How to join rows from another table with minimum value?

Question

MySQL

Given:
1) Products tableproduct

| id |  title  |
| 1  |продукт 1|
| 2  |продукт 2|

2) Supplier price tablesupplier_price

|product_id|supplier_id|price|
|    1     |     1     | 10  |
|    1     |     2     | 5   |
|    2     |     1     | 15  |

Task:
Select all product fields with minimum prices and corresponding supplier ids:

|product_id|supplier_id|price| id |  title  |
|    1     |     2     | 5   | 1  |продукт 1|
|    2     |     1     | 15  | 2  |продукт 2|

What have I tried?
1) Used JOIN

SELECT
    supplier_product.product_id,
    supplier_product.supplier_id,
    MIN(supplier_product.price)
FROM product
LEFT JOIN supplier_product ON product.id = supplier_product.product_id
group by supplier_product.product_id, supplier_product.supplier_id
#group by supplier_product.product_id

Problem:
If you add the supplier_id field to group by, products with different suppliers are doubled in the selection.
If not added (turn off full_only_group_by in sql_mode), then the supplier_id and price values ​​​​will not be consistent (the muscle randomly selects a value)
2) Used a nested SELECT

SELECT 
    product.*, 
    min_price
FROM product 
LEFT JOIN (
    SELECT 
        supplier_product.product_id,
        MIN(supplier_product.price) AS min_price
    FROM supplier_product
    GROUP BY supplier_product.product_id
  ) AS sp
ON product.id = sp.product_id

Problem: Essentially the same. I get the minimum price, but there is no supplier associated with it.
So is there a way to make the muscle group correctly?
That is, when choosing the minimum price, it takes the corresponding supplier id values

Answer 1

SELECT `p`.`product_id`, `p`.`supplier_id`, `p`.`price`, `i`.`title`
  FROM (
    SELECT `product_id`, MIN(`price`) AS `price`
      FROM `supplier_price`
      GROUP BY `product_id`
  ) AS `mp`
  JOIN `supplier_price` AS `p` ON `p`.`product_id` = `mp`.`product_id`
    AND `p`.`price` = `mp`.`price`
  JOIN `product` AS `i` ON `i`.`id` = `mp`.`product_id`

Or via ROLLUP

SELECT `p`.`product_id`, `p`.`supplier_id`, `p`.`price`, `i`.`title`
  FROM (
    SELECT `product_id`, `supplier_id`, `price`
      FROM `supplier_price`
      GROUP BY `product_id`, `supplier_id`, `price` WITH ROLLUP
      HAVING `price` = MIN(`price`)
  ) AS `p`
  JOIN `product` AS `i` ON `i`.`id` = `mp`.`product_id`