Double-magicity of proton drip-line nucleus $^{22}$Si with \textit{ab initio} calculation

New magic numbers have been discovered in the neutron-rich region of the nuclear chart. However, there has been a lack of research on proton-rich nuclei. $^{22}$O, the mirror nucleus of $^{22}$Si, is a double-magic nucleus bearing a high $E(2_1^+)$. Whether $^{22}$Si exhibits double-magic characters is an intriguing topic. To investigate this matter, we utilized \textit{ab initio} valence space in-medium similarity renormalization group for $^{22}$Si/$^{22}$O, and their nearby nuclei. Our \textit{ab initio} calculations provide good descriptions for the double magicity of $^{22}$O, as well as the shell evolution of $N=14$ and $Z=14$ through $E(2_1^+)$. The computed $E(2_1^+)$ indicate that the closure of $Z=14$ sub-shell in proton-rich nuclei is weaker than the $N=14$ sub-shell closure in their mirror nuclei. Particularly, the calculated $E(2_1^+)$ of $^{22}$Si is 800 keV lower than the one of $^{22}$O. To further explore the magicity of $^{22}$Si, the mirror energy difference (MED) of $^{26}$Si/$^{26}$Mg, $^{24}$Si/$^{24}$Ne, as well as $^{22}$Si/$^{22}$O are calculated. The results demonstrate that the calculated MEDs agree well with available experimental data, and the $E(2_1^+)$ values of $^{22,24,26}$Si are all lower than their respective mirror nuclei due to the Thomas-Ehrman shift with large $s_{1/2}$ occupation. Moreover, our calculation provides that the many-body configurations of the low-lying state of $^{22}$Si/$^{22}$O are nearly identical despite the fact that the states bearing large MED. In conclusion, our \textit{ab initio} results suggest that $^{22}$Si is a double magic nucleus, similar to its mirror nucleus $^{22}$O, albeit with a lower $E(2_1^+)$.