Understanding the Distributive Law in a Field

[gwsinger]

Suppose we're in a field F in which x,y,z are members. Consider the axiom of distribution which states that

xy + xz = x(y + z) for all x,y,z ∈ F.

Now consider this deduction:
xy + xz - xy ⟹ xy + x(z-y)

True no doubt, but to check my understanding aren't we missing a step? Shouldn't we more accurately say:
xy + xz - xy ⟹ xy + xz + x(-y) ⟹ xy + x(z-y)

I'm trying to self-study Rudin and just want to check my understanding.

 

[micromass]

Suppose we're in a field F in which x,y,z are members. Consider the axiom of distribution which states that

xy + xz = x(y + z) for all x,y,z ∈ F.

Now consider this deduction:
xy + xz - xy ⟹ xy + x(z-y)

True no doubt, but to check my understanding aren't we missing a step? Shouldn't we more accurately say:
xy + xz - xy ⟹ xy + xz + x(-y) ⟹ xy + x(z-y)

I'm trying to self-study Rudin and just want to check my understanding.

Yes, you are correct. And we could even be more accurate and say

xy+xz-xy=xy+xz+x(-y)=xy+x(z+(-y))=xy+x(z-y)