Is there a symbol for "as long as" in math?

Question

Let's say we have the expression

$$∀,∈ℤ:=⟹=$$

Which means "for all values of and in the set of integers, if equals then must equal ."

For example, if 4×6=4×6, then 4=4.

However, this does not apply to 0.

For example, 7×0=6×0, but 6≠0.

Is there a way to add to the previous expression a term that says "as long as x doesn't equal 0"?

Answer 1

There is no formal symbol for this. Even if there were, words are usually easier on your reader than formality. So write

If $x \ne 0$ then $ax= bx \implies a =b $.

You could write

$ax= bx \implies a =b $ provided $ x \ne 0$.

but that might make your reader hesitate and wonder until they finished the sentence. It's kinder to state the condition first.

The Ruby programming language has an unless keyword that can follow the statement it references, so you could code something like

you can cancel x unless x is 0

Answer 2

You could use "as long as $x$ does not equal $0$" like so:

With $\forall x \neq 0$ at the end:

$∀a,b\in\mathbb{Z}:ax=bx\implies a=b, \forall x \neq 0$

or with $\forall x\in\mathbb{Z}^*$ at the end:

$∀a,b\in\mathbb{Z}:ax=bx\implies a=b, \forall x\in\mathbb{Z}^*$

or with $\forall x\in\mathbb{Z}^*$ at the beggining:

$∀a,b\in\mathbb{Z} \wedge \forall x\in\mathbb{Z}^*:ax=bx\implies a=b$