Today trade unions are taking over the Marble Hall in the TUC's headquarters, Congress House, to show to the world what they have been doing - and will be doing - to challenge prejudice and discrimination against lesbian, gay, bisexual and transgender people, as part of LGBT History Month.

Some people thought that when the coalition government introduced the right to marry for same-sex couples last year, the battle for LGBT equality was over.

Some people in the LGBT communities have put away their marching shoes and sunk into cosy acceptance by their heterosexual neighbours and work colleagues.

But when they turn out to show anger at Vladimir Putin's ban on "homosexual propaganda" in Russia, any assumption of moral superiority because Britain is just fine now is misplaced.

It is true, and welcome, that decades of LGBT campaigning have led to a transforming of the legal landscape and a dramatic shift in social attitudes.

We have moved from two-thirds of people thinking same-sex relationships are wrong to two-thirds thinking that they are fine. But the reality is more complicated.

For a start, one-third of the population rejecting the validity of same-sex relationships is a lot of people.

They are present on football terraces, in social media, in pubs and in schools, where surveys show that homophobic bullying is rampant.

Education Secretary Michael Gove's drive to create more schools free to teach that being gay is a sin is making things worse, despite his protestations that all schools are covered by laws outlawing harassment.

What about workplaces?

Last month we saw the largest ever survey of lesbian, gay and bisexual workers' experiences.

The Manchester Business School (MBS) found that LGB workers were two-and-a-quarter times more likely than heterosexual workers to face bullying.

No less worrying was that heterosexual colleagues from the same workplaces displayed attitudes that were at best ignorant and often just prejudiced.

They routinely blame the victim for their own plight and deny any discrimination.

Stereotypes of what a gay man or lesbian looks like are dominant. Sexuality is reduced to sex.

Abuse and offensive jokes are not recognised as harassment. People suffer whether or not they decide the best defence is to be "out."

The consequences include a horrendous level of mental health issues. This was the reality for hundreds of LGB people surveyed in 2013, the year that same-sex marriage became legal.

There was no cheer to be found from managers either. Those interviewed, from large employers across six different sectors, were equally ignorant.

The MBS survey did not cover trans people. But 2013 was the year that NUT member Lucy Meadows took her own life following a monstrous press campaign led by the Daily Mail.

It's OK to be trans - but not if you teach children.

LGBT people looking for help from what was a vibrant community voluntary sector now find services slashed by austerity policies.

Those hurt most are those most in need - young people thrown out by homophobic families now homeless, victims of homophobic hate crime or domestic abuse. These are just the people at the top of a long list.

The British government flies the rainbow flag from embassies around the world and lobbies foreign governments over anti-LGBT legislation being promoted in countries such as Nigeria and Uganda as well as Russia.

This is good. But at the same time, applicants for asylum from such countries are being asked to prove their claim to be LGBT. Back in 2010, David Cameron promised to end this practice, but it continues unabated.

Applicants are asked appallingly intrusive questions about their sex lives and sexual practices - just as in the MBS survey, the understanding that sexuality is not the same thing as what you do with a sexual partner has not reached the immigration services.

LGBT people gained a toehold on the equality agenda when the 2010 Equality Act for the first time gave us equal standing with other equality "strands."

Almost immediately the new coalition government weakened the law. The best way to challenge and overturn popular prejudices is to continue a high-profile struggle for full equality for LGBT people in Britain and abroad.

Trade unions are a critical component of a battle that must be waged in workplaces and across negotiating tables, but also in schools and colleges, on football terraces and in communities - including faith communities - everywhere.

Trade unionists are well placed to lead this.

Peter Purton is the TUC LGBT/disability officer