The 4 pounds of DME would raise you to 1.059.
The priming sugar would raise you a insignificant amount. What I assume happened is that you didn't get a thorough mix of wort/water when you pulled your sample. The wort is heavier than the top up water, and it's incredibly difficult to get it to mix well. Unless you added an additional 6 pounds of sugar, your OG was NOT 1.112, and your yeast would have never fermented it that low regardless. There is no way to have such a high OG unless you didn't add water to 5 gallons, or unless you added more fermentables.
With the 4 pounds of DME, and 3.3 pounds of LME, your ABV would be 5.1%.