As far as I understand BCH uses the same address space as BTC but different transaction format, so I can spend any pre-fork BTC tx output using BCH client and it will go into BCH blockchain and essentially register as BCH.

What I don't get is if BTC network will know I spent that output into BCH?
If BTC doesn't read/recognize BCH transactions, will I still be able to spend that output in BTC network?