Metatable Making a Function Nil?

I started learning Metatables from a YouTube series on Lua this morning and I have run into a peculiar error while practicing Metatables. I made an advanced table previously and it had worked but it appears that the Metatable has caused it to break and I can't tell why. It prints out the coordinates and names correctly and then breaks when it attempts to connect to the move function. An explanation of what is going on and how it could be fixed would be greatly appreciated.

Error:

Workspace.Script:60: attempt to call field 'move' (a nil value)

Player = {
        x = 0, y = 0, z = 0,
        name = "",
 
        mt = {},
 
        new = function()
                local p = {}
                setmetatable(p, Player.mt)
                return p
        end,

		move = function(p, x, y, z)
                p.x = p.x + x
                p.y = p.y + y
                p.z = p.z + z
        end
}
 
Player.mt.__add = function(p1, p2)
        local p = Player.new()
        p.x = p1.x + p2.x
        p.y = p1.y + p2.y
        p.z = p1.z + p2.z
        return p
end
 
Player.mt.__sub = function(p1, p2)
        local p = Player.new()
        p.x = p1.x - p2.x
        p.y = p1.y - p2.y
        p.z = p1.z - p2.z
        return p
end
 
Player.mt.__eq = function(p1, p2)
        return p1.x == p2.x and p1.y == p2.y and p1.z == p2.z
end
 
Player.mt.__tostring = function(p)
        return "("..p.x..", "..p.y..", "..p.z..")"
end
 
p1 = Player.new()
p2 = Player.new()
 
p1.x = 20
p1.y = 10
p1.z = 0
p1.name = "Player1"
 
p2.x = 40
p2.y = 80
p2.z = 120
p2.name = "Player2"

print(tostring(p1), p1.name)
print(tostring(p2), p2.name)
 
p1.move(p1, 50, -40, 60)
p2.move(p2, 30, 10, -10)
 
print(p1.x, p1.y, p1.z)
print(p2.x, p2.y, p2.z)
 
p3 = p1 + p2
p4 = p2 - p1

print(p3)
print(p4)
print(p3 == p4)