If there exist rings $R$ and $S$ (meaning that its set of elements have two binary operations: multiplication and addition, under addition, ring is an abelian group, under multiplication it is associative, or otherwise known as monoid). Two new rings can be created by these rings $R × S$ and $R + S$. If $R$ and $S$ are finite rings, amount of elements in $R$ multiplied by amount of elements in $S$ is equal to amount of elements in $R × S$.
Statement above seems to be very simply understandable, then comes the "equation", If there exists A and B that are coprime, then:
As i understand, the "equation" above states that under positive integers group generated by multiplication of $A$ and $B$, is isomorphic (has the same properties) to group generated by $A$ multiplied by group generated by $B$.
This seems to be the abstract algebraic definition of multiplicative function, but i'm unable to find the proof. Also author of the answer mentions that this is the Chinese Remainder Theorem, but as a computer scientist who researches asymmetric cryptography, i thought the purpose of CRT in asymmetric cryptography was to find a new method other than $c^d$ (where d is private key, multiplicative inverse) to increase speed of decryption. But now i found out that there are different purposes as well.
Is the isomorphism above the proof that Euler's totient function is multiplicative? If so, how? If not, how can abstract algebra be utilized to prove this?