Theorem sgrpplusgaopALT

Description: Slot 2 (group operation) of a semigroup as extensible structure is an associative operation on the base set. (Contributed by AV, 13-Jan-2020) (New usage is discouraged.) (Proof modification is discouraged.)

	Ref	Expression
Assertion	sgrpplusgaopALT	$$⊢G\in \mathrm{Smgrp}\to {+}_G\mathrm{assLaw}{\mathrm{Base}}_G$$

Proof

Step	Hyp	Ref	Expression
1		simpr	$$⊢\left(G\in \mathrm{Mgm}\wedge \forall x\in {\mathrm{Base}}_G\forall y\in {\mathrm{Base}}_G\forall z\in {\mathrm{Base}}_G\left(x{+}_{G}y\right){+}_{G}z=x{+}_G\left(y{+}_{G}z\right)\right)\to \forall x\in {\mathrm{Base}}_G\forall y\in {\mathrm{Base}}_G\forall z\in {\mathrm{Base}}_G\left(x{+}_{G}y\right){+}_{G}z=x{+}_G\left(y{+}_{G}z\right)$$
2		eqid	$$⊢{\mathrm{Base}}_G={\mathrm{Base}}_G$$
3		eqid	$$⊢{+}_G={+}_G$$
4	2 3	issgrp	$$⊢G\in \mathrm{Smgrp}\leftrightarrow \left(G\in \mathrm{Mgm}\wedge \forall x\in {\mathrm{Base}}_G\forall y\in {\mathrm{Base}}_G\forall z\in {\mathrm{Base}}_G\left(x{+}_{G}y\right){+}_{G}z=x{+}_G\left(y{+}_{G}z\right)\right)$$
5		fvex	$$⊢{+}_G\in \mathrm{V}$$
6		fvex	$$⊢{\mathrm{Base}}_G\in \mathrm{V}$$
7		isasslaw	$$⊢\left({+}_G\in \mathrm{V}\wedge {\mathrm{Base}}_G\in \mathrm{V}\right)\to \left({+}_G\mathrm{assLaw}{\mathrm{Base}}_G\leftrightarrow \forall x\in {\mathrm{Base}}_G\forall y\in {\mathrm{Base}}_G\forall z\in {\mathrm{Base}}_G\left(x{+}_{G}y\right){+}_{G}z=x{+}_G\left(y{+}_{G}z\right)\right)$$
8	5 6 7	mp2an	$$⊢{+}_G\mathrm{assLaw}{\mathrm{Base}}_G\leftrightarrow \forall x\in {\mathrm{Base}}_G\forall y\in {\mathrm{Base}}_G\forall z\in {\mathrm{Base}}_G\left(x{+}_{G}y\right){+}_{G}z=x{+}_G\left(y{+}_{G}z\right)$$
9	1 4 8	3imtr4i	$$⊢G\in \mathrm{Smgrp}\to {+}_G\mathrm{assLaw}{\mathrm{Base}}_G$$