Theorem ismnd

Description: The predicate "is a monoid". This is the defining theorem of a monoid by showing that a set is a monoid if and only if it is a set equipped with a closed, everywhere defined internal operation (so, a magma, see mndcl ), whose operation is associative (so, a semigroup, see also mndass ) and has a two-sided neutral element (see mndid ). (Contributed by Mario Carneiro, 6-Jan-2015) (Revised by AV, 1-Feb-2020)

	Ref	Expression
Hypotheses	ismnd.b	$$⊢B={\mathrm{Base}}_G$$
	ismnd.p	$$⊢\underset{\dot{}}{+}={+}_G$$
Assertion	ismnd	$$⊢G\in \mathrm{Mnd}\leftrightarrow \left(\forall a\in B\forall b\in B\left(a\underset{\dot{}}{+}b\in B\wedge \forall c\in B\left(a\underset{\dot{}}{+}b\right)\underset{\dot{}}{+}c=a\underset{\dot{}}{+}\left(b\underset{\dot{}}{+}c\right)\right)\wedge \exists e\in B\forall a\in B\left(e\underset{\dot{}}{+}a=a\wedge a\underset{\dot{}}{+}e=a\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		ismnd.b	$$⊢B={\mathrm{Base}}_G$$
2		ismnd.p	$$⊢\underset{\dot{}}{+}={+}_G$$
3	1 2	ismnddef	$$⊢G\in \mathrm{Mnd}\leftrightarrow \left(G\in \mathrm{Smgrp}\wedge \exists e\in B\forall a\in B\left(e\underset{\dot{}}{+}a=a\wedge a\underset{\dot{}}{+}e=a\right)\right)$$
4		rexn0	$$⊢\exists e\in B\forall a\in B\left(e\underset{\dot{}}{+}a=a\wedge a\underset{\dot{}}{+}e=a\right)\to B\ne \varnothing$$
5		fvprc	$$⊢\neg G\in \mathrm{V}\to {\mathrm{Base}}_G=\varnothing$$
6	1 5	eqtrid	$$⊢\neg G\in \mathrm{V}\to B=\varnothing$$
7	6	necon1ai	$$⊢B\ne \varnothing \to G\in \mathrm{V}$$
8	1 2	issgrpv	$$⊢G\in \mathrm{V}\to \left(G\in \mathrm{Smgrp}\leftrightarrow \forall a\in B\forall b\in B\left(a\underset{\dot{}}{+}b\in B\wedge \forall c\in B\left(a\underset{\dot{}}{+}b\right)\underset{\dot{}}{+}c=a\underset{\dot{}}{+}\left(b\underset{\dot{}}{+}c\right)\right)\right)$$
9	4 7 8	3syl	$$⊢\exists e\in B\forall a\in B\left(e\underset{\dot{}}{+}a=a\wedge a\underset{\dot{}}{+}e=a\right)\to \left(G\in \mathrm{Smgrp}\leftrightarrow \forall a\in B\forall b\in B\left(a\underset{\dot{}}{+}b\in B\wedge \forall c\in B\left(a\underset{\dot{}}{+}b\right)\underset{\dot{}}{+}c=a\underset{\dot{}}{+}\left(b\underset{\dot{}}{+}c\right)\right)\right)$$
10	9	pm5.32ri	$$⊢\left(G\in \mathrm{Smgrp}\wedge \exists e\in B\forall a\in B\left(e\underset{\dot{}}{+}a=a\wedge a\underset{\dot{}}{+}e=a\right)\right)\leftrightarrow \left(\forall a\in B\forall b\in B\left(a\underset{\dot{}}{+}b\in B\wedge \forall c\in B\left(a\underset{\dot{}}{+}b\right)\underset{\dot{}}{+}c=a\underset{\dot{}}{+}\left(b\underset{\dot{}}{+}c\right)\right)\wedge \exists e\in B\forall a\in B\left(e\underset{\dot{}}{+}a=a\wedge a\underset{\dot{}}{+}e=a\right)\right)$$
11	3 10	bitri	$$⊢G\in \mathrm{Mnd}\leftrightarrow \left(\forall a\in B\forall b\in B\left(a\underset{\dot{}}{+}b\in B\wedge \forall c\in B\left(a\underset{\dot{}}{+}b\right)\underset{\dot{}}{+}c=a\underset{\dot{}}{+}\left(b\underset{\dot{}}{+}c\right)\right)\wedge \exists e\in B\forall a\in B\left(e\underset{\dot{}}{+}a=a\wedge a\underset{\dot{}}{+}e=a\right)\right)$$