Fact-Checking Donald Trump's First Remarks as Republican Nominee

— -- On the heels of accepting his party’s nomination, Donald Trump today held his first post-convention news conference.

ABC News decided to fact-check the newly-minted Republican presidential nominee's opening remarks of the general election race. Here are some of the claims and how they stack up:

Fact Check: War Over Wives

Claim: Donald Trump Didn’t Start the Back-and-Forth With Ted Cruz Over Their Wives. Cruz’s Friends Did

Rating: Questionable. A pro-Cruz super PAC attacked Melania first. But there’s no evidence Cruz knew about it.

Donald Trump said: “I didn't do anything. Somebody tweeted a picture of Melania and a picture of Heidi, who I think, by the way, is a very nice woman and a very beautiful woman. I have to tell you. I think Heidi Cruz is a great person. ... His people run the PAC. He said we had nothing to do, we had -- now, probably, you could trace it down with emails, but they're pretty smart. They don't even send, probably just phone calls. Look, it was a PAC with many of his friends. It was a Cruz super PAC. I think he said, well, it wasn't really meant for us. It was a Cruz PAC. It was his people.”

Background: An anti-Donald Trump super PAC, Make America Awesome, ran a Facebook ad featuring Melania Trump posing nude on a fur rug. Full-frontal nudity was not visible.

Trump then re-tweeted an unflattering photo of Heidi Cruz, the senator’s wife, next to one of Melania Trump.

Trump has repeatedly implied that Cruz was responsible for the original attack on Melania -- without evidence, according to Politifact.

The super PAC in question was run by GOP strategist Liz Mair, a former Republican National Committee staffer and Republican communications and strategy consultant who has worked on numerous campaigns, but none of them for candidates aligned strongly with Cruz’s wing of the Republican Party. Earlier in the current election cycle, Mair worked for Wisconsin Gov. Scott Walker.

The PAC reported running no pro-Ted Cruz ads to the Federal Election Commission. Although its goal in the final months of the GOP primary was to boost Cruz over Trump, all of its reported ad spending was anti-Trump, not pro-Cruz, in content.

Trump’s claim is questionable because, while the Melania ad was posted by a super PAC seeking to deliver Cruz an electoral victory, there’s no evidence that Cruz directed the ad, knew about it, or has historically enjoyed a particularly close relationship with the staffer running the super PAC. It’s not true that Trump “didn’t do anything” -- he re-tweeted an unflattering photo of Cruz’s wife -- so this claim is false and sounds more like an argument that Trump didn’t start the dispute.

Fact Check: Capturing the Sanders Vote

Claim: Donald Trump Will Get ‘a Lot of’ Bernie Sanders Voters Because They’re Aligned on Trade

Status: Questionable. Trump trails far behind Hillary Clinton among Sanders' supporters, and it’s questionable whether the gulf between Sanders’ supporters and Trump can be bridged by a similar view on trade.

Donald Trump said: “We're going to get a lot of the Bernie voters, by the way. ... I think we'll get a lot of his voters because of the trade issue.”

Background: It's unclear exactly what Trump means when he says he'll get "a lot" of Bernie Sanders' supporters to vote for him. But regardless of where he's setting the goalpost, ABC News' most recent polling shows him trailing far behind Clinton in the competition to capture Sanders' erstwhile voters. Among Democratic-leaning voters who preferred Sanders, 79 percent back Clinton over Trump.

There’s also reason to question whether Trump’s similar view to Sanders’ on trade will cause supporters of the 74-year-old Vermont senator to gravitate to the New York billionaire. First, like Trump and Sanders, Hillary Clinton also now opposes the Trans-Pacific Partnership, albeit after initially supporting the deal. Yet perhaps more significantly, there’s a broader gulf between Trump and Sanders that a shared view on trade is unlikely to bridge, according to an essay in Politico by Bill Scher, a writer for the Campaign for America's Future, a nonprofit organization that advocates progressive policies:

"The problem for Trump is that the few areas of ideological overlap don’t come close to outweighing the long list of issues where Sanders and Trump are practically opposites. Sanders supports a carbon tax; Trump calls global warming a hoax. Sanders wants a $15 minimum wage; Trump has said 'our wages are too high.' Sanders wants to jack the top income tax rate up to 54 percent; Trump wants to slash it to 25 percent," Scher wrote in his essay.

"Their foreign policies do not dovetail that neatly. Sanders’ anti-imperialist fans would not echo Trump’s call to 'take the oil' in Iraq. Nor would they want to 'authorize something beyond waterboarding' for suspected terrorists, let alone 'take out their families,'" Scher added.

This cycle has also seen apparent signs of bad blood between Sanders' supporters and Trump. For example, followers of the Sanders have been credited with forcing Trump to shut down a rally in Chicago over public safety concerns.

Fact Check: President Trump’s Super PAC

Claim: If He’s Elected President, Donald Trump Can Set Up a Super PAC to Attack Ted Cruz

Rating: Mostly False. This would probably not be feasible under current election law and regulations, but there might be some creative ways around it.

Donald Trump said: “Maybe I'll set up a super PAC if he decides to run [in 2020]. Are you allowed to set up a super PAC, Mike, if you are the president, to fight somebody?”

Background: Trump made this aside after dismissing the idea that he would even want Ted Cruz’s endorsement. In Trump’s supposition, he is a first-term president, and he is setting up a super PAC to attack Cruz, who is running against him.

This would likely be impossible for a president to do, said Larry Noble of the Campaign Legal Center. Candidates are not allowed to establish their own super PACs under current regulations.

The only way Trump could do it is if he were not running for re-election. It’s possible that Trump would not have begun his re-election campaign yet, but he can become a candidate by engaging in political activity, and establishing a PAC to attack an opponent for office would likely be considered such activity.

But candidates get around super PAC rules quite frequently, as evidenced by the volume of complaints that campaign-finance reformers (like the Campaign Legal Center) file against them every year with the Federal Election Commission.

If Trump were to tell an aide to set up the PAC, that would not be okay, but it’s conceivable he could do so secretly without being found out.

Another potential option would be to form a super PAC that only engaged in issue advocacy -- for instance, telling voters that “Ted Cruz is wrong on jobs,” but not to vote against him. Even then, it would be a big stretch.

“He can’t just ignore the basic rules surrounding a candidate,” Noble said.

Fact Check: Ted Cruz: Debate Season's Biggest Loser?

Claim: Cruz “Lost” Every Single Debate

Rating: Highly questionable. Few credible polls have numbers on debate performance, but those that do exist do not show Cruz in last place.

Background: Trumps said of Cruz, “And he was a good debater but he didn't do well in the debates against me. According to every poll, I mean, every poll, you know, he lost in every single poll, in every single debate.”

A CNN/ORC poll in September did show that those who watched a Sept. 16 CNN debate believed that Trump performed better than Cruz. Trump was ranked third, behind Carly Fiorina (52 percent) and Marco Rubio (14 percent), with 11 percent of respondents saying that they believed Trump did the best job in the debate. Cruz came in fifth, tied with Ben Carson, at 3 percent. Although he polled lower than Trump, Cruz came out in front of Lindsey Graham, Jeb Bush, John Kasich, Rand Paul and Mike Huckabee. In this sense, it is not true that Cruz “lost” in “every” poll.

It is also important to note that most “polls” on debate performance are unscientific, asking readers or viewers to dial in or vote online.