How to keep login status with Flutter - firebase

Question

Assumptions/what you want to achieve
Currently, I am learning about application development using Flutter+Firebase.
I want to implement the following functions as a part of the login function.
Transition to the login page and login only when logging in for the first time
Unless you log out, opening the app after exiting the app does not bring you to the login page
I did some research on my own and tried them, but they didn't work.
As a concrete method, how can the above functions be implemented?
Also, I am currently implementing a function that uses Firebase's Email and Password.
Is it a common method to implement the above functions even for Google login and Facebook login?
Since I am a new student of Flutter, I have many things that I do not understand, but I hope you can teach me.
Currently implemented
Login function using Firebase Auth
Database using Firebase Cloud Store
What I tried
Check the login status using Current User of Firebase Auth
Additional information (FW/tool ​​version, etc.)
Used
FirebaseAuth
FirebaseCloudStore
class MyApp extends StatelessWidget {
// This widget is the root of your application.
final UserState user = UserState();
#override
Widget build(BuildContext context) {
return ChangeNotifierProvider<UserState>.value(
value: user,
child: MaterialApp(
// Hide debug label
debugShowCheckedModeBanner: false,
title:'Flutter Demo',
theme: ThemeData(
primarySwatch: Colors.blue,
visualDensity: VisualDensity.adaptivePlatformDensity,
),
home: LoginCheck(),
initialRoute: "/",
routes:<String, WidgetBuilder>{
// "/": (BuildContext context) => LoginPage(),
AddRecord.routeName: (BuildContext context) => AddRecord(),
"/login":(BuildContext context) => LoginPage(),
"/home":(BuildContext context) => PageManager()
},
)
);
}
}
class LoginCheck extends StatefulWidget{
LoginCheck({Key key}): super(key: key);
#override
_LoginCheckState createState() => _LoginCheckState();
}
class _LoginCheckState extends State<LoginCheck>{
#override
void initState(){
super.initState();
checkUser();
// TODO: implement initState
}
void checkUser() async{
final UserState userState = Provider.of<UserState>(context);
final currentUser = await FirebaseAuth.instance.currentUser();
print(currentUser);
if(currentUser == null){
Navigator.pushReplacementNamed(context,"/login");
}else{
userState.setUser(currentUser);
Navigator.pushReplacementNamed(context, "/home");
}
}
#override
Widget build(BuildContext context) {
// TODO: implement build
return Scaffold(
body: Center(
child: Container(
child: Text("Loading..."),
),
),
);
}
}