под - It's mean under, below and nothing else.
твёрдый - You are right, hard and solid.
ить - Makes verb from adjective. For example: твёрдый (adjective) - твердить (verb, means to repeat), мягкий (soft) - смягчить (to make softer), мокрый (wet) - мочить (wet like verb).
подтвердить - confirm, to declare about your position / decision firmly.