Help with loop printing too many times [Solved but wondering if there's a more elegant solution]

Question

The task is:
/*
Write a program that takes in a string representing a zip code. Output true or false if it is a valid zip code or not.
A valid zip code is only numbers, must be 5 characters in length, and contain no spaces.

Input Format:
A string containing a zip code.

Output Format:
A string: true is the input is a valid zip code, or false, if it is not.

Sample Input:
752f78

Sample Output:
false
*/

Code works but "true" prints too many times :

#include <cctype>
#include <iostream>
using namespace std;

int main() {
string str;
cin >> str;
int check;
int strlen = str.length();
for (int i = 0; i < strlen; ++i)
{
check = isalpha(str[i]);

if ((check > 0) || (str.length() > 5)){
cout << "false";
break;
}
else cout << "true";
}
}
return 0;
}

Any suggestions? Should I be using a different type of loop like while or do... while? Boolean?
https://code.sololearn.com/c843d1ksTwqx/?ref=app

Answer 1

The logic is not whole okay. This can be solved with a variable, which can be set depending of character test. After the for loop can be evaluated.

Answer 2

JaScript
Odd, I did not get a notification you had replied, I just saw it while updating the discussion title [Solved etc.]. Nonetheless I did find a solution (the above code now says "working", it passed all tests) .

But I still wonder if there is a better/cleaner solution. All advice welcome since it would not just be giving me the solution (as I found one possible answer) but would help me improve☺

I realize hints are more valuable than just being given the solution!