Theorem sgrp0

Description: Any set with an empty base set and any group operation is a semigroup. (Contributed by AV, 28-Aug-2021)

	Ref	Expression
Assertion	sgrp0	$$⊢\left(M\in V\wedge {\mathrm{Base}}_M=\varnothing \right)\to M\in \mathrm{Smgrp}$$

Proof

Step	Hyp	Ref	Expression
1		mgm0	$$⊢\left(M\in V\wedge {\mathrm{Base}}_M=\varnothing \right)\to M\in \mathrm{Mgm}$$
2		rzal	$$⊢{\mathrm{Base}}_M=\varnothing \to \forall x\in {\mathrm{Base}}_M\forall y\in {\mathrm{Base}}_M\forall z\in {\mathrm{Base}}_M\left(x{+}_{M}y\right){+}_{M}z=x{+}_M\left(y{+}_{M}z\right)$$
3	2	adantl	$$⊢\left(M\in V\wedge {\mathrm{Base}}_M=\varnothing \right)\to \forall x\in {\mathrm{Base}}_M\forall y\in {\mathrm{Base}}_M\forall z\in {\mathrm{Base}}_M\left(x{+}_{M}y\right){+}_{M}z=x{+}_M\left(y{+}_{M}z\right)$$
4		eqid	$$⊢{\mathrm{Base}}_M={\mathrm{Base}}_M$$
5		eqid	$$⊢{+}_M={+}_M$$
6	4 5	issgrp	$$⊢M\in \mathrm{Smgrp}\leftrightarrow \left(M\in \mathrm{Mgm}\wedge \forall x\in {\mathrm{Base}}_M\forall y\in {\mathrm{Base}}_M\forall z\in {\mathrm{Base}}_M\left(x{+}_{M}y\right){+}_{M}z=x{+}_M\left(y{+}_{M}z\right)\right)$$
7	1 3 6	sylanbrc	$$⊢\left(M\in V\wedge {\mathrm{Base}}_M=\varnothing \right)\to M\in \mathrm{Smgrp}$$