I know that the Multiclassing Proficiencies are listed on p. 164 of the Player's Handbook:
When you gain a level in a class other than your first, you gain only some of that class's starting proficiencies, as shown in the Multiclassing Proficiencies table.
[...] Cleric — Light armor, medium armor, shields [...]
Cleric proficiencies are listed as:
Armor: Light armor, medium armor, shields
Weapons: All simple weapons
So it's clear that simple weapons are not available to one multiclassing into Cleric.
However, Cleric's Tempest and War Domains both have the following Bonus Proficiency on p. 62 and p. 63 of the Player's Handbook, respectively:
At 1st level, you gain proficiency with martial weapons and heavy armor.
And of course, specific rules beat general rules, as noted on p. 7 of the Player's Handbook, but it's not clear to me which of the two is more specific.
Do I get these Domain Bonus Proficiencies when I multiclass into Cleric and use the War or Tempest domain?