I was reading about the 9 or 12 basic properties of 'fields' (if that's what they're called) in a book called Spivak's Calculus, 3rd Edition, and got quite befuddled by dealing with as basic stuff as assuming $1 \neq 0$ (although I get that now).
But I got stuck when I wanted to conclude stuff $ a \cdot c = b \cdot c $ from $ a = b $ where $\cdot$ means addition, multiplication or w/e. There was no property dealing with these cases!
Someone on IRC told me that a property stating that operations give equal results given equal inputs (note: the order of the inputs is preserved; we aren't dealing with commutativity here) was implicitly assumed in the book.
Also, last year in school, we learnt about Euclid's axioms which explicitly stated stuff like this (and in a rather wordy and arbitrary way too, it seemed; unlike the rest of this group and field stuff).
My questions are:
- What is this property actually called?
- Does it apply to every single operation definable (except stuff like RNGs)?
- Why wasn't this mentioned along-side other 'basic' facts in the book?