower; in rising air (e.g. thermals, cumulus cloud, ridge lift) it will be a few knots faster.
- Higher gross weight slows you down at the same power setting (the rule of thumb — very approximate — is 1 knot per 100 lb for a PA-28 or C172).
- A forward CG costs you a knot or two (but makes your pitch more stable in turbulence); an aft CG buys you a knot or two (but makes your pitch less stable in turbulence).
The caveats are why we use RPM rather than IAS to set power with a fixed-pitch prop.