Someone could have logged in from inside your home network, that is one way but won't be possible to prove most likely.
The other way is they spoofed your IP. This means that they changed their IP to look like yours. This is not easy to do and someone would need a large amount of skill in networking to do this.
Again I do not know how you would go about proving you didn't do this. In the same way they cannot prove with 100% certainty that you did this you cannot prove that you didn't. There would be no way.
You said above that all staff have been to your home so it would be possible that someone posted from your house I suppose but again you wouldn't really have anyway to prove it.
I think if you did take legal action on this they would not have anyway to prove 100% that you did this considering the information you told me.
But again based on this information you have stated there is no technical way to prove that you did not do this.
The facts are someone appearing to be from your home IP address did this. But I don't thing you or them can prove without a doubt that one of you is telling the truth.
I can only tell you the technical end of things and the short answer is neither of you can say 100% yes or no that you did or did not do this based off the info.
Your next step depending on how far you want to push this would be to speak to a lawyer. This website does have employement lawyers you can speak to herewww.justanswer.com/employment-law
You may want to ask them a question as it is much cheaper than speaking to a lawyer in person.
So in conclusion, yes it is possible for someone to use your IP from a technical stand point and this is why having someones IP address is not proof positive that a specific person performed an action online. But on the other hand you have no way to technically prove that it was not you.