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General notes:

• use if __name__ == '__main__': to avoid the main() function to be executed when the module is imported
• I'd pass around the credit card number as a string instead of converting it to string in every single validation step
• add docstrings to each of the defined functions

Regarding check_provider() function:

• you can check the length to be in range in one go:

  if not(13 <= len(str(number)) <= 16):
  

• I would improve the way you distinguish between cards by introducing a mapping between brands and regular expressions (like it was done in pycard module). Then, match each of the expressions one by one until you find a match or a brand was not found:

  import re
  BRANDS = {
   'Visa': re.compile(r'^4\d{12}(\d{3})?$'),
   'Mastercard': re.compile(r'''
   ^(5[1-5]\d{4}|677189)\d{10}$| # Traditional 5-series + RU support
   ^(222[1-9]|2[3-6]\d{2}|27[0-1]\d|2720)\d{12}$ # 2016 2-series
   ''', re.VERBOSE),
   'American Express': re.compile(r'^3[47]\d{13}$'),
   'Discover': re.compile(r'^(6011|65\d{2})\d{12}$'),
  }
  def check_provider(number):
   """Checks a credit card number and returns a matching brand name, or INVALID if no brand matched."""
   for brand, regexp in BRANDS.items():
   if regexp.match(number):
   return brand
   return 'INVALID'
  

Regarding implementing the Luhn algorithm, check the pycard's implementation - it is quite clean and understandable.

General notes:

• use if __name__ == '__main__': to avoid the main() function to be executed when the module is imported
• I'd pass around the credit card number as a string instead of converting it to string in every single validation step
• add docstrings to each of the defined functions

Regarding check_provider() function:

• you can check the length to be in range in one go:

  if not(13 <= len(str(number)) <= 16):
  

• I would improve the way you distinguish between cards by introducing a mapping between brands and regular expressions (like it was done in pycard module). Then, match each of the expressions one by one until you find a match or a brand was not found:

  import re
  BRANDS = {
   'Visa': re.compile(r'^4\d{12}(\d{3})?$'),
   'Mastercard': re.compile(r'''
   ^(5[1-5]\d{4}|677189)\d{10}$| # Traditional 5-series + RU support
   ^(222[1-9]|2[3-6]\d{2}|27[0-1]\d|2720)\d{12}$ # 2016 2-series
   ''', re.VERBOSE),
   'American Express': re.compile(r'^3[47]\d{13}$'),
   'Discover': re.compile(r'^(6011|65\d{2})\d{12}$'),
  }
  def check_provider(number):
   """Checks a credit card number and returns a matching brand name, or INVALID if no brand matched."""
   for brand, regexp in BRANDS.items():
   if regexp.match(number):
   return brand
   return 'INVALID'
  

Regarding implementing the Luhn algorithm, check the pycard's implementation - it is quite clean and understandable.

General notes:

• use if __name__ == '__main__': to avoid the main() function to be executed when the module is imported
• I'd pass around the credit card number as a string instead of converting it to string in every single validation step
• add docstrings to each of the defined functions

Regarding check_provider() function:

• you can check the length to be in range in one go:

  if not(13 < len(str(number)) < 16):
  

• I would improve the way you distinguish between cards by introducing a mapping between brands and regular expressions (like it was done in pycard module). Then, match each of the expressions one by one until you find a match or a brand was not found:

  import re
  BRANDS = {
   'Visa': re.compile(r'^4\d{12}(\d{3})?$'),
   'Mastercard': re.compile(r'''
   ^(5[1-5]\d{4}|677189)\d{10}$| # Traditional 5-series + RU support
   ^(222[1-9]|2[3-6]\d{2}|27[0-1]\d|2720)\d{12}$ # 2016 2-series
   ''', re.VERBOSE),
   'American Express': re.compile(r'^3[47]\d{13}$'),
   'Discover': re.compile(r'^(6011|65\d{2})\d{12}$'),
  }
  def check_provider(number):
   """Checks a credit card number and returns a matching brand name, or INVALID if no brand matched."""
   for brand, regexp in BRANDS.items():
   if regexp.match(number):
   return brand
   return 'INVALID'
  

Regarding implementing the Luhn algorithm, check the pycard's implementation - it is quite clean and understandable.

General notes:

• use if __name__ == '__main__': to avoid the main() function to be executed when the module is imported
• I'd pass around the credit card number as a string instead of converting it to string in every single validation step
• add docstrings to each of the defined functions

Regarding check_provider() function:

• you can check the length to be in range in one go:

  if not(13 <= len(str(number)) <= 16):
  

• I would improve the way you distinguish between cards by introducing a mapping between brands and regular expressions (like it was done in pycard module). Then, match each of the expressions one by one until you find a match or a brand was not found:

  import re
  BRANDS = {
   'Visa': re.compile(r'^4\d{12}(\d{3})?$'),
   'Mastercard': re.compile(r'''
   ^(5[1-5]\d{4}|677189)\d{10}$| # Traditional 5-series + RU support
   ^(222[1-9]|2[3-6]\d{2}|27[0-1]\d|2720)\d{12}$ # 2016 2-series
   ''', re.VERBOSE),
   'American Express': re.compile(r'^3[47]\d{13}$'),
   'Discover': re.compile(r'^(6011|65\d{2})\d{12}$'),
  }
  def check_provider(number):
   """Checks a credit card number and returns a matching brand name, or INVALID if no brand matched."""
   for brand, regexp in BRANDS.items():
   if regexp.match(number):
   return brand
   return 'INVALID'
  

Regarding implementing the Luhn algorithm, check the pycard's implementation - it is quite clean and understandable.