It's ironic that in a dispute over correct writing, you've split your infinitive. "Properly to play" or "to play properly" are correct English grammar.
'Merica.
That is an archaic, prescriptivist opinion, not an actual grammar rule in English. There are many style guides but those are opinions, not rules.
Just because a couple hundred years ago, the British had a Latin fetish and tried to shoehorn in new rules doesn't mean anyone should ever follow them.
Yeah, this is an odd quirk of some British grammars that wanted to imitate Latin. It's not an actual rule, and those grammars have been retired long ago.
I'm sorry. I'm not a native English speaker and probably I watch too many american movies. Or should I say films?
Don't worry, some stuffed shirts made up the rule wholesale in the 1830s and a particular kind of pedant has been pushing it ever since. It's never been true.