Theorem mgmplusgiopALT

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

	Ref	Expression
Assertion	mgmplusgiopALT	$$⊢M\in \mathrm{Mgm}\to {+}_M\mathrm{clLaw}{\mathrm{Base}}_M$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢{\mathrm{Base}}_M={\mathrm{Base}}_M$$
2		eqid	$$⊢{+}_M={+}_M$$
3	1 2	mgmcl	$$⊢\left(M\in \mathrm{Mgm}\wedge x\in {\mathrm{Base}}_M\wedge y\in {\mathrm{Base}}_M\right)\to x{+}_{M}y\in {\mathrm{Base}}_M$$
4	3	3expb	$$⊢\left(M\in \mathrm{Mgm}\wedge \left(x\in {\mathrm{Base}}_M\wedge y\in {\mathrm{Base}}_M\right)\right)\to x{+}_{M}y\in {\mathrm{Base}}_M$$
5	4	ralrimivva	$$⊢M\in \mathrm{Mgm}\to \forall x\in {\mathrm{Base}}_M\forall y\in {\mathrm{Base}}_{M}x{+}_{M}y\in {\mathrm{Base}}_M$$
6		fvex	$$⊢{+}_M\in \mathrm{V}$$
7		fvex	$$⊢{\mathrm{Base}}_M\in \mathrm{V}$$
8	6 7	pm3.2i	$$⊢\left({+}_M\in \mathrm{V}\wedge {\mathrm{Base}}_M\in \mathrm{V}\right)$$
9		iscllaw	$$⊢\left({+}_M\in \mathrm{V}\wedge {\mathrm{Base}}_M\in \mathrm{V}\right)\to \left({+}_M\mathrm{clLaw}{\mathrm{Base}}_M\leftrightarrow \forall x\in {\mathrm{Base}}_M\forall y\in {\mathrm{Base}}_{M}x{+}_{M}y\in {\mathrm{Base}}_M\right)$$
10	8 9	mp1i	$$⊢M\in \mathrm{Mgm}\to \left({+}_M\mathrm{clLaw}{\mathrm{Base}}_M\leftrightarrow \forall x\in {\mathrm{Base}}_M\forall y\in {\mathrm{Base}}_{M}x{+}_{M}y\in {\mathrm{Base}}_M\right)$$
11	5 10	mpbird	$$⊢M\in \mathrm{Mgm}\to {+}_M\mathrm{clLaw}{\mathrm{Base}}_M$$