It will work fine. If any thing the CPU might bottleneck the GPU slightly in a few games but it won't be a whole lot.
The Radeon HD 7870 is quite a lot faster than the built-in Radeon HD 6550D on the APU.
The built-in graphics on the APU will be disabled when you install the 7850. However, Radeon HD 7870 cards are 2 slot cards so make sure that your motherboard and case have enough space.