With homemade signs and spontaneous Mexican waves, it was an audience fit for the most famous musician. Speaking to a 15,000-strong audience of mostly women at the O2 Arena in London, the former first lady Michelle Obama called on those who were unhappy with the Trump White House to “roll up your sleeves”.

She was welcomed on stage at the event on Sunday night – part of an international book tour – with a standing ovation and screams. Asked by the US television host Stephen Colbert how she liked her reception, she said it gave her hope in difficult political times.

“I think it’s a testament to how much we all have in common around the world,” said Obama. “The fact that people are finding themselves in the story of this little girl, Michelle Robinson, on the south side of Chicago … is not a testament to me and my story, but it’s a reminder that we’re OK, folks. We’re going to be OK.”

Obama’s memoir, Becoming, tells the story of her journey from growing up in a working-class family in Chicago to living in the White House. It has sold more than 10m copies worldwide since it was published in November.

Asked for her views on the state of US politics, she said: “It’s time for us to roll up our sleeves and if we are not happy with the state of things then, in democracies, we have votes. We have to pay attention and we have to be engaged and we can’t take our rights and liberties for granted. Because if we don’t vote, somebody will.

“I have to remind people that Barack Obama was elected twice in the United States. That really did happen,” she said. “That wasn’t make-believe. The country actually did accomplish it and half the people who voted in the last election, if they could have, they would have voted for him for a third term.

“We have to remember that what is happening today is true, but what happened before was also true … that should give us some solace at some level.”

She added: “Let’s just stop and think about – for anybody who had problems with Barack Obama – what we were troubled by. There were never any indictments.”

“He wore a tan suit once,” joked Colbert.

The talk at the O2 followed another London event to promote the book, at the Southbank Centre in December, which sold out in minutes.

The Guardian reported on Friday that more than 100 tickets for the event on Sunday were being advertised for a significant markup. Professional touts listed one pair for more than £2,000 apiece, more than three times face value.

The book tour has taken Obama to Amsterdam, Copenhagen, Paris, Oslo and Stockholm, as well as a string of events across the US and Canada.

According to Nielsen BookScan, overall UK sales of her book stand at more than 600,000, ranking it 11th on the list of bestselling memoirs and biographies since sales records began in 1998.

In the wide-ranging interview, Obama talked about how her family’s lives changed beyond recognition as soon as her husband was named president-elect in November 2008.

“Your whole life changes in that moment. You get a presidential motorcade and it is a 20-car motorcade,” she said. “There’s an ambulance carrying pints of your husband’s blood type because you realise the threat that he is under.”

Obama said that the US was living through “one of many chapters”.

“It may feel like a dark chapter but any story has its highs and lows, but it continues. Yeah, we’re in a low, but we’ve been lower,” she said.

“We’ve had tougher times with more to fear. We’ve lived through slavery and the Holocaust and segregation and we’ve always come out on the other end, better and stronger.”