soft question - Looking for Proofs Of Basic Properties Of Real Numbers

I have just begun my study of complex numbers and I learned where imaginary numbers came from and their importance. However there's one thing that I need to clarify and that is the properties of real numbers and their proofs.

1. Closure Laws
   For all $a,b \in \mathbb{R}$, $a+b$, $a-b$, $ab$, $a/b$ are real numbers. Thus $\mathbb{R}$ is closed under four fundamental operations.




2. Commutative Laws
   For all $a,b \in \mathbb{R}$ $a+b = b+a$ and $ab = ba$.




3. Associative Laws
   For all $a,b,c \in \mathbb{R}$ $a+(b+c) = (a+b)+c$ and $a(bc) = (ab)c$.




4. Additive Identity
   For all $a \in \mathbb{R}$ there exists $0\in \mathbb{R}$ such that $a+0 = 0+a = a$.




5. Additive inverse
   For all $a \in \mathbb{R}$ there exists a $b \in \mathbb{R}$ such that $a+b = b+a = 0$, the additive identity $b = -a$ is called the additive inverse or the negative of $a$.

and similarly Multiplicative Identity, Multiplicative inverse, Distributive Law, Trichotomy Law, Transitivity of order, Monotone Law of Addition, Monotone law of multiplication.

I understand that the above laws hold good throughout mathematics. Should these laws be accepted as being true "on faith" or are there proofs?
If yes, I am curious to know the proofs. As per my understanding no textbook has ever talked about proofs for these.

Answer

If some book states them like that, you should NOT take them on faith, NOR believe that they can get proven. The set of all real numbers is NOT closed under the operation of division, as the above statement of the closure laws indicates, since there does not exist division by 0 on the set of all real numbers.