jquery or json form stay on page when submitted

I have a form that uses Jquery to show a message for

*field required error message

I am trying to get it to show a success message if the form is submitted on the same page.
I know json can be used, but do not know where to start with json.

The form submits as long as the req fields are filled in.

Does anyone know how I can modify this code to show the "success" div if all the "req" fields are filled out just like the ".req" shows if the fields are empty.

** notes
in the php, I currently redirect to thanks.php for valid and spam.php for honeypot filled in

My brain is fried today.

Thanks


form.js

$(function() {
       function validateform() {
          var valid = true;
          $(".req").css("border","1px solid #ccc");
          $(".req").each(function() {
             if ( $(this).val() == "" || $(this).val().replace(/\s/g, '').length == 0 ) {
                $(this).css("border","1px solided");$(".required").css("display","block");
                valid = false;
             }
          });
          return valid;
       }    
       $("#submit").click( function() {
          $('#myform').submit( validateform );
          $('$name').submit();
       });  
   });

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process.php

<?php
    // Clean up the input values
    foreach ( $_POST as $key => $value ) {
      if ( ini_get('magic_quotes_gpc') )
        $_POST[$key] = stripslashes( $_POST[$key] );

      $_POST[$key] = htmlspecialchars(strip_tags($_POST[$key]));
    }

    // Honeypot don't send
    if ( !empty($_POST["confirm_email"] ) ) {
      header("location:spam.php");exit;
    }

    // Assign the input values to variables for easy reference
    $name = $_POST["name"];
    $email = $_POST["email"];
    $confirm = $_POST["confirm_email"];
    $phone = $_POST["phone"];
    $message = $_POST["message"];
    $mls = $_POST["mls"];


    // Send the email *****************************
    $to = "email@mywebsite.com"; 
    $subject = "Website message: $name";
    $message = "MLS Listing:\n$mls\n\nFrom:\n$name\n\nEmail:\n$email\n\nPhone:\n$phone\n\nMessage:\n$message";
    $headers = "From: $email";

    mail($to, $subject, $message, $headers);
    
echo("<script>location.href = 'thanks.php';</script>");

?>

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form.html

<!--form-->
<form class="myform" id="myform" name="myform" action="process.php" method="post">

<div class="required">Red Highlights are required</div>

<div class="success">This is where the Succcess message is!</div>

<label>Name</label><input id="name" name="name" class="req" type="text"/>

<label>Email</label><input id="email" name="email" class="req" type="email" />

<label class="confirm_email">Confirm Email</label><input id="confirm_email" name="confirm_email" class="confirm_email" type="email" class="req2"/>

<label>Phone</label><input id="phone" name="phone" type="tel" />

<label>Message</label><textarea id="message" name="message" type="text" class="req"></textarea>

<input id="mls" name="mls" class="mls" value="MLS1234567" />

<input type="submit" id="submit" value="Submit Message" />
</form>
<!--form-end-->
<script src="http://code.jquery.com/jquery-1.10.1.min.js"></script>
<script src="form.js"></script>

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