Transmit coaxial signal wirelessly

And would a hack work? Basically, connecting a coaxial antenna to the coaxial socket

and a small indoor antenna to the TV's coaxial socket

since they are 4-5 meters apart, could it work?

I'm sorry... but I don't know of any systems that only transmit the antenna signal wirelessly, and I can't find any.
Aside from the proposed solutions?

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 It's very difficult to catch a black cat in a dark room.
                                Especially when it's not there...

the powerline adapters in the same apartment should work just fine...
If I understand correctly, one box is connected to the antenna and to a power socket, while the second box (which acts as the TNT decoder) is connected to the television and also to a power socket...
The antenna signal is thus processed by the first box, transmitted to the second via the electrical sockets before being decoded by the second box and sent to the television via an HDMI cable.

However, forget about the indoor antenna unless the TV transmitter you rely on is in sight...
The tinkering... sorry but what you're considering doesn't exist.

At 4 or 5 meters, a coaxial cable remains; there are various lengths available and you can even make one yourself.
But it remains to be seen if it's possible without crossing the entire room. You can follow the baseboards, go around the doors or windows
--

 Very difficult to catch a black cat in a dark room.
                              Especially when it's not there...