One of those unfortunate situations where two people are heading into the same lane, one moving left, and one moving right. (Both getting in correct lane for their destination.) It's easy to get caught out by this. You wouldn't have been visible in his mirror when he started his move, unless he had a wide-angle addon. And even though he didn't react to your horn, bear in mind he was further forward than you meaning you had more information to work on and increased responsibility. That's balanced by the fact you started moving into the lane first, but the key word is moving - you hadn't completed the move, you hadn't gained right of way, so to speak.
Situations like this are exactly why I got a wide angle add on mirror for the passenger side. The normal mirror genuinely gives the impression the adjacent lane is clear, even when it isn't.
Edit> If you look closely, his left mirror is badly adjusted - so you were definitely invisible to him when it mattered.