The starting point is the formula (1) which is the characteristics of the operational amplifier and is no problem.
And then we make two important assumptions:
The first is that the output voltage of the operational amplifier is limited. The output cannot exceed the power supply, so this hypothesis is absolutely true.
The second one is to say that the open loop gain of operational amplifier
A is very large. A of common operational amplifiers is often up to the sixth or seventh power of ten, even higher, so the hypothesis is generally no problem. But don't forget that the operational amplifier's practical open loop gain is also related to the work state. If out of the linear area, A may be not big, so the second hypothesis is conditional. We remember this first.
When the open loop gain A is large, the operational amplifier can have "Virtual Short". But it's just a possibility, and it cannot be automatically implemented. No one will believe if we take an operational amplifier and say its two inputs is "Virtual Short". "Virtual Short" can only be achieved in a particular circuit.
The conditions for the existence of "Virtual Short" are:
1) A, the open loop gain of the operational amplifier, is large enough;
2) Have negative feedback circuits.
In fact, the conditions (1) are ok for most of the operational amplifiers, and the key is the working area. If it's a circuit in the book, we should judge by calculation; if it is an actual circuit, we can judge by instruments whether the output voltage is reasonable. There is also a condition associated with "virtual short", which is called "virtual ground". It is a kind of "virtual short" when there is an input terminal grounding. It is not a new situation. In some books, "virtual short" can be used only under the condition of deep negative feedback. I think it's not accurate, and in my view, the potential thinking of this opinion is that in the case of deep negative feedback, operational amplifiers are more likely to work in the linear area. But it is not absolute, when the input signal is too large, the operational amplifier in deep negative feedback will also get into saturation.
Therefore, the output voltage value inference should be the most reliable.