试图在一个简单的电子邮件表单中找出错误-仍在学习php

我正在努力找出一个网站的电子邮件代码中的错误。这是一张预订表格，但我填写后无法发送信息。我还是PHP的新手，所以任何帮助都将不胜感激。提前谢谢你。

以下是PHP代码：

<?php
$err = '';
if(isset($_POST['submit'])){        
    $first = addslashes(trim($_POST['first'])); 
    $last = addslashes(trim($_POST['last']));   
    $email = addslashes(trim($_POST['email']));
    $date = addslashes(trim($_POST['date']));
    $women = addslashes(trim($_POST['women']));
    $men = addslashes(trim($_POST['men']));
            $message = "Name: $name" . PHP_EOL;
            $message .= "Phone: $phone" . PHP_EOL;
            $message .= "Email: $email" . PHP_EOL;
            $message .= "Date: $date" . PHP_EOL;
            $message .= "Women: $women" . PHP_EOL;
            $message .= "Men: $men" . PHP_EOL;
            $header = "From: reservations@bryanlv.com";
            mail("sales@pinnaclewebsitedesigns.com", "New Club Reservation", $message, $header);
            header('location:thank-you.html');      
        } else {    
        if($name=='') $err.='<p style="color:red">Your Name is empty</p>';  
        if($phone=='') $err.='<p style="color:red">Your Phone is empty</p>';    
        if($email=='') $err.='<p style="color:red">Your E-mail is empty</p>';   
        if($date=='') $err.='<p style="color:red">Your Reservation Date is empty</p>';  
        if($women=='') $err.='<p style="color:red">Number of Women is empty</p>';
        if($men=='') $err.='<p style="color:red">Number of Men is empty</p>';
    }

？>

这是表格：

<form name="reservation" method="post" action="book.php">
                    <label>Name</label>
                    <input type="text" name="name" id="name" placeholder="name" />
                    <label>Phone</label>
                    <input type="text" name="phone" id="phone" placeholder="phone" />
                    <label>E-mail</label>
                    <input type="text" name="email" id="email"  placeholder="email" />
                    <label>Date of Reservation </label>
                    <input type="date" name="date" id="date"  placeholder="date" />
                    <label>Number of Women </label>
                    <input type="text" name="women" id="women"  placeholder="women" />
                    <label>Number of Men </label>
                    <input type="text" name="men" id="men"  placeholder="men" />
                    </label>
                    <br />
                    <button type="submit" class="btn btn-primary">Book now</button>
                 </form>

我知道一定是我错过了一些简单的东西。