quantum chemistry - Why replace c with v in Broglie hypothesis

I am in XI and I am currently studying atomic orbitals, I stumbled up and am stuck at the following step of de Broglie's hypothesis.

I did a little search on-line and found almost same derivation everywhere, the only place I found a more complex and supposedly accurate formula was the Wikipedia itself, though it did not explain the derivation.

Few such sites:

In the derivation of the formula

$$\frac{1}{\lambda}=\frac{h}{m\color{green}c}=\frac{h}{m\color{red}v}$$

we replace $c$ with $v$. I want to ask why?

As of Wikipedia it gives a better formula with the Lorentz factor $\gamma$ :-

$$\frac{1}{\lambda}=\frac{h}{\color{red}\gamma m_oc}=\frac{h}{m_ov} \sqrt{\color{red}1\color{red}-\frac{\color{red}v^\color{red}2}{\color{red}c^\color{red}2}}$$

Can anyone give the derivation or a hint or a logic that helps understand why the derivation from $E=mc^2$ and $E=h\nu$ gives the De Broglie equation.