bear out — To prove right or justified; confirm [thefreedict]

This is another way to say “confirm” — for example, the results bears out our claims.

not bat an eyelid — not show any shock or surprise [thefreedict]

When talking about a new model of phone with a rich set of features, someone said: “I hardly bat an eyelid“. While technologies are advancing fast, so are people’s expectations — today we are no longer easily impressed by claimed “new” things.