The way patriarchy is usually defined by feminists, women putting other women down (at least to the extent that that is because the women being put down are acting in a way that is outside of the fixed gender roles held by the women doing the putting down) is an example of patriarchy, not an effect of it.

I can't tell if those are rhetorical questions or not. Part of it might be patriarchy, but I'm pretty sure for the most part it's just nature at work. Women competing with other women for the attention of men doesn't sound new to me.