"British gymnastics has been growing and growing," said Smith. "Now everyone will be more motivated than ever.
Mitch FennerBBC gymnastics commentator
"There are people all over this arena who have been part of a 40-year legacy building up to this.
"To think a British team is in contention with Japan and Russia - that cannot be put into perspective. Those are nations GB could not even look at a decade ago, and now they are fighting with them.
"Nine years ago, Britain's men were 23rd in the world. Now they are the Olympic bronze medallists."
"This means so much and we've still got finals left. It's a beautiful day for the sport and for British gymnastics."
Though Britain were denied silver following an appeal from eventual runners-up Japan, the team insisted their bronze medals - unprecedented for GB in modern gymnastics - would have a huge impact on the sport's future.
Smith added: "The beauty about what we've got is that this team isn't just a one-hit wonder.
"Two of these guys [Sam Oldham and Max Whitlock] are 19 years old, not even at their peak yet, with Rio 2016 coming up.
"The juniors we've got coming up - we've been junior European champions for the last six years - there's so much depth."
Day Three Gymnastics round-up
In an Olympic Games notable so far for a comparative dearth of British medals up against expectations in some quarters, gymnastics has provided positive headlines and a real success story.
"Really, the road started six years ago," explained Eddie van Hoof, the technical director of the men's programme.
"There was a real depth of gymnasts over that period and also of coaching staff. We've been a very tight bunch for the last six years.
"We created belief. The belief came when Louis got the first medal in Beijing, that really set us on the right road and it's an amazing result. A lot of hard work well-rewarded."
Former GB gymnast Craig Heap, watching the final for BBC Radio 5 live, believes this team bronze medal will have a far greater positive impact on his sport than even Smith's breakthrough bronze in 2008.
"It transforms gymnastics in this country," said Heap, who won two Commonwealth team titles with England at Kuala Lumpur in 1998 and Manchester in 2002, in the infancy of the GB team's development.
"It makes Louis's achievement from four years ago look quite insignificant. It proves we've got the depth of a team now.
"There'll be kids all over the country thinking, 'I don't have to be as good as Louis, we can do it as a team or as individuals.' These guys have proved that with heart, determination and the right crowd, everything is achievable."
Kristian Thomas, the unheralded engine room of the British team alongside Dan Purvis for several years, seized his chance to carve his name into the London Olympics with a series of superb routines which, time and again, kept Britain in the reckoning for a medal.
His vault, high bar and floor routines all delivered significant scores - a breathtaking 16.550 in the case of the vault - and ultimately propelled Britain past Ukraine, who finished fourth on 271.526 points to Britain's 271.711.
"It was a fantastic feeling, drilling my feet into the ground and knowing, 'Right, we're onto a good day here,'" said Thomas, 23, of his all-important vault midway through the final.
"Our results over the past few years just seem to have got better and better. We're starting to get a bit more recognition from the outside world and it's what we need.
"The more people that get involved, it can only do British gymnastics good. Hopefully this can put gymnastics right at the front with all the other sports, and show the rest of the nation and the world we're a force to be reckoned with. Long may it continue."
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