The ∀𝑎 ∈ (Base‘𝑔) expression, or in ascii syntax "A. a e. (Base`g)" is the beginning of the restricted forall quantifier "A. x e. A ph" where you have highlighted just the "A. x e. A" part. It is read "for all x in A, ..." and denotes that some property "ph" holds for every x such that x e. A holds. In this case, the property in question is the remainder of the expression ∃𝑚 ∈ (Base‘𝑔)(𝑚(+g‘𝑔)𝑎) = (0g‘𝑔). In words, the expression says this:
The class "Grp" is defined to be the set of all *g* in "Mnd" (i.e. *g* being a monoid) such that for all *a* in the base set (carrier) of *g*, there exists some *m* in the base set of *g* such that *m* + *a* = 0, where + and 0 are the monoid operations on *g*. On Wed, May 31, 2023 at 7:46 PM Humanities Clinic < humanitiescli...@gmail.com> wrote: > Please pardon me for this rather basic question. > > I have read https://us.metamath.org/mpeuni/mmset.html, especially on the > sections "The Axioms", "The Theory of Classes". I have basic knowledge on > set theory and classical logic, so I understand all the black symbols in > prepositional and predicate, but I still find it difficult to understand > some expressions in definitions. > > For example, in https://us.metamath.org/mpeuni/df-grp.html: > > Grp = {𝑔 ∈ Mnd ∣ ∀𝑎 ∈ (Base‘𝑔)∃𝑚 ∈ (Base‘𝑔)(𝑚(+g‘𝑔)𝑎) = (0g‘𝑔)} > > I know that Mnd, Base, +g, 0g etc. are all classes. But I don't get what > it means for ∀𝑎 ∈ (Base‘𝑔) to be next to ∃𝑚 ∈ (Base‘𝑔)(𝑚(+g‘𝑔)𝑎) > = (0g‘𝑔) > > What background knowledge am I still missing out which I should be > reading, or did I miss out some material already on > https://us.metamath.org/mpeuni/mmset.html? Please help me by pointing me > to relevant reading material.. > > -- > You received this message because you are subscribed to the Google Groups > "Metamath" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to metamath+unsubscr...@googlegroups.com. > To view this discussion on the web visit > https://groups.google.com/d/msgid/metamath/ef11fbc4-7c45-42bd-a982-06b714fa9aecn%40googlegroups.com > <https://groups.google.com/d/msgid/metamath/ef11fbc4-7c45-42bd-a982-06b714fa9aecn%40googlegroups.com?utm_medium=email&utm_source=footer> > . > -- You received this message because you are subscribed to the Google Groups "Metamath" group. To unsubscribe from this group and stop receiving emails from it, send an email to metamath+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/metamath/CAFXXJSsyni4-B%3DUjRE3V9sOoN9Xa6WWsRROaT3TC5ZZU4RzMZQ%40mail.gmail.com.
