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I do not understand one point in the proof from user68061 of this statement

There is a similar question (here) but this time with no unicity condition for the solutions x and x'.

Henning Makholm provided an answer but I do not understand his first statement:

If e and a are given such that ea=a, then e is a left identify for every element. Proof: Given b let x be such that ax=b. Then eb=eax=ax=b.

What he says is really this: $$\forall a \forall b \exists x (b = a \ast x) \text{(hypothesis: $a \ast x = b$ admits a solution)}$$ $$\forall e \forall a \forall b \exists x (e \ast b = e \ast ( a \ast x )) \text{(binary operation allows left multiplication by a same element e)}$$ $$\forall e \forall a \forall b \exists x (e \ast b = ( e \ast a ) \ast x) \text{(associativity)}$$

At this point we would need to use $\exists e \forall a ( e \ast a ) = a$ in order to move to the next step, but this is precisely what you want to show!

Can someone provide a formal proof of that statement?

Community
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Christophe
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1 Answers1

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Rewording Henning Makholm's statement, we want to prove the following:

Let $G$ be a set with an associative binary operation $*$ such that for all $a,b \in G$, there is some $x \in G$ such that: $$ a * x = b \tag{$\star$} $$ Now suppose that there exists some $e \in G$ and that there exists some $c \in G$ such that $e * c = c$. Then for all $d \in G$, we know that $e*d = d$.

Proof: Choose any $d \in G$ and suppose that there exists some $e,c \in G$ such that: $$ e * c = c \tag{1} $$ Now take $a = c$ and $b = d$ so that (using $(\star)$) there exists some $x \in G$ such that: $$ c * x = d \tag{2} $$ Then observe that: \begin{align*} e*d &= e * (c * x) &\text{by substitution via $(2)$}\\ &= (e * c) * x &\text{by associativity}\\ &= c * x &\text{by substitution via $(1)$}\\ &= d &\text{by substitution via $(2)$}\\ \end{align*} as desired. $~~\blacksquare$

Adriano
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