Correction:
You stated: Hagana is Tzahal, or Tzva Hagana Leumi
As far as I know Hagana was Hagana. It was the predecessor of The IDF (Israel Defense Forces or the acronym ZH”L)
Tzahal stands for Tzva Hahagana L’Yisrael
The rest of your statement seems right on.
Yes, and so was the Hagana, only called Hagana for short. Upon the 1948 declaration, Lehi and Etzel merged with the Hagana into Tzahal, so as to eliminate all the internecine squabbling, and win the imminent war. It didn’t work, according to Begin’s Revolt!. The Hagana faction of Tzahal in many instances stiffed the Etzel and Lehi-nicks on the battlefield in terms of manpower and supplies, and actually destroyed munitions being brought in on the Altelena, mistakenly believing the Irgun was attempting a hostile takeover. Had the munitions on the Altelena not been destroyed, we could have kept the Jewish Quarter of the Old City of Jerusalem, and much more.