Vichy France join the Axis 1940

Could Vichy France join the Axis after the attack on Mers-el-Kébir?

What would be the impact of this?

How would this effect France post war?

IIRC one of the British official histories said that it was touch and go whether Vichy France would declare war on Great Britain for the week after Mers-el-Kébir but I won't be able to look for the page reference until this evening.
Here's the quote.
For a few days after the incident at Mers-el-Kébir it was touch and go whether the Government, now installed at Vichy, would be provoked into declaring war on Great Britain. A few French bombs were in fact half-heartedly dropped over Gibraltar and the French Government broke off diplomatic relations. A strategical appreciation of the implications of French hostility was clearly called for, and on July 16 the Chiefs of Staff issued a report on this subject, assuming the worst possible cause, of a France actively hostile.
Source: Page 230 of Grand Strategy, Volume II, Chapter X - The Immediate Consequences of the French Collapse.
 
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