I need some help with the bra-ket notation. Suppose we have a normalized wavefunction for a two-qubit system (where $A$ and $B$ denote the two qubits respectively), like: $$|\psi\rangle_{A B} = a(|0\rangle_A \otimes |0\rangle_B) + b (|0\rangle_A \otimes |1\rangle_B) + c (|1\rangle_A \otimes |0\rangle_B) + d (|1\rangle_A \otimes |1\rangle_B)$$
Then what is the correct notation for writing the probability of wavefunction to collapse to a state where the state of qubit $A$ is $|0\rangle_A$ ? Is it $|_A\langle 0|\psi\rangle_{AB}|^2$ (which should be $|a|^2 + |b|^2$) ?
Also it would be very helpful if someone could link me a webpage or resource which discusses the properties of this operation i.e. multiplying a bra with the tensor product of two ket vectors. Does multiplying $\langle 0|_A$ with $|0\rangle_A \otimes |0\rangle_B$ or $|0\rangle_A \otimes |1\rangle_B$ equal $1$ ? And does multiplying $\langle 1|_A$ with $|1\rangle_A \otimes |0\rangle_B$ or $|1\rangle_A \otimes |1\rangle_B$ equal $0$ ? I'm not very sure about this.