Inverse of an element of a linear vector space For a linear vector space $\mathbb{V}$, the inverse $|\phi\rangle$ associated with an element $|\psi\rangle$ is defined as $$|\phi\rangle+|\psi\rangle=|0\rangle$$ where $|0\rangle$ is the additive identity and is not same as number $0$. In that case, is it justified to write $|\phi\rangle=-|\psi\rangle$? All I can see from the definition is that $|\phi\rangle=|0\rangle-|\psi\rangle$. But how is $|0\rangle-|\psi\rangle=-|\psi\rangle$? I used Dirac notation which physicists use all the time.

Yes, $-\left|\psi\right\rangle$ is the additive inverse of $\left|\psi\right\rangle$.
The symbol $-$, when used as unitary prefix operator ($-x$), is universally understood as the function mapping an element $x$ to its additive inverse. It doesn't matter if $x$ is $1$, $i$, $2^{100}$, $\vec v$ or $\left|\psi\right\rangle$.
Note that the use as binary infix operator, $x-y$, is actually equivalent to addition of the inverse, $x+(-y)$.
A short note on your use of $\left|0\right\rangle$ to denote the zero vector: While this is not directly wrong, it is highly unusual in physics. When physicists use $\left|0\right\rangle$, this usually denotes a nonzero (indeed, normalized) vector corresponding to some specific quantum state of the system under consideration; most often the ground state (state of lowest energy), in quantum field theory the vacuum.
