To determine the total cost of a spell, start with the mana cost or alternative cost you're paying (such as the alternative cost of Invigorate), add any cost increases, then apply any cost reductions. The mana value of the spell remains unchanged, no matter what the total cost to cast it was.

If an effect says that an opponent can't gain life, you can't have that player gain life to pay Invigorate's alternative cost. If an effect instead replaces life gain, such as that of Sulfuric Vortex, you may choose to pay the cost even though the cost's action is replaced with nothing.