In mathematical writing, how should I emphasise an equation together with its simplification?

Question

I always have this problem where I want to write an equation then immediately its simplification after it. I used to use the imply or equivalent symbol such as:

Therefore we end up with $$ \frac{a^2}{c^2} + b^2 =1 \implies a^2 + b^2c^2 = c^2. $$

But I was told this was not good in formal settings. Instead I should do something like:

Therefore we end up with $$ \frac{a^2}{c^2} + b^2 =1, $$ and hence $a^2 + b^2 c^2 = c^2$.

I understand why this would be better. Sometimes I can skip the penultimate equation and that would be fine. But other times I do find it a lot more clear for the reader to see that extra step. And yet the words in between the equations seems very distracting and not needed. Is there a symbol that I can put between the equations that doesn't carry a lot of meaning and that is also formal? Or is this just a bad idea altogether and I should always connect the two equations with words?

Edit: BTW I have looked at this question but it didn't help. I would never write a proof with imply symbols. This is purely about a step between two equations that is trivial enough not to require explanation yet the text would be less clear without it.

Answer 1

Outside of the context of specifying theorems, where statements like $$\text{“for each real $x,\quad$ <conditions> $\implies$ <conclusion>”}$$ are commonplace, usually the connective that is actually meant is “hence/therefore/thus” rather than “implies”, in which case the correct symbol is $\,\therefore\;$ instead of $\implies.$

(Published work spell “therefore” out because peppering mathematical prose with symbols decreases readability.)

For example,

Therefore we end up with $$\frac{a^2}{c^2} + b^2 =1,$$ and hence $$a^2 + b^2 c^2 = c^2.$$

or something like

Therefore we end up with $$\frac{a^2}{c^2} + b^2 =1;\\\therefore a^2 + b^2 c^2 = c^2.$$

or even

Therefore we end up with $$\frac{a^2}{c^2} + b^2 =1\\ a^2 + b^2 c^2 = c^2.$$

each—as intended—asserts both equalities, whereas

Therefore we end up with $$\frac{a^2}{c^2} + b^2 =1 \implies a^2 + b^2c^2 = c^2.$$

merely asserts that $a^2 + b^2c^2 = c^2$ is a consequence of $\frac{a^2}{c^2} + b^2 =1,$ while stating neither the former nor the latter to be true. This last presentation is momentarily confusing—and at least jarring— because I do literally read it as “Therefore we end up with P implying (i.e., being a sufficient condition for) Q.”

---

Addendum

Page 17 of this style guide directly addresses your question: